I hadn’t intended to publish another math-heavy article so soon, but when the muse strikes you have to follow it…

To be a top-class GM, you need to have an almost instinctive understanding of probability.

Such understanding rarely comes naturally; you have to work at it, exploring different ways of looking at odds and outcomes. These build up into an experience bank that forms the foundations of an instinctive awareness of the subject.

Counter-intuitive Probabilities

This is made far more difficult by the fact that an incomplete understanding of probability – or a poorly-applied understanding – leads to intuitive results that are wrong. For example, imagine a game show. Let’s say that there are three cards – one that wins something valuable, and two that yield nothing. You, as the contestant, are then required to choose one of the cards.

The host then turns one of the cards that you didn’t choose around, revealing that it’s one of the ‘no prize’ card, and offers you the choice of staying with your original choice or changing to the other unrevealed card.

Should you change or not?

Those with a deficient understanding of probability would say that it makes no difference, the chance is still one-in-three that you made the right choice. Those that think this way are then impacted by a confirmation bias that makes it almost certain that they will stay with their first choice.

But the reality is that by swapping to the other unrevealed card, they double their chances of winning. You see, there was originally a 2-in-3 chance that the card they chose was the wrong one – and once one of the two remaining cards is eliminated, that means that there is now a 2-in-3 chance that the unrevealed card they didn’t choose is the winning card.

Counter-intuitive, right? That’s why it’s sometimes known as the Monty Hall Paradox, or the Monty Hall problem.

The existence of counter-intuitive results when your intuition is giving you a bum steer is a problem that has to be overcome in order to train your intuition properly. It’s often helpful to break situatons down into their simplest form, then introduce refinements.

So let’s do just that.

A simple roll

Almost every roll – be it a saving roll or a skill check or an attack roll – can be expressed by the simple proposition of success or failure.

It’s normal for one of those to be more likely than the other, but that’s a complication beyond a first-cut analysis.

That defines our simplest form as a 50-50 chance, success or failure – or any other contrasting outcome, for that matter, such as high or low.

The simplest die

That defines the simplest die as a d2, also known as a coin, with heads and tails as the outcomes. But actually flipping coins is a noisy and inconvenient process – at least it is if you are trying for true randomness – so I’m actually going to simulate a perfect coin with dice.

This is better than actually tossing dice because there’s always a finite possibility of a real coin landing on it’s edge. With simulated coins, that’s no longer a potential outcome.

The memory of rolls past

If you’ve flipped ten ‘perfect coins’ and they’ve all come up heads, what’s the likelihood that the eleventh flip will also be a head?

The answer is, 50%, the same as always – but even though we know this, intellectually, emotionally we feel that a tail is more likely to occur.

I was thinking about this and wondering what the average length of any string of similar results would be. My suspicion is that it would be the average of the longest possible string (n) and the shortest possible string (1), where (n) is the number of coin-flips – but I don’t have any maths or logic to back up that suspicion, which assumes a linear probability. For all I know, it could be the square root of (n × 1), a decidedly non-linear

So, let’s try and create some.

First flip

The first flip, quite obviously is going to be either a head or a tail.

Second flip

The second flip is also going to be either a head or a tail. That gives four possible combinations of outcomes so far – HH, HT, TH, or TT.

Number of combinations

If we’re talking about ultimately getting to eleven flips, that means that we’re going to have to deal with 2-to-the-11th-power combinations – 48,828,125 of them. There’s no way that’s practical.

This only confirms in my mind that analyzing a simpler set of combinations and extrapolating is the only way to go.

Analysis: two flips

From two flips, we have two outcomes with strings of 2 similar results (HH and TT), and two with dissimilar results (HT and TH). So the average length of result strings is 1.5, exactly what my intuition was suggesting. So far, so good.

Third Flip

This doubles the number of possible results to eight, and for the first time, introduces the possibility of result strings of intermediate length. The eight combinations are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. (Double check, counting them up – yep, that’s all eight).

Analysis, three flips

• We have two combinations of length 3 – HHH and TTT.
• We have four combinations of length 2 – HHT, HTT, THH, and TTH.
• That leaves two combinations of length 1 – THT and HTH.

(2 × 3) + (4 × 2) + (2 × 1) = 6+8+2 = 16, so the average length is 16/8=2. Still supporting the instinctive measure – but this suggests something I didn’t expect, our old friend the standard probability curve. It’s too soon to confirm that, but it’s definitely a pattern to watch for.

Fourth flip

With the fourth flip, we’re looking at 16 possible result combinations: HHHH, HHTH, HTHH, HTTH, THHH, THTH, TTHH, TTTH, HHHT, HHTT, HTHT, HTTT, THHT, THTT, TTHT, and TTTT.

I generated that list the easy way: copy the previous list twice, add heads to the first set, and tails to the second set.

Analysis, four flips

• Combinations of length 4: HHHH and TTTT = 2.
• Combinations of length 3: THHH, TTTH, HHHT, and HTTT = 4.
• Combinations of length 2: HHTH, HTHH, HTTH, TTHH, HHTT, THHT, THTT, and TTHT = 8 (actually, I counted seven and thought, that doesn’t seem right – and sure enough, I’d missed one).
• Combinations of length 1: 16-2-4-8=2.

But wait – should TTHH and HHTT count as one or two strings of length 2 results? Answer: only if HTHT and THTH also count as four strings of length 1 results, and HHTH counts as one string of length 2 and two strings of length 1. That could mean that my entire methodology is flawed, because I haven’t been counting the length of strings of results, I’ve been counting combinations that contain a string of results of given length. And that’s not necessarily the same thing at all!

Anyway, lets push on, and then revisit the results using the other, more complicated approach.

(4 × 2) + (3 × 4) + (8 × 2) + (2 × 1) = 8 + 12 + 16 + 2 = 38, and 38/16 = 2.375.

Wait, what?

Not only does this not match up with the instinctive approach expected, it doesn’t look much like a standard distribution, either. There would need to be a second set of outcomes with a result count of 4 somewhere in between length 2 and length 1, and we don’t have one – can’t possibly have one. But it’s possible that this is due to a “rounding error” in the number of length 2 results, in which case, sanity should be restored with an odd number of flips (which would permit something to be in the middle of one and three – in fact, requires something, length 2, to be in between). Until this gets resolved, let’s set aside the length-of-string analysis and go for a fifth flip.

Fifth Flip

32 possible result combinations: HHHHH, HHTHH, HTHHH, HTTHH, THHHH, THTHH, TTHHH, TTTHH, HHHTH, HHTTH, HTHTH, HTTTH, THHTH, THTTH, TTHTH, TTTTH, HHHHT, HHTHT, HTHHT, HTTHT, THHHT, THTHT, TTHHT, TTTHT, HHHTT, HHTTT, HTHTT, HTTTT, THHTT, THTTT, TTHTT, and TTTTT.

That’s starting to get to the point where the results are swimming together and I can no longer visualize the full range of results all at once. You might be more capable than I, but that point will inevitably be reached for most of us eventually.

Analysis, 5 flips

• Combinations of length 5: HHHHH and TTTTT = 2.
• Combinations of lengrh 4:THHHH, TTTTH, HHHHT, and TTTTH = 4.
• Combinations of length 3: HTHHH, TTHHH, TTTHH, HHHTH, HTTTH, THHHT, TTTHT, HHHTT, HHTTT, and THTTT = 10.
• Combinations of length 2: HHTHH, HTTHH, THTHH, HHTTH, THHTH, THTTH, TTHTH, HHTHT, HTHHT, HTTHT, TTHHT, HTHTT, THHTT, and TTHTT = 14.
• Combinations of length 1: HTHTH and THTHT = 2.

Check that I haven’t missed anything: 2+4+10+14+2 = 32.

This is definitely NOT standard distribution.

(5 × 2) + (4 × 4) + (3 × 10) + (2 × 14) + (1 × 2) = 10+16+30+28+2 = 86.
86 / 32 = 2.6875.

Ummm – if there’s a pattern here, I’m not seeing it. I would hope that the increase in the product of results would show something by 86-38=48 and that doesn’t leap out at me as meaning anything. Nor does there seem to be a pattern in the number of results of different length – 2, 4, 10, 14 is not a series that makes sense to me.

The one thing that I can say for certain is that this is NOT “(n +1)/2”.

So much for intuition then. Unless the length of string results yield something more useful, of course.

Let’s go back to the set-aside alternative, then.

Length of string, 1 flip

H or T. That’s two outcomes of length 1. And (2 × 1) / 2 = 1, exactly as you would expect.

Length of string, 2 flips

HH, HT, TH, TT.

• Length 2: HH and TT = 2.
• Length 1: HT and TH = 2 × 2 (one for the H and one for the T in each) = 4.
• Total: (2 × 2) + (1 × 4) = 4+4 = 8;

8/6 = 1.333333….

Hmmm….

Length of string, 3 flips

HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

• Length 3: HHH and TTT = 2.
• Length 2: HHT, HTT, THH, and TTH = 4.
• Length 1: there’s 1 in each of the length 2 listings, and 3 in each of HTH and THT, for a total of 4+6=10.

(3 × 2) + (2 × 4) + (1 × 10) = 6+8+10 (now that’s a pattern! But it’s just a coincidence.) = 24
24 / (2+4+10) = 24 / 16 = 1.5.

Hmmm again….

Length of string, 4 flips

HHHH, HHTH, HTHH, HTTH, THHH, THTH, TTHH, TTTH, HHHT, HHTT, HTHT, HTTT, THHT, THTT, TTHT, and TTTT.

• Length 4: HHHH and TTTT = 2.
• Length 3: THHH, TTTH, HHHT, and HTTT = 4.
• Length 2: HHTH, HTHH, HTTH, TTHH (2), HHTT (2), THHT, THTT, and TTHT = 10.
• Length 1: HHTH (2), HTHH (2), HTTH (2), THHH, THTH (4), TTTH, HHHT, HTHT (4), HTTT, THHT (2), THTT (2), and TTHT (2) = 24.

If you aren’t sure of what I’m doing, it might help if I wrote the combinations “HH-T-H” – there are two strings of length 1, so I put a (2) after the combination.

Hmmm: 2 + (1 × 2) = 4; 4 + (2 × 3) = 10; 10 + (3 × 4) = 22. Close, but no banana.

(4 × 2) + (3 × 4) + (2 × 10) + (1 × 24) = 8 + 12 + 20 + 24 = 64

64 / (2+4+10+24) = 64 / 40 = 1.6

I’m not seeing a pattern here, either. I don’t think I need to go to the 5-flip results, I think the point is established.

What point is that? That intuition and probability are not all that compatible!

From these results, I can say that the average is increasing with each flip, but quite slowly, simply because the number of 1-length strings continually outnumbers everything else put together, the number of 2-length strings continually outnumbers everything higher put together, and so on.

A long string of flips

So, let’s used some dice to generate a longer string of flip results and see what we get.

HHHH-T-HH-T-H-TTT-H-TTTTT-HH-TTTTT-H-T-H-TT-HH-TT-H-T-HH-T-HHH-TTTTTT-HHHH-T-H-TT-H-TT

That’s 50, by my count. I’ve inserted a dash every time a string of like results comes to an end. Let’s translate the resilts into a more convenient form- HHHH to H4.- which gives me

H4-T1-H2-T1-H1-T3-H1-T5-H2-T5-H1-T1-H1-T2-H2-T2-H1-T1-H2-T1-H3-T6-H4-T1-H1-T2-H1-T2

The numbers indicate the length of the string of like results, and that means that statistical analysis becomes easy:

• 6-long: 1
• 5-long: 2
• 4-long: 2
• 3-long: 2
• 2-long: 8
• 1 long: 13

1+2+2+2+8+13 = 7+8+13 = 15+13 = 28.

(6 × 1) + (5 × 2) + (4 × 2) + (3 × 2) + (2 × 8) + (1 × 13) =
6 + 10 + 8 + 6 + 16 + 13 = 30 + 29 = 59

59 / 28 = 2.107

That seems completely in line with the results suggested by the smaller analysis. What’s more, it seems to suggest that the increases with each successive flip added to the chain keep getting smaller – if that weren’t the case, the average with this many additional flips would be a lot higher than just 2.1.

By the way, there’s nothing in this analysis to say that improbably results can’t or won’t happen; I’ve seen them happen too many times for that!

Three reels on a poker machine

Let’s take it up a gear. A typical poker machine has three reels, each of which bears symbols representing Ace, King, Queen, Jack, and 10 (symbolized by a zero). There may be others as well; for convenience I will assume that most of these are “null” characters, symbolized by Ø for the purposes of this article.

Let’s assume that there are 4 of each of the main symbols on a single ring, one for each suite. Let’s also assume that there are 11 Ø symbols on each reel and one wild card, which will be symbolized by ☆ in this article. Various combinations give a payout – three of a kind (except three nulls), or two of a kind plus a ☆.

Ring one: AAAAKKKKQQQQJJJJ0000ØØØØØØØØØØØ☆ (probably not in that order).
Ring two: same as ring one.
Ring three: same as rings one and two.

	A:	4
	K:	4
	Q:	4
	J:	4
	0:	4
	Ø:	11
	☆:	1

Total, 32 symbols on each reel.

• 21 of these on reel 1 yield a payout if the right things come up on reels 2 and 3. That’s 21/32 = 65.625%.
• Only 5 of the results on reel 2 will match what’s on reel 1 – 5/32 = 15.625%.
• Only 5 of the results on reel 3 will match what’s on reels 1 and 2 = 15.625% again.

Put all of those together, and you get a 1.6% chance of a payout.

Because that tends to frustrate players, various other combinations may be allocated a lesser payout – two of a kind, or a single Ace on any reel. This complicates the chances, but increases them substantially – two of a kind = +8.65% chance of a payout, and any ace = +9.57%. Total = 19.82%.

More reels?

So, let’s contemplate adding 2 more reels. There are two effects: first, the possibility of getting four or even five of a kind now exists, but it’s very improbable, and so you would get a much larger payout. Second, there are now 5 reels and that increases the chances of getting three of them to match, so the chance of success goes up considerably. There are now ways to win with Ø showing on any two of the reels.

How much better? Let’s see:

First, any two reels can be showing Ø so long as the others are right. That means that we can multiply the number of combinations of Ø and non-Ø reels by the chance of one specific configuration to get the total.

	ØØCCC
	ØCØCC
	ØCCØC
	ØCCCØ
	CØØCC
	CØCØC
	CØCCØ
	CCØØC
	CCØCØ
	CCCØØ

A systematic examination of the combinations lists 10 of them. Now, the chances of any one of them: We already know that the first three reels showing CCC has a 1.6% chance of appearing. We need to adjust for the chances of Ø showing up on the other two reels – or, in fact, anything other than the specific matching card symbol. That’s 27/32 for reel 4 and 27/32 for reel 5 – a total chance of 1.139%. But there are 10 of those combinations – so ten lots of 1.139% = 11.39% of getting three of a kind.

Answer: a lot better.

Multiple Lines on a slot machine

Your chances get even better if you can match along different lines. The minimum that I’ve seen in this respect is three lines.

How much better?

At first glance, three times as good. But that ignores the possibility of multiple wins from the same spin – and this is where the exact configuration of each reel becomes a factor as well. On top of that, there is absolutely no reason why the designer needs to follow the rather simplistic pattern that I set up as an example – reel 3 might have fewer aces and more tens, fewer kings and more jacks, fewer queens and more Øs. Do the same across all five reels, and you can see that designers of slot machines have almost total control over the likelihood of any given payout, and can set the house percentage to whatever they think they can get away with.

It’s a fairly default assumption – that a machine is “honest” in the chances that it offers. Design is a totally above-board, totally legal, way of distorting the odds.

So it is with RPGs – GMs have to assume that a player’s dice are “legit”, and players have to assume that the GM’s adjudication, and settings for the chance of success, are fair. If this trust ever breaks down, it almost certainly spells a confrontation, strained relationships, and potentially the end of friendships.

Simulating A Slot Machine

Let’s think about hypothetical approaches to simulating a slot machine with standard RPG dice.

I’ll pick three reels and five lines – three straight across and two at an angle.

The reels we defined earlier had 32 entries per reel, and that doesn’t comfortably fit any standard die. We can get close by externalizing the chance of a null result – eleven of the 32 thus get excluded, leaving 21. Defining a special mechanism for the ‘wild card’ result gets us down to 20, which works.

Instead of the even chances listed earlier, let’s bias things toward the lower end.

	A:	2
	K:	2
	Q:	3
	J:	5
	0:	8
	Ø:	0*
	☆:	0*

or, to put it another (more familiar way:)

	01-02	A
	03-04	K
	05-07	Q
	08-12	J
	13-20	0
	xx-xx	Ø
	xx-xx	☆

So, three d20s will give us our middle line. As shown by the “xx-xx” results listed, though, there’s still work to do.

Next, we need a d6:

	1-2	Ø
	3-6	As shown on d20

And then we need a wild card mechanism, using the same d6 roll, so let’s replace the above with:

	1-2	Ø
	3-5	As shown on d20
	6	☆ if d20 reads “20”, otherwise as shown on d20

This reduces the chances of getting a 10 very minutely, and fills the resulting probability void with a wild card. How minutely? To get ☆, you need a 6 on d6 (1/6) and a 20 on d20 (1/20) – multiply those together and you get 1/120, or a little less than 0.85%.

So, that’s got our main results line sorted. Next, we need a way to simulate the results before and after – above and below – the result showing on the middle line. I could work it with d6s, but to keep the rolls obviously distinct, let’s use d8s instead.

	1-4	+1
	5-6	+2
	7	+3
	8	+4
	fresh d20 roll if Ø and +3 or +4 showing;
	☆ if ‘0’ and +4 showing and no ☆ already shown.

Note that these adjustments are to the indicated results of the d20, not to the roll, so:

• ‘Ø’+1=’Ø’
• ‘0’+1 = ‘J’
• ‘J’+1 = ‘Q’
• ‘Q’+1 = ‘K’
• ‘K’+1 = ‘A’
• ‘A’+1 = ‘0’
   
• ‘Ø’+2=’Ø’
• ‘0’+2 = ‘Q’
• ‘J’+2 = ‘K’
• ‘Q’+2 = ‘A’
• ‘K’+2 = ‘0’
• ‘A’+2 = ‘J’
   
• ‘Ø’+3= new d20 roll
• ‘0’+3 = ‘K’
• ‘J’+3 = ‘A’
• ‘Q’+3 = ‘0’
• ‘K’+3 = ‘J’
• ‘A’+3 = ‘Q’
   
• ‘Ø’+4= new d20 roll
• ‘0’+4 = ☆ if no &star showing on this reel, otherwise ‘A’
• ‘J’+4 = ‘0’
• ‘Q’+4 = ‘J’
• ‘K’+4 = ‘Q’
• ‘A’+4 = ‘K’

The same technique gives us the row of results below the middle row.

Interpreting the results is probably most easily done by actually laying out playing cards in an appropriate 3×3 grid. So, below, we have the results from the die rolls, and below them, an illustration of the resulting ‘display window’ on our simulated poker machine:

	d20:	6,	14,	1
	d6:	6,	6,	2
	Middle row:
		Q	10	Ø
	Row above:
	d8:	7,	3,	8
		Q+3=10	10+1=J	Ø+4=d20;
				Reroll:10 Result: J
	Row below:
	d8:	3,	6,	4
		Q+1=K	10+2=Q	Ø+1=Ø

Looking at the result, there are two winning combinations – a pair of Jacks on the top row and a pair of tens on the top-left-to-bottom-right diagonal. So it’s just a matter of knowing how much those particular combinations will pay out.

But my, that’s a lot of palaver!

In Search Of A Simpler Simulation

The big advantage of the approach above is that you don’t need to know the probability of any given result coming up, any sort of reasonable guess will be good enough.

But the simplest dice-based simulation removes that comfort, producing a set of percentile tables that directly spits out not just the paying combination, but every combination of paying combination.

Generating such tables involves a lot of tedious number-crunching. So much so that you might well be tempted to say “bugger this” and simply make up the numbers.

But if you’re going to do that, why not skip the entire act of simulation of results and simply tell the players what the payout is? Using mathematical functions to generate the tables so that the size of a payout is proportionate to it’s improbability, less a house percentage – 5%, 10%, 15%, 20% or even 22 1/2% – is probably going to be quicker and easier.

But what’s the price of that simplicity?

It’s my opinion that this sucks all the excitement out of the process – as does the die rolling simulation given above. And you want the players, and hence their characters, to feel that excitement.

In Search Of A Better Simulation

A far better approach would be to create three suitable decks of cards – one for each reel – shuffle each, and then deal them out, one reel at a time.

Certainly, if such a simulation were needed for an in-game setting, that’s by FAR the better approach.

It also gives you a chance to practice assessing the timing needed to build tension. Done improperly, this has all the impact of wet spaghetti; done perfectly, and the PCs will be sweating on every turn of the cards.

As a learning tool

But I started talking about these things as a tool for GMs to learn to feel probabilities, and none of these methods is perfectly suited to that, for the simple reason that the GM is subconsciously aware of the makeup of each deck (assuming that he uses the most efficient simulation method) and this gives him a leg-up on assessing the probability of results.

The full benefit only comes from something close to the real thing. As a general rule, the best method is playing an online slot machine – preferably a free one, but (having sampled those), they often fall back on the same solution rejected in our simulation discussion, of simply guesstimating the probabilities and leaving it at that.

What’s more, most of them are single-line simulations, which simplifies the problem and reduces the benefits to be obtained.

And that only leaves an online casino, where they spend a huge amount of time and money making the simulations as perfect as possible – a site such as Novibet, for example.

You shouldn’t just play games of this type; you should try to get a sense of the odds that have resulted from all those ways of manipulating the odds that I described earlier as they apply to this particular (virtual) machine.

The objective should be to get familiar enough with probability that you can return to those coin-flips and instinctively know what happens to the average length of results if you alter the odds of a head.

With a coin, that’s virtually impossible short of somehow distorting the shape or the weight of the coin or something. But if the coin is just a metaphor for success or failure of a die roll, this is the sort of assessment that GMs have to be able to make on a regular basis – what happens with a bonus of +1 or a penalty of -1? Or -2?

This is a simple assessment with a linear die roll, like a d20; it becomes more complicated with multiple dice in a compound roll, like 3d6 or 4d8.

A lot of this stuff is intuitive, but there are surprising corners every now and then that are strongly counter-intuitive.

For example, there’s Luck in the Hero System.

Feeling Lucky?

The way this game mechanic works is that a character buys a certain number of dice and then rolls them at the start of each game session. Every ‘6’ that comes up contributes to a level of luck, which can be used by the character’s owner to reshape outcomes and induce improbable events favorable to them. One level of luck is a minor benefit, 2 is a bit more significant, and 3 is almost reality-distorting. All clear? good.

The base Hero System limits the number of dice of luck that you can buy to 3d6, charging a fixed amount for each.

Right away, that seems wonky – the benefits of a third die of luck are far more than the benefits of a second die of luck. In any reasonably-realistic schema, the price of each die would increase dramatically.

But what happens to the chances of successfully rolling 3 levels of luck if you increase the number of dice of luck?

Well, for a while, everything increases more or less as you would expect, and everything is fine. But there comes a point – from memory, 15d6 – at which the probability of three levels of luck overtakes the probability of one or two levels. Or maybe it was 21d6 – the point is, though that it happens.

There’s also the question of what to do if a character with, say, 6d6 in Luck rolls four sixes? Do they get a three-level luck result and a one-level? Or do you define additional reality-altering capabilities that are only accessible with higher levels of luck?

Some readers may be wondering why you would want to permit more than three dice of luck in the first place. The first answer is that comics have long had characters whose power is “super-luck” – there is the DC Villain “Amos Fortune,” who gave the Justice League of America bad luck by stealing their luck for his own use, and there’s “Longshot,” a marvel hero.

Additionally, I found that the “luck” mechanic was a wonderful way of incorporating nuance into all sorts of all-or-nothing game mechanics.

The discovery of the distorted probability situation described above brought an end to that, and the unpredictability, unreliability, and wide range of possible outcomes eventually led to the game mechanism being eliminated from the rules completely in favor of a different system.

It was, in fact, thinking about the ‘luck phenomenon’ that initially started me down the road toward what became The Sixes System.

The Improbability Of Success

Let’s look at an example of a practical benefit from the sort of intuitive understanding that we’re talking about.

What are the chances of success in a task requiring more than one roll? And what if there are modifiers – positive or negative – to some of those rolls, but not all? And what if the roll is to be made on 3d6?

Each of those parameters raises the complexity and difficulty of the problem. The best approach is to simplify it again, then reintroduce the complications one at a time.

First Principles

Let’s start by working out how to proceed using a d20. Because this gives a linear probability of any given result, it makes the problem a lot easier to solve.

When you have multiple rolls, all of which need to succeed, you can get the end probability by multiplying the individual probabilities together.

So, starting with a 10/- needed for success on any individual roll, which is to say a 50% chance of success (yes, I’m aware that some of this is so basic and obvious that it’s blatantly obvious):

• On one roll, the chance of success is 50%.
• On two rolls, the chance of success is 50% of 50%, or 25%.
• On three rolls, the chance of success is 50% of 50% of 50%, or 50% of 25%, which is 12½%.

 
Applying a positive modifier to one of the rolls increases that individual chance of success.

• For a +1 modifier:
  • On one roll, the chance of success is 50%+5% = 55%.
  • On two rolls, only one of them modified, the chance of success is 50% of 55%, or 27.5%.
  • On three rolls, only one of them modified, the chance of success is 50% of 50% of 55%, or 50% of 27.5%, or 13.75%.

  This shows that the power of the +1 is considerably reduced – from +5% chance of success to +1.25%. Unsurprisingly, this is 1/4 of what it was.
   

• So, how about +2 on two rolls?
  • On one roll, +2 translates to +2/20, or +10%. So 50% becomes 60%.
  • On two rolls, both at +2, the chance of success is 60% of 60%, which is 36%.
  • On three rolls, two of them at +2, the chance of success is 50% of 36%, or 18%.

  So the matching +2s don’t yield a +20%, or even a +10%; they increase the chance of success overall by 5.5%. NOT 5%, as might have been suspected.
   

• And what if there was a -2 on the third roll, in addition?
  • Minus 2 translates to -10%, so the chance of success on the third roll becomes 40%.
  • Which means that the chance overall is now 40% of 36%, or 14.4%. So, overall, there is an increase of just under 2% from the combination of all these modifiers.

   

• Which raises the question, what negative modifier would cancel out the net benefit of the two +2s?
  • That means that instead of defining the third roll, we are defining the result of the first two (36%) and the net result (12.5%).
  • 12.5/36 = 0.3472222222 = 34.72222222%.
  • So, starting with a base chance of 50%, a modifier of -15.27777778% is needed.
  • ….and that translates to a modifier of -3.055555555.

  ….which means that it would be reasonably close to the truth to say that +2, twice, is equal to -3.

 
This demonstrates exactly how counter-intuitive all this can be at first glance.
 
Next question: what happens with a change in the base chance? What if the base chance was 8/- on d20 and not 10 or less?

• Well, this is exactly the same as applying a -2 modifier to all three rolls.
  • Which is to say, the base chance is 40% of 40% of 40%, or 40% of 16%, or 6.4%.
  • So that small change has roughly halved the chances of overall success.

   

• And if we apply +2 to the first two rolls of the set of three?
  • Then we are talking about 40% of 50% of 50% – which is the same thing as 50% of 50% of 40%, or 50% of 20%, which is to say, 10%.
  • 10% is not very different from the all-50% base result of 12.5%.

   

• And then apply a -2 to the last of the three rolls?
  • Now we’re talking about 50% of 50% of (40-10)%, or 50% of 50% of 30%. Which works out to 7.5%.
  • Which is a small improvement on the 6.4%.

   

• And if we make that -2 a -3, which is what we calculated would just about neutralize the two +2’s?
  • So, 50% of 50% of 25% is 6.25%.
  • The -3 therefore has overwhelmed the two +2s – not by a lot, but by a sufficient amount that the putative truism determined earlier is no longer accurate, because the 0.15% difference in chance is a far larger margin than the error of -0.055555555% that was unaccounted for.

 
Another illuminating question might be, how do the two +2s on two rolls of three, compare to a single +4 on a single roll of a set of three?

• The minimum chance of success on a basic d20 roll comes at 1 or less to succeed (or 20 or more, if you prefer; it’s exactly the same thing).
  • Chance of success (base) = 5% of 5% of 5%. or 0.0125%.
  • +2 on two rolls = 5% of 15% of 15%, or 0.1125%. Which is a substantial increase over the base chance, but doesn’t feel all that generous.
  • +4 on one roll = 5% of 5% of 25% = 0.0625%. Pretty close to bang in the middle of the two numbers. Which means that a single +4 appears to be roughly half as effective as two +2s.

   

• Let’s up the base chances of success to 4 or less.
  • Base chance of success = 20% of 20% of 20%, or 0.8%. Still less than 1% net chance, then.
  • +2 or two rolls = 30% of 30% of 20%, or 1.8%.
  • +4 on one roll = 20% of 20% of 40%, 1.6%.

  That’s not close to half-way between the two – it’s very close to the pair of +2s!
   

• So, let’s up the ante again, to 8 or less base chance.
  • Base chance = 40% of 40% of 40%, or 6.4%.
  • Two +2’s = 50% of 50% of 40%, or 10%.
  • One +4 = 40% of 40% of 60%, or 9.6%.

  So the +4 is now even closer to the two +2s, but still mot quite there.
   

• So, what happens at a 12 or less base chance?
  • Base chance = 60% of 60% of 60%, or 21.6%.
  • Two +2’s = 70% of 70% of 60%, or 29.4%.
  • One +4 = 60% of 60% of 80%, or 28.8%.

  Still not quite on parity terms.
   

• A base chance of 16/-, and we’re running out of maneuvering room.
  • Base chance = 80% of 80% of 80% = 51.2%. That’s right, this is how high you need to set the base rolls to end with a roughly 50-50 chance of success overall!
  • Two +2s = 90% of 90% of 80% = 64.8%.
  • One +4 = 80% of 80% of 100% = 64%.

 
Strange things happen if we go any higher, because the chance of success is capped at 100%. If your base chance of success is 19 or less, a +2 doesn’t make it 21 or less, the chance can’t go above 20 or less.

That doesn’t mean that a +2 modifier is worthless; it just means that we need to individually track each possible result and then work out the overall chances, a lot more work.

Rather than spend time on that, let’s look at what we can learn from the totality of what’s above.

1. Two +2s are always just a little more beneficial than one +4.
2. The greater your base chance of success, the greater the impact of bonuses.
3. It might be less obvious because I haven’t explicitly calculated any examples, but there is enough information there to show that the same is true of penalties. But this effect tends to get swamped by another factor:
4. It takes a ridiculously large base chance to get even a moderate chance of overall success on three rolls. This effect is only exacerbated and amplified by requirements of 4 rolls.
5. A base chance of X with a +Y modifier is the same thing as a base chance of X+Y. Yes, I know this is obvious.
6. Multiple rolls with a base chance of X and a modifier to one of the rolls of Y yield the same chance of success as the same number of rolls with a base chance of X+Y and a modifier of -Y on all but one roll. Think about that for a moment.
7. Lastly, we have now determined a basic technique and employed it often enough that it is almost routine.

One of my players and occasional contributors applied this principle of point 1 to D&D and started asking his GMs for +1 items instead of +2, +3, or even +4 items. The latter requests frequently fell on dead and uncooperative ears, while the smaller requests were more often granted.

So, how many +1s does it take to equal a +4?

x+4 = (x+1)^n

Take the log of both sides:

log(x+4) = n log(x+1)

Rearrange to get n on one side of the equality:

n = log(x+4) / log(x+1)

so:

	x = 1, n = 2.32
	x = 2, n = 1.63
	x = 3, n = 1.4
	x = 4, n = 1.29
	x = 5, n = 1.23
	x = 6, n = 1.18
	x = 7, n = 1.15
	x = 8, n = 1.13
	x = 9, n = 1.11
	x = 10, n = 1.1
	x = 11, n = 1.09
	x = 12, n = 1.08
	x = 13, n = 1.07
	x = 14, n = 1.07
	x = 15, n = 1.06
	x = 16, n = 1.06

If we’re talking D&D combat, then X would be your required roll or less to overcome a particular armor class – or, more accurately, twenty minus the required roll or more to overcome that armor class.

So the answer to the question is inherently variable depending on factors not specified. What is beyond doubt is that the number is a lot smaller than most people would expect.

Another way of looking at the above table is to assume that × basically tracks upward with character level, and that as a general rule of thumb two synergizing +1s are more powerful than a +4 item.

I was discussing this online with someone the other day, and they suggested that a different reality could be perceived by assuming that n has to be multiplied by 4, in this case (because we’re comparing with a +4 item)..

His suggestion was that the results would be an estimate of the synergistic total benefit of four +1’s vs a single +4 given that at higher class levels, natural capability increases would tend to be more significant than bonuses. I can kind of see what he was getting at, but I’m not convinced by his formulation.

What can be said for certain is that four +1s at a low character level are far more likely to be granted than a single +4.

Going to 3d6

So, let’s get a bit messier. With 3d6, not all results are created equally, in terms of probability of result.

If you convert those likelihoods of result to percentages, you get:

	3		0.46%		11	62.5%
	4		1.85%		12	74.07%
	5		4.63%		13	83.8%
	6		9.26%		14	90.74%
	7		16.2%		15	95.37%
	8		25.93%		16	98.15%
	9		37.5%		17	99.54%
	10		50%		18	100%

 
So, let’s put together another suite of results, comparing two +2s with one +4.

• Start with the minimum possible result, 3/- (three or less) chance of success.
  • 3/- = 0.46%, so base chance on three rolls is 0.0046 × 0.0046 × 0.46%, or 0.000 009 733 6% – so that will happen once in 10,273,691-and-a-fraction attempts. It’s as close to impossible as you can get.
  • +2 on 2 rolls makes them 5/-, which is 4.63%. So the overall chance of success becomes 0.0463 × 0.0463 × 0.46% = 0.000 986 097 4%. That’s 101.3 times more likely than the base level but the chances of success are still only one in 101,410, so confidence would be a bit premature.
  • +4 on one roll makes it 7/-, which is 16.2%. Right away, I can see that this will be (16.2 / 0.46) times as likely as the base level, or 35.2 times. Better, but still not great. The actual chance is 0.0046 × 0.0046 × 16.2 = 0.000 342 792%, or a 1 in 291,722 chance. Clearly, base level is a heavily-dominant factor, at least when it’s low.

   

• let’s try 5/- base chance. This is still very low compared to a typical roll in any 3d6 system, it should be noted.
  • We already know that 5/- is 4.63%, so the base chance is 4.63% × 4.63% × 4.63%, or 0.009 925 284 7%, or 1 in 10,075.3 chance. Still not very likely to happen, you will not be surprised to observe. What is more startling is the comparison with the 3/- base level – this is 1019.7 times more likely to succeed, a huge ramping up of the probability.
  • +2 to 5/- gives 7/-, which we already know is 16.2%. So two +2s gives an overall success chance of 16.2% × 16.2% × 4.63%, which calculates out to 0.121 509 72%, or 1 in 823 attempts. Still the longest of long-shots, in my book.
  • +4 to one roll gives 9/-, which is 37.5%, so the base level here is 4.63% × 4.63% × 37.5%, or 0.080 388 375%, the equivalent of 1 in 1,244. Two +2s still yields a much better chance of success.

   

• At 7/-, things should start to get interesting.
  • 7/- is 16.2%; base chance = 16.2% × 16.2% × 16.2%, or 0.425 152 8%, equivalent to about 1 in 235.
  • +2 is 9/-, which is 37.5%. So two +2s = 37.5% × 37.5% × 16.2%, which equals 2.278 125%, or a 1 in 44 chance.
  • +4 is 11/-, which is past the peak of the probability curve, at 62.5%. So the chance of success would be 0.625 × 0.162 × 16.2 = 1.640 25% – so the +4 gives a success chance of 1 in 61.

   

• At 9/-, the base roll is just before the probability peak, while both +2 and +4 modifiers push the chance beyond that peak.
  • 9/- = 37.5%, so 37.5% of 37.5% of 37.5% = base chance of 5.273 437 5 – ever-so-slightly better than a 1 in 20 chance.
  • +2 = 11/- = 62.5%, so two +2s gives a chance of .625 × .625 × 37.5 = 14.648 437 5%, almost 3 in 20.
  • Base +4 = 13/- = 83.8%, so this would yield a chance of success overall of 37.5% × 37.54% × 83.8% = 11.784 375%, more than 2 in 20. The margins between the +4 and the two +2s are shrinking, but two +2s still outweighs a single +4.

   

• At 11/-, the base roll is past the hump. From now on, the base chance should rocket up.
  • 11/- on three rolls is a not at all uncommon in real gameplay, so this is an important result. We already know that 11/- = 62.5%, so the base chance of three rolls = 62.5% of 62.5% of 62.5%, or 24.414 062 5%.- just shy of a 25% chance.
  • 11/- +2 is 13/-, or 83.8%, as noted above. Two +2s therefore give an overall chance of 83.8% of 83.8% of 62.5%, which equals 43.890 25% – quite close to a 9-in-20 chance. Arguably, this is a threshold, above which you could begin to feel reasonably confident.
  • 11/- +4 = 15/-, or 95.37%, so a single +4 gives a net chance of 62.5% of 62.5% of 95.37%, or 37.253 906 25%, just under 7½ out of 20. Once again, the higher the base roll gets, the smaller the gap between the two +2s and a single +4.

   

• 13/- is the last result (going up by pairs) before chance calculations start hitting the cap of 18/- (100%). It also means that our individual-roll probabilities are no longer rising as quickly, so this is going to be getting close to the best result, the point at which further improvements in base chance have (comparatively) little impact.
  • Base chance, 1 roll at 13/-, = 83.8%; so the net chance on three rolls = 83.8% × 83.8% × 83.8% = 58.848 047 2%. So that additional +2 to the base roll more than doubles the net chance over three rolls!
  • 13/- +2 = 15/-, which is 95.37%, so two rolls out of 3 at +2 gives an effective chance of 95.37% of 95.37% of 83.8%, or 76.219 761 222%, or better than a 15-in-20 chance. Perhaps it would be more illuminating, though, to compare it to a single 3d6 roll – this chance is just a little better than 12/- on 3d6, which means that the net effect of the two additional rolls at +2 is essentially a -1 modifier on a single 3d6 die roll – at least at this base chance.
  • 13/- +4 = 17/-, which is 99.54%, or a virtual certainty. Does this mean that you can’t roll box cars on 3d6? Absolutely not, in fact you would expect such a result once every 216 rolls, on average. The net chance is therefore going to be a teeny-tiny whisker under 83.8% of 83.8%; when you do the math, you get 69.901 367 76%. For convenience, use 70%. Again, translating this to a single 3d6 roll is quite instructive – it comes out to a bit below 12/-, call it a ‘theoretical’ 11.7 or 11.8, on 3d6. The two +2s gave us a translated result of about 12.5 on 3d6 – so the difference between the two is is really marginal, in fact it’s within the practical rounding error of using a 3d6 scale!

   

• 15/- starts to give us problems with the +4 modifier, because there’s no such thing as 19/- on 3d6. But the two +2s and base.result should still be illuminating:
  • Base chance at 15/- = 95.37%, so the net chance over three rolls is 95.37%^3, or 100 × 0.9537^3 – which is just another way of writing the usual expression. To the mathematician, this is a more elegant phraseology, and somehow feels more accurate (though it isn’t); to a practical mathematician, an arithmetician, it’s easier to grasp 95.37% of 95.37% of 95.37%. In either case, you end up with a result of 86.743 181 715 3%; note that, as predicted, growth in the base chance has started to slow.
  • 17/- is equivalent to a 99.54% chance as already observed; so two +2s gives a net chance of 99.54% × 99.54% × 95.37%; I predict a value in the low-to-mid 90s even before reaching for my calculator app! After doing so, the result of 94.494 614 029 2% seems right on expectations.
  • simply to demonstrate the addition to the toolkit, let’s look at the +4 answer.
    • The other two rolls give a combined chance of success of 95.37% of 95.37%, or 90.954 369%. That’s the easy part.
    • That means that whatever the chances of success are on the last 3d6 roll, the net chance of success will be 90.95% of it.
    • At first glance: Rolling anything more than 11 is an automatic failure. Rolling 10 or better, with the +4, succeeds. This first glance is incorrect; this is applying the +4 the wrong way around, as though it were a penalty, reducing the chances of success.
    • In fact, anything less than 15 rolled will succeed even without the +4. Rolling a 16 succeeds only because of the +4, and the same is true of rolling 17 or 18. So it doesn’t matter What we (hypothetically) roll, we succeed. That’s what +4 means on a base 15/- chance.
    • So the final probability of success is 90.95%.

   

• A couple of special cases are worth examining, using a nice middle of the road base chance of 11/-. The first of these compares a +2 / -2 modifier combination with the established values.
  • Base chance, from above: 24.414 062 5%
  • Two +2’s, from above (for comparison purposes): 43.890 25%
  • One +4, from above (for comparison purposes): 37.253 906 25%
  • 11/-+2 = 13/- = 83.8%; 11/- (base) = 62.5%; 11/- -2 = 9/- = 37.5%.
  • Calculation: 83.8% of 62.5% of 37.5% = 19.640 625%.

   

• Same base roll (permitting the same results for comparison), Two +2s and one -1:
  • 11/- -1 = 10/- = 50%.
  • Calculation: 83.8% × 83.8% × 50% = 35.1122%. This is very close to a single +4 – at this base roll.

   

• Same base roll, for the same reasons; Two +3s and one -2:
  • 11/- +3 = 14/- = 90.74%; 11/- -2 = 9/- = 37.5%.
  • Calculation: 90.74% × 90.74% × 37.5% = 30.876 553 5%. Despite seeming more generous in doling out the bonuses, this is actually a harder combination than Two +2s and one -1.
  • To understand why, you need to look at the individual rolls relative to the probability peak – the 14’s are well past the peak, but (obviously) below the 100% mark, but the base roll is below the peak, and the -2 applied to it shifts it to well below the peak.
  • That means that we have two numbers close to, but a little below, 100%, and one that is a long way below 100%; if the first two were 100%, the last would be faithfully extended to cover the whole set of rolls, as things stand, they can only make a bad situation worse. So the “-2” is strongly dominant in the final result.

Binomial, Trinomial, and Quadronomial expansions

This section will make a mathematical analysis of everything that’s going on. If you’re not especially interested in that, you can skip it (but I don’t recommend doing so) or skim it (a better choice).

Two rolls can be expressed as a binomial formula:

Three rolls can be expressed as a trinomial formula:

…and, unsurprisingly, Four rolls can be expressed as quadrinomial formula:

These all use the same nomenclature. P is the net probability of success, B is the base roll, ƒp simply means “convert result to a percentage probability”, and a, b, c, and d are the bonuses / penalties to each roll.

Things get more interesting if you replace the ƒp function with a more complicated but useful structure – ƒ1[B] + ƒ2[a/b/c/d]. To simplify, let’s call the ƒ1 formula “X” and the ƒ2 formula “Y1”,- 2, -3 ,and -4 for a, b, c, and d, respectively. So X defines the base probability and Y the change in that base probability.

In practical terms, Yp(n) has to be calculated with a conversion expression to allow for non-linear rolls:

Formulating the expression in this way means that our binomial expression can be written

P = (x + y1) × (x + y2) / 100

or even,

P = x^2 + (y1+y2)•x + (y1•y2) /100

The trinomial expansion can be derived in a similar way, first by expanding two of the terms and then expanding the combination with the third:

	100^2•P= [x + y1] • [x^2 + (y2+y3)•x + (y2•y3)]
	= x • [x^2 + (y2+y3)•x + (y2•y3)] + y1 • [x^2 + (y2+y3)•x + (y2•y3)]
	= x^3 + (y2+y3)•x^2 + (y2•y3)•x + y1•x^2 + y1•(y2+y3)•x + y1•(y2•y3)
	= x^3 + (y2+y3)•x^2 + y1•x^2 + (y2•y3)•x + y1•(y2+y3)•x + y1•y2•y3
	= x^3 + (y1+y2+y3)•x^2 + (y1•y2 + y1•y3+ y2•y3)•x + y1•y2•y3

Similarly the quadrinomial expression (or expressions describing even longer chains of rolls) can be derived – but I’m not going to bother with that right now; instead, let’s move on.

Think about typical values and what these expressions tell us about those typical values.

For a start, we can say that base values are likely to be somewhere in the 8-15 range. This is true whether we’re talking about 3d6 or d20. Next, we can state that the typical modifiers are going to be around the +2 to -2 range.

That means that x is going to be roughly between 4 times and 8 times any of the y values.

Our binomial expansion makes the significance of that clear: x^2 is going to be between 16 and 64 times as significant as y1•y2, with the bit in the middle somewhere in between.

Similarly, the sequence of significance in the trinomial expansion is going to be:

• The x^3, which is between 64 and 512 times as important as the y1•y2•y3 term;
• The •x^2 term, which is between 16 and 64 times as important as the y1•y2•y3 term;
• The •x term, which is 4-8 times as important as the y1•y2•y3 term.

The exception to this truism occurs when a positive modifier is common to all individual rolls, because these effectively raise the base roll. Plus 1 on every roll is the same as setting B one higher. And the lower the base value of B is, the more significant that increase is.

To put it another way, +2 on 15/- is nice to have but not as significant as +2 on 10/-, or even +2 on 5/-.

And that means that one more comparison is worth making: two +2s vs three +1s vs two +1s and one +2. For simplicity, let’s use a d20 roll.

• Low: Base 5/- =25%; 5/- +1 = 6/- = 30%; 5/- +2 = 7/- = 35%.
  • Base chance = 25% × 25% × 25% = 1.5625%.
  • Two +2s: 35% × 35% × 25% = 3.0625%.
  • Three +1s: 30% × 30% × 30% = 2.7%.
  • Two +1s & one +2: 30% × 30% × 35% = 3.15%

   

• Middle: Base 10/- = 50%; +1 = 11/- = 55%; +2 = 12/- = 60%.
  • Base Chance = 50% × 50% × 50% = 12.5%.
  • Two +2s: 60% × 60% × 50% = 18%.
  • Three +1s: 55% × 55% × 55% = 16.6375%.
  • Two +1s & one +2: 55% × 55% × 60% = 18.15%

   

• High: Base 15/- = 75%; +1 = 16/- = 80%; +2 = 17/- = 85%.
  • Base Chance = 75% × 75% × 75% = 42.1875%.
  • Two +2s: 85% × 85% × 75% = 54.1875%.
  • Three +1s: 80% × 80% × 80% = 51.2%.
  • Two +1s & one +2: 80% × 80% × 85% = 54.4%

The important observation here is that three +1s is never quite as good as two +2’s and a base roll, while two +1s & one +2 are even more effective than two +2s and a base roll.

The 9d6 / 3d20 question

The three sets of 3d6 raise the question of comparisons with a single 9d6 roll. The d20 equivalent raises a similar question with respect to a single 3d20 roll.

But we have a LOT of results from preceding sections to compare, so I’m going to make this as minimalist as possible.

To start with, we need the basis of comparisons – statistical analysis of the two sets of rolls, listing the percentage equivalents. For this, I turned to my usual source, Anydice.

I used their service to produce a couple of very pretty graphs, presented below. Unfortunately, to get them to fit the available screen space at Campaign Mastery, they had to be shrunken from the original size, and that has compromised the legibility of the percentages – so I’m going to have to supplement each with a table of the sort already presented.

If you would like to examine the actual graphs as Anydice produces them, I’ll be providing links to those, as well.

First, 3d20:

Link to actual results table: Anydice 3d20

Results:

	1	n/a		16	7.00%		31	50.00%		46	93.00%
	2	n/a		17	8.50%%		32	53.75%		47	94.31%
	3	0.01%		18	10.20%		33	61.15%		48	95.45%
	4	0.05%		19	12.11%		34	64.75%		49	96.42%
	5	0.13%		20	14.25%		35	64.75%		50	97.25%
	6	0.25%		21	16.63%		36	68.25%		51	97.94%
	7	0.44%		22	19.25%		37	71.63%		52	98.50%
	8	0.70%		23	22.10%		38	74.85%		53	98.95%
	9	1.05%		24	25.15%		39	77.90%		54	99.30%
	10	1.50%		25	28.38%		40	80.75%		55	99.56%
	11	2.06%		26	31.75%		41	83.37%		56	99.75%
	12	2.75%		27	35.25%		42	85.75%		57	99.87%
	13	3.58%		28	38.85%		43	87.89%		58	99.95%
	14	4.55%		29	42.52%		44	89.80%		59	99.99%
	15	5.69%		30	46.25%		45	91.50%		60	100%

Analysis, multiple d20 rolls vs 1 roll of 3d20:

• 10/- (base) chance d20
  • One roll = 50% = 31/- on 3d20
  • Two rolls = 25% = 24/- on 3d20
  • Three rolls = 12.5% = 19/- on 3d20
• +1, d20
  • One roll = 55% = 32/- on 3d20
  • Two rolls, one at +1 = 27.5% = 25/- on 3d20
  • Three rolls, one at +1 = 13.75% = 20/- on 3d20
• +2, d20
  • One roll = 60% = 34/- on 3d20
  • Two rolls, both at +2 = 36% = 27/- on 3d20
  • Three rolls, two at +2 = 18% = 22/- on 3d20
  • Three rolls, two at +2, one at -2 = 14.4% = 20/- on 3d20
  • Three rolls, two at +2, one at -3 = 12.6% = 19/- on 3d20
• Three rolls at -2, or base chance 8/- on d20 = 6.4% = 16/- on 3d20
• Three rolls, one at -2 = 10% = 18/- on 3d20
• Three rolls, one at -4 = 7.5% = 16/- on 3d20
• Three rolls, one at -5 = 6.25% = about 15½/- on 3d20
• comparing two +2s on three rolls vs a single +4 on one of three rolls:
  • Base roll 1/- = 0.0125% = 3/- on 3d20
  • Base roll 1/-, Two +2s = 0.1125% = 5/- on 3d20
  • Base roll 1/-, One +4 = 0.0625% = 4/- on 3d20
  • Base roll 4/- = 0.8% = 8/- on 3d20
  • Base roll 4/-, Two +2s = 1.8% = 11/- on 3d20
  • Base roll 4/-, One +4 = 1.6% = 10/- on 3d20
  • Base roll 8/- = 6.4% = about 16/- on 3d20
  • Base roll 8/-, Two +2s = 10% = 18/- on 3d20
  • Base roll 8/-, One +4 = 9.6% = around 17/- on 3d20
  • Base roll 12/- = 21.6% = 23/- on 3d20
  • Base roll 12/-, Two +2s = 29.4% = about 25½/- on 3d20
  • Base roll 12/-, One +4 = 28.8% = 25/- on 3d20
  • Base roll 16/- = 51.2% = 31/- on 3d20
  • Base roll 16/-, Two +2s = 64.8% = a fraction over 35/- on 3d20
  • Base roll 16/-, One +4 = 64% = 35/- on 3d20
• Low: Base 5/-
  • Base chance = 1.5625% = 10/- on 3d20
  • Two +2s = 3.0625% = 12/- on 3d20
  • Three +1s = 2.7% = 12/- on 3d20
  • Two +1s & one +2 = 3.15% = 13/- on 3d20
• Middle: Base 10/-
  • Base Chance = 12.5% = 19/- on 3d20
  • Two +2s = 18% = 22/- on 3d20
  • Three +1s = 16.6375% = 21/- on 3d20
  • Two +1s & one +2 = 18.15% = 22/- on 3d20
• High: Base 15/-
  • Base Chance = 42.1875% = 29/- on 3d20
  • Two +2s = 54.1875% = 32/- on 3d20
  • Three +1s = 51.2% = 31/- on 3d20
  • Two +1s & one +2 = 54.4% = 32/- on 3d20

Next, 9d6:

Link to actual results table: Anydice 3d20.

Results:

	1	n/a		16	0.11%		31	50.00%		46	99.89%
	2	n/a		17	0.24%		32	57.61%		47	99.95%
	3	n/a		18	0.46%		33	64.96%		48	99.98%
	4	n/a		19	0.85%		34	71.81%		49	99.99%
	5	n/a		20	1.49%		35	77.96%		50	100%
	6	n/a		21	2.47%		36	83.28%		51	100%
	7	n/a		22	3.92%		37	87.72%		52	100%
	8	n/a		23	5.96%		38	91.29%		53	100%
	9	0.00%		24	8.71%		39	94.04%		54	100%
	10	0.00%		25	12.28%		40	96.08%		55	n/a
	11	0.00%		26	16.72%		41	97.53%		56	n/a
	12	0.00%		27	22.04%		42	98.51%		57	n/a
	13	0.01%		28	28.19%		43	99.15%		58	n/a
	14	0.02%		29	35.04%		44	99.54%		59	n/a
	15	0.05%		30	42.39%		45	99.76%		60	n/a

Notice that rounding error has crept into the table – if the result is less than 0.01%, it has been listed as “0.00%, and if more than 99.99%, as 100%. The probabilities of these results are so low that they might as well not exist. It will only matter on one roll out of 10,000 – or less.

Analysis, multiple 3d6 rolls vs 1 roll of 9d6:

• base roll 3/- = 0.000 009 7336% = 9/- on 9d6
• +2 on 2 rolls, base 3/- = 0.000 986 0974% = 9/- on 9d6
• +4 on 1 roll, base 3/- = 0.000 342 792% = 9/- or maybe 10/- on 9d6
   
• base roll 5/- = 0.009 925 2847% = 13/- on 9d6
• +2 on 2 rolls, base 5/- = 0.121 509 72% = 16/- on 9d6
• +4 on 1 roll, base 5/- = 0.080 388 375% = about 15½/- on 9d6
   
• base roll 7/- = 0.425 1528% = 18/- on 9d6
• +2 on 2 rolls, base 7/- = 2.278 125% = 21/- on 9d6
• +4 on 1 roll, base 7/- = 1.640 25% = 20/- on 9d6
   
• base roll 9/- = 5.273 4375% = 23/- on 9d6
• +2 on 2 rolls, base 9/- = 14.648 4375% = about 25½/- on 9d6
• +4 on 1 roll, base 9/- = 11.754 375% = 25/- on 9d6
   
• base roll 11/- = 24.414 0625% = 27/- on 9d6
• +2 on 2 rolls, base 11/- = 43.89025% = 30/- on 9d6
• +4 on 1 roll, base 11/- = 37.253 906 25% = 29/- on 9d6
   
• +2 on 1 roll, -2 on another, base 11/- = 19.640 625% = about 26½/- on 9d6
• +2 on 2 rolls, -1 on a third, base 11/- = 35.1122% = about 29/- on 9d6
• +3 on 2 rolls, -2 on another, base 11/- = 30.876 5535% = about 28/- on 9d6
   
• base roll 13/- = 58.848 0472% = 32/- on 9d6
• +2 on 2 rolls, base 13/- = 76.219 761 222% = 35/- on 9d6
• +4 on 1 roll, base 13/- = 69.901 367 76% = 34/- on 9d6
   
• base roll 15/- = 86.743 181 7153% = 37/- on 9d6
• +2 on 2 rolls, base 15/- = 94.494 614 0292% = 39/- on 9d6
• +4 on 1 roll, base 15/- = 90.95% = 38/- on 9d6

Reflections

If you study the results from anydice, it should strike you that the 3d20 rise more gradually and evenly than the 9d6. In a nutshell, the more dice, the faster the attack on the average values and the more remote the extremes of the range.

The shape of Lucky

Dice are at the heart of tabletop RPGs. They are the weapons and instruments of both the Players and the GM. Like any tool, they are more powerful and useful in the hands of an expert who has mastered them tham they are in the hands of an amateur.

Such mastery is not easily come by. I have known people who have gamed for 30 years who couldn’t tell you how the chances of rolling successive successes on 3d6 change with different bonuses.

Every time you think you have a grasp on the subject, remember that +2 × 2 = +3 × 1, and you will find any overconfidence quickly undermined.

Once you have mastered the convoluted shape of Luck, however, you will begin to think of rolls not in terms of their chances of success or failure but as navigational markers through your plotlines.

It’s at that point that you can finally know, almost instinctively, what the chances are, and how you can use that knowledge to everyone’s benefit as GM.

This article contains material generated as background reference in Mike’s Doctor Who: A Vortex Of War campaign, but it holds relevance to most campaigns including those of the Fantasy genre.

Introduction

Space is big – really, really, big.

I’m sure most readers will have come across that phrase, or something very like it, on numerous occasions, and have taken its lesson to heart.

But I would be equally certain that comic book writers and sci-fi authors and scriptwriters would also have done so – especially given the practice of vetting for scientific accuracy inherent in the last category.

And yet, I have been let down repeatedly in this respect by those very groups, so sometimes you have to wonder…

Part of the problem is undoubtedly because the scales concerned are epic beyond our ability to comprehend them directly. Of necessity, they have to be abstracted and we have to learn to think in those abstract scales.

But doing so leaves us vulnerable when we have to step up to another scale again; we understand the first scale and think that gives us a handle on the second. And that confidence is frequently misplaced.

Today’s article is intended to bridge that gap.

And my chosen starting place is one of my favorite comics as a kid: Green Lantern, specifically, the Green Lantern Corps.

3600 Sectors Of Trouble

Part of the canon of the Green Lantern Corps is that there are 3600 Green Lanterns, each of whom patrols a different sector of the galaxy, and who are usually drawn from one of the inhabited worlds within that sector.

And, if you don’t appreciate how big the galaxy actually is, that sounds perfectly reasonable. But one look below the surface reveals trouble brewing.

The size of the galaxy

In the course of a previous article on astrophysics (both for within games, and in general), A Game Of Drakes and Detectives: Where’s ET?, I reported on the size of the milky way, and gave various other parameters that will be useful in this discussion.

Let’s start with the cross -section of the milky way.

To quote from the accompanying article:

The milky way is roughly 150,000-200,000 light years in diameter, giving it a radius of 75-100,000 light years. But most of that is outlying material; in terms of the parts we’re interested in, it’s about 100,000 light-years across and about 1,000 light-years thick. But that thickness is the average for the whole thing, and the core noticeably bulges; about three times the thickness of the arms. We also need to exclude that core from our calculation of the plan area of the disk if we hope to get a volume. Looking at the galactic cross-section, the core is about 1/5th of the total diameter across, so about 20,000 light-years.

When I do that, I get an average thickness of the disk section of 926 light years, and a torroidal area of 2,400 million pi square light years, so the arms contain roughly 7 million million cubic light years.

The size of a sector

For the moment, let’s ignore the central bulge. That means that out 3600 sectors contain 7 million million cubic light years. If each sector holds equal volume, then will have a volume of 1944444444 cubic light years.

While it would be inaccurate to do so, let’s ignore that inaccuracy, and project this volume as a square section of the milky way that’s 926 light years thick. That means that our square has to have an area of roughly 2099832 square light years, which is a square of sides 1449 light years to a side.

That means that a hypothetical ‘sector’ would look like the diagram to the right:

Appreciating the size of a sector

To get a real handle on how big this is, let’s assume that our hypothetical green lantern is based at the extreme bottom front left, labeled α, and that some emergency occurs just barely within his area of responsibility at the extreme top right back. In other words, he has to cross the sector from one corner to the other, the distance between the points α and γ.

We know that β is a right angled corner, so what we have is a simple triangle in which we know only one length – βγ, defined as 926 light years.

But, we can also see that αβ is also a triangle with a right-angle, and we know the length of both sides (1449 light years), so good old pythagoris tells us what we need to know:

αβ^2 = 1449^2 + 1449^2 = 2 &Times; 2,099,601 = 4,199,202; therefore,
αβ = √4199202 = 2049.195.

Now we have two sides of the triangle αβγ – 2049.195 and 926. So:

αγ^2 = 2049.195^2 + 926^2 = 4199202 + 857476 = 5056678;
αγ = √ 5056678 = 2248.706 light years.

Now, the green lantern corps can travel at FTL speeds, but the actual speed is rarely if ever stated out loud; it’s “as fast as their will permits”. So, let’s throw some reasonable FTL speeds around and see how long this theoretical corner-to-corner trip will take.

At 10c, 2248.706 / 10 = 224.87 years.
At 100c, 22.48706 years.
At 1000c, 2.248706 years.

Hmm, that’s not working too well. So, let’s press on to more radical speeds:

At 10,000c, 0.2248706 years = 82.13398665 days (defining a year as 365.25 days).
At 100,000c, 0.02248706 years = 8.213398665 days.
At 1,000,000c, 0.8213398665 days = 19hrs 42m 44s.

Based on those numbers, to make a sector patrolable in any practical sense, speeds of between 50,000 and 1,000,000 times the speed of light are required.

But, by scaling the problem to numbers that we can all comprehend, we start to get a real impression of just how big a region of space we’re talking about.

The Size of a sector, part 2

But wait a moment – how are the people of gamma supposed to tell alpha that there’s a problem? A radio message will take over 2,200 years to get there – and be very hard to even detect, as the earlier article points out. So it’s not just the green lanterns that have to travel at ridiculous speeds, it’s everyone else.

The alternative, since not all the people being protected even have space flight, is for the Green Lantern to visit regularly, showing the flag and looking for trouble. And that points us back toward those ridiculous travel speeds.

The size of the galaxy, part 2

Let’s imaging a green lantern on the outer rim of the galaxy. Every now and then he has to report back to Oa. Most of the time, he creates a space warp that conveniently gets him there, but every now and then, there can be reasons for doing the trip the long way around.

Before we can assess that, however, we need to know where Oa is located.

Well, there are three logical possibilities – it’s either at the outer edge of the galaxy, it’s in close to the galactic core, or it’s somewhere in the middle of the disc.

This diagram illustrates the worst-case that results. The three proposed locations for Oa are labeled β, γ, and ε, respectively, while α remains our point of origin. Even without the black hole at the center (4), there would be enough radiation sources that travel straight through the core would be inadvisable. So, to safely get to β, we need to go to point 1 first. Similarly, to get to γ we need to go to point 2 first; and to get to ε, we need to go to 2, then to 3. Five and Six denote the ‘edges’ (top and middle, respectively) of the bulge.

A little thought will show that α to 1 is the hypotenuse of a triangle, with 4 it’s other corner, and that β-to-1 will be exactly the same length, and so will α to 2. It’s only once past the dangerous central galaxy that the course is altered by the different locations of Oa.

According to the cross-section diagram shown earlier, distance 1-4 is going to be 10,000 light years, and alpha to 4 will be 10,000 + 40,000 = 50,000 light years. That means that the first-leg distance is:

α-to-1 = 1-to-β = &alpha-to-2 = √ (10,000^2 + 50,000^2) = 50990 light years.

Therefore, α to β is twice this, or 101,980 light years.

2-4-γ forms a triangle with the same 4-2 measurement as 4-to-1, 10,000 light years, but the long axis is 20,000 light years less than the 50,000. So the distance from 2 to γ is about 31623 light years. So the total trip from α to γ will be 50,990 + 31,623 = 82,613.

α-to-two-to-three-to-ε is a more complicated problem, but we can easily calculate the distance direct from 2 to epsilon; while the additional deliverance to 3 will add to that, it would be a relatively small error. So, the length to ε from 4 is going to be 40,000 less than the 50,000, or 10,000; and therefore the direct distance from 2 to ε will be about 14142. Round it up to 14400, and that should be more than enough to compensate for the more complex course; and the total trip from α to ε is going to come to roughly 50,990 + 14,400 = 65,390 light years.

Now let’s apply those earlier speed estimates (50,000 and 1,000,000 times the speed of light, respectively, and calculate some travel times:

α-to-β @ 50,000c = 2.0396 years.
α-to-γ @ 50,000c = 1.65226 years, or about 20 months..
α-to_ε @50,000c = 1.3078 years, or about 15½ months.

α-to-β @ 1,000,000c = 0.10198 years = 37.248195 days.
α-to-γ @ 1,000,000c = 0.082613 years, or 30.2 days – call it a month.
α-to_ε @50,000c = 0.06539 years, or about 24 days.

The more ridiculously fast we make the travel, the less of a problem this becomes.

The Forest

There’s another saying – that you sometimes can’t see the forest for the trees. How many stars are likely to be present in a single sector?

In that earlier article, I calculated as a very rubbery best-guess that there were 220,000 million stars in the disc-region of the milky way. If there are 3600 sectors, that means that on average, each will contain 61,111,111 stars. From the earlier calculation of the volume of a sector (1,944,444,444 cubic light years), that means that each would occupy roughly 31.82 cubic light years, or a sphere 1.966 light years radius, on average. So the average gap between stars will be twice that, or one star every 3.933 light years.

Corner-to-corner in a sector? 2248.706 light years? That means running into (on average) 572 stars – but one is our departure point, and one our destination, so that’s 570 in the way, en route.

At 10c, that would be one every 224.87/570 = 0.3945 years.= 1 every 144 days. That’s doable.
At 100c, that becomes one every 14.4 days.
At 1000c, 1.44 days.
At 10,000c, 0.144 days = 3.456 hours.
At 100,000c, 0.3456 hours = 20.736 minutes.
At 1,000,000c, 2.0736 minutes. Constantly. For 20 hours or more.

I submit that with size, and radiation output, and potentially hostile residents, that anything faster than about 7,000 times the speed of light involves impossible speed of navigation – that would be a course correction every 5 hours or so, giving at least half-a night’s sleep. Drillers and fishermen have been operating on a four-hours-on, four-hours off schedule for years, and it’s not exactly unfamiliar territory for the military, either.

But if that’s our top speed, then the corner-to-corner sector trip will take about 117 days. And that’s far too long for a green lantern to be able to respond to an emergency.

But what’s the alternative?

Challenging assumptions

Okay, so let’s start by chucking the idea of 3600 sectors, and allow there to be more – many more. In fact, let’s look at stellar populations, make a few sci-fi-valid assumptions, and derive an estimate for just how big a sector should be – and use that to determine how many sectors there should be.

Let’s start by thinking about systems of significance – because some of them won’t be.

For a start, one of the inherent assumptions is that if life is possible, it will find a way; inhabited systems will be common. Next, let’s assume that for every inhabited system, there will be 1½ systems containing significant resources, but no life, giving those inhabited systems something to fight over, and something to kick-start interstellar expansion. And, because a system can have no significance other than being innately interesting for some reason, let’s say that such ‘scenic’ worlds are another ½

How many inhabited systems can one Green Lantern protect? Well, 1/3 aren’t advanced enough, technologically, to get themselves or anyone else into trouble; but that makes them an easy target for conquerors and would-be exploiters. 1/3 would be advanced enough to fend for themselves and enlightened enough not to try and exploit others (but they can still get into trouble occasionally). That leaves 1/3 as potential troublemakers.

Let’s assume that each of the troublemakers have to visited every year to keep an eye on them, and that such inspections take at least 3 days, not counting travel time. The more advanced and enlightened worlds might need to be visited once every 5 years for a day; and the primitive worlds once a year for a day.

So 1/3 of the stars need 3 days attention a year; 1/3 need 1 day’s attention; and 1/3 need 1/5 of a day. Add those up, and you get 4.2 days per interesting star. Throw in a couple of days of travel between them, and you get 8.2 days per star system of interest.

365 days in a year, divided by 8.2 days, gives 44.5 systems of interest. But there’s an assumed inefficiency here – sometimes you will be able to deal with one thing while en route to deal with another. So let’s increase that workload 300% and then allow for a little time off each year – giving 120 or so star systems.

With those numbers as a rough starting point, I get 61 inhabited systems, 93 worlds with significant resources, and 30 systems of other galactic significance, and a net stellar population of 1200 stars under one Green Lantern – on average.

Based on that premise, I divided the galaxy up so that green lanterns only had one galactic arm each within their sectors, and used stellar densities to divide the galaxy up into 305 regions, each of which would contain 400 sectors. I also found that I needed multiple strata or layers. In fact, when I counted them up, I got 350. Put those together, and you end up with 8,200,000 sectors, as the diagram below makes clear (the dots were my method of counting them, each color is 50 regions or strata).:

Click on the image for an even larger (more legible) version in a new tab.

That really puts into perspective just how far wide of the mark that 3600 sectors was, doesn’t it?

Enhanced functionality

But this defines an average sector – as noted, some regions could have as many as 20 times these numbers, while others have less.

It can be presumed that 20 times the standard number of inhabited systems – 1220 of them – there would be twenty times the number of systems capable of provide a Green Lantern to the corps. Instead of one Green Lantern, they might have ten or twenty. Add in the fact that as stellar densities go up, travel time from one star to another goes down because the stars are closer together. Which means that fewer Green Lanterns are actually needed in such dense Sectors.

What about the sectors with fewer inhabited systems? Potentially, one Green Lantern could look after multiple adjacent sectors, but travel times form a significant restriction, so there are limits to this sort of thing. Fortunately, there’s an excess of Green Lanterns from the more densely-populated sectors, so a few of those can be “exiled” to the galactic periphery, perhaps as a temporary tour, eventually rotating back to their more-populated home sector.

The size of a sector, revisited

Instead of 3600 sectors, dividing the galaxy up into 8,200,000 makes them significantly smaller – so much so that it’s worth revisiting the physical size of a typical sector, and recalculating the corner-to-corner (worst case) travel times.

There are two possible approaches to the calculation: we could use the density of stars derived earlier, multiply by 1200, and get one answer for the volume; or we could take the estimated volume of the milky way and divide that by the number of sectors. In theory, both should give the same answer.

But I have the suspicion that the packing problem might be a source of significant error with the first approach.

Not familiar with the Packing problem? Consider a box of oranges. Your job is to arrange them to get as many as possible into the box, i.e. to minimize the wasted space.

If you simply stack them one on top of another (as shown above), there is a huge amount of empty space – each orange is taking up a cube of sides “2 orange-halves” long, a volume of 8o^3, but each orange only fills 4/3πr^3 = 4.19o^3. Almost half the space taken up by an orange is empty.

Instead, each row nests in the hollow created by the oranges of the layer below, effectively interleaving the layers of oranges. Calculating the difference isn’t particularly relevant, but ANY improvement is significant. And you can improve packing density even more by choosing slightly smaller oranges for the ‘indented’ layers.

I’m concerned that taking the spherical volume controlled by each star and simply multiplying by the number of stars might assume perfect stacking, or might assume linear stacking like the example shown, and any rounding error multiplied by 1200 is going to be significant.

So let’s do it in exactly the way we derived the size of a 3600th-sector.

7 million million cubic light years divided by 8,200,000 = 853658.5366 cubic light years each, =
a cube of sides 94.86 light-years across. Call it 95 light years for convenience.

Aside from the dimensions and proportions, the diagram representing a sector hasn’t changed.

α-to-β ^2 = 95^2 + 95^2 = 2 &Times; 9025 = 18050;
α-to-β = 134.35 light years.

α-to-γ ^2 = 95^2 + 134.35^2 = 9025 + 18050 = 27075;
α-to-γ = 164.545 light years.

At 10c, that’s 16.4545 years.
At 100c, that’s 1.645 45 years = 20 months..
At 1000c, that’s 0.164 545 years = 2 months.
At 10,000c, that’s 0.016 4545 years.= 6.010006125 days.
At 100,000c, that’s 0.001 645 45 years = 0.6 days = 14.424 hrs.
At 1,000,000c, that’s 16.4545 years = 0.06 days = 1.44240147 hours, = 86.544 minutes.

At 60,100c, that’s exactly 24 hours.

The Starfleet Problem

So, we have 8.2 million sectors that need Green Lanterns. Most need only one, but a significant number need between 1 and 20, and a significant number can’t supply even one, and so need to “borrow” one from one of the sectors with multiple GLs. Which means the average of those higher sectors isn’t going to be 10.5, it’s going to be more like 11.5 or 12.

If 20% of the sectors need to provide 12 GLs and 20% provide none, on average, that’s a total of 24.6 million GLs that need recruitment and training. Once trained, they need to maintain their proficiency, so that’s a further training burden.

How long does the average Green Lantern last? Maybe 20 years, maybe less? That means that 1.23 million need to be trained every year. And, if they have to renew their qualifications every 5 years, but that takes a fiftieth as long as the training, that’s another 0.0984 million ‘trainees’ a year. Total: 1.3284 million.

How many trainers are there to a trainee? How much allowance has to be made for trainees that wash out? How many administrators and other support staff are needed?

This brings us headlong into the Starfleet problem.

There is an episode of the Next Generation which follows Wesley Crusher to his being tested for entrance into the Starfleet Academy. Four gifted students have been preselected, but there’s only one space available. The other three are out of luck – for this year’s intake.

If you have an organization like Star Fleet, you are going to get millions upon millions of applicants per year – if not Billions. If there are 3,000 inhabited star systems in the Federation (a number plucked out of thin air) with an average of 1,000,000 inhabitants each (another number plucked from the ether with absolutely no justification), that’s 3000 million people. Earth alone, even after the calamities in the Star Trek history, is likely to have at least that number, and so are a number of other worlds. Kronos (the Klingon home world) and Vulcan come to mind, for example. All up, a minimum population of at least 12 billion people, and potentially considerably more.

If one percent a decade apply, that’s 120,000,000 applications, or 12 million a year. And if only 1% pass pre-application screening, that’s 120,000 applications. For how many openings? 30,000? 20,000? Ten?

It’s clear that the producers and writers of the episode in question had thought about this, and hence the 1-in-4 cut-off.

But here’s the rub: There is no certainty that the applicants from Moomba-III that are accepted are better than the applicants from Nonga-II that were rejected.

Starfleet is not an elitist organization, it’s not geared to recruit the best of the best – it’s geared to reject the excess while distributing it’s representation as broadly as possible.

And yet, in virtually every episode of TNG, and DS9, and Voyager, and more, Starfleet is portrayed as being the best of the best. So, while the portrayal of the recruitment process is logical, but flawed, it is also inconsistent with the portrayal of the organization outside of this episode.

The Starfleet problem is how do you recruit the best of the best when they are scattered throughout the Federation?

If instantaneous communications galaxy-wide are possible, as shown in Star Trek’s various incarnations, it becomes possible to do so – but that invalidates the entire premise of the drama within the episode in question. For this reason, I’ve never considered the episode as canonical; it falls through a logic hole.

The Green Lanterns – do they have such instantaneous communications? Some adventures suggest yes, others suggest no.

A bigger problem, though is the logistics required to actually train that many recruits. And house them. And feed them.

The Logistics Of Galactic Organizations

And therein lies the problem. These calculations, for the first time, create a practical appreciation of the size of the galaxy, and hence of the size of any galaxy-wide organization. And the results just don’t fit with the descriptions of those organizations in science fiction and other media.

What’s more, the questions scale – they apply just as reasonably to am organization like Star Fleet, even though that organization only operates in somewhat less than one quadrant of the galaxy.

They would scale to the local interstellar region, where small empires of 50-100 star systems might exist.

You can even scale them to be appropriate to an empire or kingdom in D&D terms – the questions are similar (small communities instead of stars), and the results are just as valid.

Once you can get a handle on the scale of your organization – be it a thief’s guild or a multinational church or the political organization of a nation – you can start to properly consider the logistics that are necessary for that organization to function.

There’s going to be an inherent logic that makes obvious sense to you. The consequences may well be surprising – who saw 8,200,000 sectors coming? – but they will be valid, and that will show.

Or, more accurately, the flawed extrapolations of incorrect assessments of scale will no longer be visible – romantic notions like 3600 sectors that look good on paper but make no sense in reality.

Questions Of Scale

But what, you may be wondering, if my assessments of the frequency of population of inhabited worlds is wrong? What if there aren’t 60-odd inhabited systems in a collection of 1200 stars, but only 30, or 20?

Obviously, the size of sectors would increase somewhat – but not be very much; distance between solar systems is unaffected, and that imposes a hard limit on what sounds plausible. Even 60,100 times the speed of light is pushing credibility to the limit.

Distance matters far more than most people appreciate. That’s why improvements in the technology of moving things around tend to have massive national and international repercussions; this is one of the most under-appreciated pillars of society.

If there’s one lesson from history that should be learned by all, it’s this: When people can do in days what would have taken weeks or months previously, society begins to change. When people can move freight around at the same pace, the transformation of society becomes inevitable.

• When humans had to carry everything on their own or their animal’s backs, mobility was limited, and so was the size of society.
• When the Romans introduced roads, it became far more efficient to move goods and people around. While carts had already existed, this was the change that enabled Empires to form.
• The age of Sail made international travel and commerce possible beyond one’s immediate neighbors.
• The age of Steam brought profound social impacts that altered every aspect of society, either directly or indirectly.
• The aircraft completely changed the rules of such trade. We’re still discovering and reacting to the ramifications of that – the most recent lesson being disrupted supply chains.
• But already, we can see the age of air freight coming to an end – not because of a lack of fuel, as was once thought to be the likely problem, but because of the climatic consequences. It seems likely that some reversion for cargoes of lesser importance will take place – unless we invent some sort of teleportation, of course.

Distances matter, and distances are a reflection of the proper appreciation of scale. This article has given everyone the basic tools that they need, and shown how to apply them; I consider that to be a very good day’s work.

I came across a remarkable mathematical fact the other day, which immediately gave me the idea for this post.

Yet, while I noted the fact, and roughed out a structure for this article, when the time came to actually write it, the gaming relevance that had been so obvious and self-evident that I had not written it down completely escaped me!

I can only hope that by the time I get to the end of my notes, it will have come back to me!

Introduction / Preface

I should begin by thanking Peter-3699 of Quora, who posted the remarkable mathematical fact that inspired this article – I’ll link to it when it becomes relevant.

Every non-Wikipedia link in this article is either from him or from a comment to his post, or from a page so linked, and so (arguably) would not exist without his post.

Any readers with visual impairment should note that I have gone to some trouble to quote most of the mathematical formulae discussed in this article as Alt-text, so you won’t be left out. I can’t make it any easier for you, I’m afraid, but I hope that it will be better than nothing.

The remarkable property of Pi

The properties of Pi have long fascinated mathematicians – it is what is called a Irrational Number, a number with a never-ending number of decimal places that never repeat. There are a boatload of these known to maths these days. An irrational number is one that can’t be precisely defined as a fraction of two whole (integer) numbers (though approximations are possible).

It’s conjectured (and widely believed) that the various decimal digits (0, 1, 2, and so on up to 9) are evenly and randomly distributed, but this has never been proven.

Pi is one of the earliest constants known to reflect a physical property of our reality – the Circumference of a circle is 2πr and it’s area is πr².Those mean that the properties of cylinders and spheres also use π. But π shows up in trigonometry, and electrical formulae, and in formulas about spings, and all sorts of other places, too.

When I was a young high-school student (aged 12 or 13), I was fascinated by two facets of pi and spent many hours attempting to understand them.

The first was inspired by my discovery that you can get the logarithm to any base by dividing the logarithm in a known base of the number desired by the logarithm in that same base of the desired base. Spelling it out in words is not as elegant as showing it as a formula:

I routinely use y=10.

I’ve found this to be useful in RPG rule analysis and construction many times, mostly for bases of 2 and 5.

(Another pair of formulas of value, while I’m in the vicinity of the subject, are

It doesn’t matter what the base of the logarithm is so long as it is the same both times.

…and…

It doesn’t matter what the base is so long as it is the same in all three cases.

At the time, though, i didn’t even know that RPGs existed (and to be fair, at that time, they didn’t exist in any form that we would now recognize, this was the mid-70s). Instead, I was captivated by other concepts.

I already knew that logarithm bases could be irrational, having discovered a reference to natural logarithms (logs to the base of e, which is physical constant defined as approximately equal to 2.71828. (e also shows up in all sorts of unexpected places, for example in modeling compound interest). In fact, it’s relevant to all sorts of exponential growth and decay, including half-lives and biological population growth.

But I couldn’t find anything anywhere about logs to the base of pi, and whether or not this was a useful or practical concept. Short answer – it is, but perhaps less than you might think.

The other question was inspired by a Scientific American whose cover story focused on attempting to find patterns in various geometric representation of the distribution of prime numbers, or the results of plugging prime numbers into various formulae such as n=(P(a)-1)/2, or n=[P(a) – P(a-1)] (where P(a) is a given prime number, like 11, and P(a-1) is the preceding prime number.

Aside from being fascinating in and of itself (and endlessly time-consuming), I wondered if there was some relationship between the digits of an irrational number like pi and the distribution of prime numbers. Instead of a lattice for example, what if the numbers were organizes in growing concentric rings with 0 or 1 in the center?

Short answer: I could never find one, but that doesn’t really prove anything. It was a fun diversion, though.

Quite obviously, there have been many and ongoing attempts to calculate pi, first for its practical value and second because it’s nature makes it a gateway drug into some of the most abstruse realms of higher mathematics.

Babylonian mathematicians usually approximated the value to 3, which was good enough for the archaeological projects of the time. This value was also used in astronomical calculations in India. By the 6th century BCE, Indians were using 339/108 as an approximation.

A thousand years earlier, in a text that was itself stated to be a copy of an even older document in ancient Egypt, the same fractional approximation of 339/108 was described.

Archimedes proved that pi lay somewhere in between 223/71 and 22/7 using geometry of regular polygons within a circle, which would give a circumference of ever-increasing accuracy with more ‘faces’ or ‘gons’ (“poly” means “many”, so “polygon” means “many gons”). For some unknown reason, he stopped at a 96-sided polygon even though his technique required only patience to be extended a considerable distance further.

So, pi is important, and that has led to many attempts to calculate it, to get back to the point.

In fact, the Pi Formulas page of Wolfram Mathworld lists no less than 135 different formulas for calculating Pi! Most of them are too exotic to explain here; I’ll get to some of those that are not in that category in due course.

But this answer on Quora got me thinking about the nature and representation of decimalized numbers…

Whole Numbers

The simplest such numbers are whole integers, with no decimals to worry about at all. The approximations of pi as “3” are representative of this. (Integers, when you dig into them, can be just as fascinating as irrational numbers. For example, there are an infinite number of them, but for every single one of them, there are an infinite number of numbers that aren’t integers – which is a gateway into the very strange world of the mathematics of infinity.

Simple Fractions

As soon as you come up with the concept of measuring some objective reality, you start discovering the world of simple fractions. For example, if you have an object of a particular length, the midpoint is found by dividing that length by 2. If the length as measured happens to be evenly divisible, this is easy; but if it is not, you end up with either a remainder (not useful) or a fraction, 1/2, included in the answer.

Divide something into 3, and you get the fractions of 1/3 and 2/3 being defined, and so on.

Some fractions that are technically “simple” go beyond what I would consider “simple” in an everyday interpretation of the word. “22/7” is simple in both interpretations, and is perhaps the simplest real approximation of pi; the fractional approximations given earlier, like 339/108, may technically be simple, but are pushing the limit of the everyday sense of the word.

Fractions are inherently bound up in geometry and lead into angles and trigonometry. But they remain a finite tool until something else is added to the mix: the invention of a zero.

The Invention of Zero

Zero makes positional notation possible. Without it, you can’t have decimals. “10” is positional notation; the position of the “1” is meaningful, with the character ‘0’ being used to describe that position.

Ancient Egypt had a zero concept for use in accountancy, but did not use positional notation; each number was represented by one or more hieroglyphs. The ancient Babylonians came close, with a symbol used as a placeholder for a zero in their base-60 system.

Modern representations of time that would be familiar to all readers – 3’59” for example – preserve this base-60 system, with 60 seconds equating to a minute and 60 minutes to an hour. The symbols ‘ and ” identify the significance of the 3 and the 59 that – in this example – precede those symbols, respectively. This is a somewhat more refined version of the Babylonian system.

The ancient Greeks had no symbol for zero, and no positional notation. In fact, Greek philosophers opposed the concept of zero as a number very strongly for a very long time, going so far as to translate their numbers into the Babylonian number system for calculations and then translating the results back into Greek to give their results, just so that they could avoid contaminating their number system with those pesky zero-equivalents. Ptolemy broke with this trend and started using a zero-symbol as both a placeholder and a digit, but this did not catch on.

So it was that ancient Romans weren’t able to inherit a zero from the Greeks, and the whole Roman Numerals thing happened instead. “MMCCCXVI” is partially positional (the “V” and “I” mean different things depending on their order, and “IX” applies this to the “I” and the symbol for 10, “X”). But M, C, and X were not used in a purely positional manner; instead, each represented “one” of whatever units were used. So “MM” stands for “two thousands”, and “CCC” for “three centuries”. “MMCCCXVI” is “2,316”. Romans did have a digit that represented “no remainder” after mathematical division.

Slowly, the twin concepts of zero and positional notation within numbers were built up by different societies until a Persian mathematician synthesized his own mathematics from Hindu, Greek, and Arabic sources, unifying concepts from each into a single structure of numbers. The word “Algoritmi” was the Arabic translator’s Latinization of Al-Khwarizmi’s name, and has developed into the modern word “Algorithm”. Al-Khwarizmi wrote (and taught) that “if no number appears in the place of tens in a calculation, a little circle should be used ‘to keep the rows’.: This circle was called Sifr, and it was in every practical respect the forerunner what we know of as zero today.

From these beginnings, the concept of zeros and base-ten mathematics spread to Europe by way of the Spanish Moors, and in particular, Gerbert of Aurillac, and it is from his name that the term “Arabic numerals” derives.

Mathematical calculations prior to the zero were at the level used to teach basic arithmetic to kindergarten children and other early-year students. When I was going to school, the highest form of such math was the memorization of the times tables; which used rote learning to embed concepts into applied mathematics without explanation for why numbers worked the way they did. But the fact is that every advance in arithmetic above elementary addition, multiplication, division and subtraction only works thanks to the zero and the positional notation that it makes possible.

Simple Decimals

Once you have zero and positional notation, you can have simple decimals, essentially writing a number like “2 and 3/10ths” as “2.3”, and a number like “4 tens, 3, and 57 one hundredths” as “43.57”.

Non-repeating Long Decimals

Somewhere beyond two or three decimal places, you enter the realm of “Long Decimals”. These are numbers that include fractions whose decimal conversion can be fully shown, no matter how long and complicated. “Ten thousand Seven Hundred and Forty Two one millionth 48 thousandths and 576ths” can be written “10742 / 1048576” as a fraction, or 0.0102443695068359375. For convenience, long fractions sometimes use a space after every third decimal point, just as “1048576” is sometimes written “1,048,576” – so “0.0102443695068359375” becomes “0.010 244 369 506 835 937 5” – but this decimal representation is the exact number represented by that particular fraction.

Simple Repeating Decimals

Long before you’ve worked these out, however, you have discovered simple repeating decimals. “1/3” is the simplest of these – it’s 0.333333333333… and the decimals continue on indefinitely.

One quarter and one fifth don’t have these properties, but one-sixth does – 0.166666666666666… and so does one ninth, or 0.11111111111111… and, in fact, every fraction whose denominator is evenly divisible by three. So one seventy-second is “0.01388888888888…”.

These are frequently denoted by putting a dot on top of the decimal place that is repeated – so:

Complex Repeating Decimals

One seventh is even messier, as are any fractions whose denominators are evenly divisible by 7. One 63rd, for example, is “0.015873 015873 015873 015873 015873…”, in which a string of 6 decimals is repeated an infinite number of times.

These are usually written with a dot over the first and last decimal in the repeating string, so

These clearly represent a whole new order of complexity when it comes to decimals, but we’re still not at the complexities represented by the digits of pi.

Non-Repeating Decimals as Fractionated Series

And that brings me back to the answer on Quora by David in response to the question, Can π be expressed by a series?.

In response, Peter offered up the following simple series:

But I think it becomes even more obvious when written,

I had encountered a few of these before, but they were more complicated. For example, there’s this one:

The primary source referred to by David, the Pi Formulas page of Wolfram Mathworld, as mentioned earlier, has a great many more. The series listed above is almost as elegant as David’s (only the addition-subtraction perpetual series prevents it from equalling that mark). There are others that are a lot more complicated.

These define a number not in terms of its actual value, but in terms of a process that can be used to calculate it. The problem is that to extend the number of digits of pi, you have to calculate every term up to the depth of your required decimal places, and the number of terms to be calculated grows faster than the decimal places do.

For example, in the formula above, it’s a sure bet that eventually, you will get to 1/81 – that will be somewhere around the 40th term. But 1/81th is 0.012345679 012345679 0123456790… – so that’s 40-or-so terms and we’re still only on the second decimal place!

There are some formulas that converge more quickly on pi; for example, this one…

…but by increasing the complexity of the terms of the series and using factorials, an even better method is possible:

Factorials, for those who don’t know (or don’t remember) are a series of numbers that are multiplied by each other:

So:

• 3! (described as “Factorial three” or “The Factorial of three”)= 3 × 2 × 1 = 6,
• 5! = 5 × 4 × 3 × 2 × 1 = 120, and
• 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800.

Rolling Non-Repeating Decimal Functions

There used to be a monthly magazine called Science Digest, which I quite enjoyed reading.

In the January 1990 issue, it reported on a mathematical breakthrough by two brothers, Gregory and David Chudnovsky, who extended the calculation of pi to over a billion decimal places using a new algorithm that they had developed for the purpose.

It was the sheer brilliance of how this algorithm worked that really caught my attention, even more than the feat itself. In essence, if you fed it 14 digits of pi, it would spit out the next 14 digits of pi. The formula itself is a fairly ugly thing, but it works.

Their work (and that of several subsequent researchers) was actually built upon the brilliance of an Indian mathematician, Srinivasa Ramanujan, who developed a number of innovative formulas for the calculation of pi in 1914.

To me, Ramanujan’s technique is more elegant:

…but there is no arguing with results. It’s entirely likely, however, that without Ramanujan’s formulations, the Chudnovsky brothers would not have been able to make their own breakthrough.

In fact, for technical reasons, the approach used by the Chudnovsky brothers is used for all record attempts these days, and the current record (set on my Birthday this year by Emma Karuka Iwao of Japan, and announced after verification just two months ago) extends the record to an astonishing 100 trillion digits (10^14, or 100,000,000,000,000)

Digit Extraction Algorithms

Astonishingly, this is not the last word on the subject! In 1997, David H Bailey, Peter Borwein, and Simon Plouffe published a paper describing a new formula for π, now known as the BBP formula.

This was capable of extracting any given digit of pi without calculating the preceding digits – in base-16.

You heard me. Base-16, better known as hexadecimal.

Hexadecimal uses A, B, C, D, E, and F to signify the decimal numbers 10, 11, 12, 13, 14, and 15, respectively.

I chose the hexadecimal code pretty much at random, so I was astonished to discover I had selected a named color!

If you were to count to 36 in hexadecimal, it would be “1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22, 23.” Because hexadecimal was often used in computer hardware programming, it is traditional to pad the leading values with “Ø,” signifying zero (and distinguishing it from “O” which could cause all sorts of problems in computer programs if incorrectly substituted for a zero). Each character in hexadecimal occupies one byte in a computer’s memory or disk space.

The range 00-FF in hex is particularly significant, because of the RGB color schema, in which each component in a color is specified by just such a two-byte character. “FF0000” thus specifies Red, “FFFFFF” is white, “000000” is black, and “C4D14A” is named “Americium” but is actually a medium-light yellowish-green in color: Most software these days would use the decimal number from the user’s point-of-view (196, 209, 74) – but rest assured that the number is stored in hex in the computer!

So…. hexadecimal.

This is an example of what is now referred to as a Digit-Extraction Algorithm. Mathworld defines these as an algorithm or expression that “allows digits of a given number to be calculated without requiring the computation of earlier digits.” and adds, “The BBP formula for pi is the best-known such algorithm, but an algorithm also exists for e.”

In 1996, Plouffe derived an algorithm to extract the nth digit of π using base-10 math to derive base-10 digits. It can even be used with a pocket calculator!

The problem is that this calculation is quite slow, in fact several of the earlier calculations offered are faster, notably that devised by the Chudnovsky brothers. Yet, the fact that one base-10 formula has been found, however inefficient, implies that there will be more to be found, so the question of whether or not one can approach the BBP formula in speed remains open..

The Golden Ratio

I was also intrigued to notice, amongst the many formulas listed on the relevant Wolfram Mathworld page, a couple of formulas that referenced the Golden Ratio. This is yet another irrational number, symbolized by the Greek letter phi (φ) and defined as the ratio for which this expression:

…is true.

Which sounds really esoteric, an intellectual exercise. Here’s another way to look at it, provided by Wikipedia:

This rectangle has one side of length a and another or length a+b. If you cut the long side to create a square of size a × a, you are left with a rectangle of size a × b, with a now the long side, and which has the same exact proportions as the original rectangle. If you calculate the ratios for which the above is true, you get a value of approximately 1.618.

Again, this seems like an interesting bit of trivia, but nothing important.

But the golden ratio keeps showing up in all sorts of unexpected places. Some of them are man-made, and represent ideals of aesthetics that might be self-fulfillment of standards.

• For example, the most popular size of postcards (and postage stamps for that matter) are in the Golden ratio.
• If you calculate the ratio of entries in a Fibonacci sequence* – the next term in a sequence is the sum of the two preceding numbers – the average ratio will be the Golden Ratio.
• Sunflower florets form natural spiral patterns which are said to contain Fibonacci sequences, and which therefore involve the Golden ratio.
• Ditto the arrangement of leaves on a plant stem.

I previously wrote about Fibonacci Sequences in The Meta-Physics of Magic (I thought I had looked at the subject even more extensively, because it’s very useful for RPG design, and usually overlooked, but evidently not – so that’s something I’ll have to do at some future point).

There are others, some confirmed, some disputed.

The last place that I expected one to show up, though, was in a formula to calculate the value of π!

References

Before I get into the concept that I think I intended to broach, I thought that I should list the references that I used in compiling the above information. In no particular order:

• Peter’s answer to “Can pi be expressed by a series” [Quora]
• π Formulas [Wolfram Mathworld]
• Pi [Wikimedia]
• Approximations of π [Wikipedia]
• Chronology of computation of π [Wikipedia]
• 0 (Zero) History [Wikipedia]
• Integer [Wikipedia]
• Infinity Mathematics [Wikipedia]
• Irrational Number [Wikipedia]
• e (mathematical constant) [Wikipedia]
• Circle, Analytic Results [Wikipedia]
• Series that converge to π quickly [Mathematics Stack Exchange]
• #c4d14a- Turmeric Hex Color Code [hexcolor.co]
• Factorial [Wikipedia]
• Digit Extraction Algorithm [Wolfram Mathworld]
• Bailey-Borwein-Plouffe formula [Wikipedia]
• Golden Ratio [Wikipedia]

I think that’s all of them!

Games

In some respects, the increasing complexity of decimals is synonymous with the increasing complexity of RPG plotlines. Well, it’s at least a metaphor, one that’s worth exploring.

The simplest possible plot is something like “PCs see bad guy. Bad guy sees PCs. Bad Guy attacks. Someone wins.” – or, “PCs are hired to deliver a package. PCs deliver package. PCs get paid.”

This is akin to having no decimal places at all, within this analogy.

As soon as you introduce a decimal place, you are introducing a complication. “PCs are hired to deliver a package. Someone attempts to steal the package.” Suddenly, there are two paths for the adventure to take – either the PCs win, and get to deliver the package, or the thieves make off with it and the PCs have to get it back, then deliver it.

A longer decimal is akin to a complication being a gateway to a longer chain of events. “PCs are hired to deliver a fabulous gem. Someone attempts to steal it, but is beaten off. PC discovers that the gem is a fake – is it possible that the real gem was stolen during the earlier attempt, which may have been just a distraction, or was it always a fake? Is their whole mission to be a stalking horse, a lightning rod for trouble while the real gem is smuggled in by some more secret route? Or are they part of a plot to replace the real gem with this fake?

Perhaps there are multiple groups involved, with different intentions and agendas, so that more than one of these speculations is true. Or perhaps the GM decides that whichever plot the PCs choose to investigate third is true. This is akin to a longer repeating decimal string, except that a cap has been placed on the number of times the string will repeat – call it a rounding error! And the ‘true plot’ is positionally significant.

Superficially, several investigative sub-plots like the ones implied by these “theories of the crime” might be similar, but the clever GM will take active measures to differentiate between them. Different tones, different moods, different oppositions with different rules of engagement, settings that are at least somewhat different, NPCs with different personalities.

When the PCs actions have repercussions into the future, such that these investigations are each the beginning of a long road, the campaign (and possibly the adventure) have become recursive, and the role of the GM has changed from that of ringmaster to that of agent provocateur. He is no longer directing the campaign, he is creating a landscape for the players to explore, or not, as they choose.

And, of course, in the long term, the campaign therefore becomes – or should become – more like an irrational number, a series of decimals that never repeats (though at times it might seem to – a decimal string “141592” can occur thousands of times within the length of π for example! The fact that one of those occurrences happens to be at the very beginning of the decimal series of digits is completely irrelevant.

It might seem at first that sandboxing is more akin to the notion of Digit Extraction, in which a given digit is extracted only when it is needed, but I would argue that it more closely resembles the Chudnovsky approach, because the content is inevitably derived from, and dependent on, the “terrain” that has already been explored by the players.

Having at least constructed a basic outline of everything, with embedded plot hooks and (metaphoric) landmines waiting for the PCs to step on them, which can be expanded upon at need, is far more accurately described by the digit extraction analogy; the digits of π don’t change, if you extract the same digit by several different methods they will all give the same answer. You may not know what that digit is when you start, but it’s not like Schrodinger’s Cat, it doesn’t exist in some quasi-metastable state until actually determined.

A Mnemonic Device

Aside from being at least somewhat interesting in its own right, that means that an understanding of decimals makes them a mnemonic device for reminding the GM how to construct plots.

You start with the simplicity of “The PCs are hired to deliver a package. The PCs deliver the package. The PCs get paid,” and build complications and permutations and choices – and yes, a little randomness and chaos – upon that foundation. Where you stop is up to you; this could be the introduction to an entire campaign or it could be the teaser before the title sequence, with the main movie (which may or may not be related in some way to the teaser) still to follow.

Okay, so this won’t get posted on time – as I write this, I still have a lot of formulas and equations to edit and upload, and the text has to be spellchecked and edited, and all those references converted to hyperlinks.

If the graphics were done already, it might just have been possible; without that, it’s not.

So this is being made public a day late. Sorry, everyone; I’ll try to do better next time!

It happens to everyone eventually – you look at your plot and realize that one of your PCs is going to have to interact with an NPC in a one-off scene, an NPC with whom they might never come into contact again.

There are many different ways of handling this. Some GMs will use a random generator to create a personality and let it go at that. Others will look at the momentum of the scene and decide how easy the interaction is going to be. Still others will focus on making the interaction unforgettable or challenging, even if that means that the personality makes absolutely no sense in terms of the NPC’s in-game role. A fourth group will take the functionalist approach, defining the personality as something appropriate to the in-game position of the NPC, even if that makes the personality a little (or a lot) cookie-cutter predictable.

Today, the plan is to show you a better way. I’m going to use a modern setting for the purposes of this discussion, but the technique works for any genre of game.

At the heart of the technique is an eight-step process, so let’s spell that out first, and then look at the elements in detail, and how to enhance and refine the results.

The goal is to make even your throwaway characters sufficiently interesting and well-rounded that they are distinct individuals who are capable of sustaining ongoing appearances within the campaign – because you never know what PCs will decide to do, and their choices can transform the status of that NPC from one-off to recurring guest-star.

The last thing you want, therefore, is for any NPC to be boring cipher.

The eight-step process

1. Think about the previous character interaction that will have been experienced by this PC.
2. Do the circumstances of this encounter forecast or mandate a particular interaction between the PC and NPC? If yes, proceed to step 5.
3. Think about the preceding in-game PC-NPC interaction within the planned narrative, even if it was not an interaction with either this PC or this NPC.
4. Choose an interaction mode that will contrast with both the interactions identified in steps 1 and 3.
5. Given the in-game situation in which the interaction will take place, and the known personality profile of the PC, choose a basic personality for the NPC that will yield the desired interaction mode.
6. Given the in-game occupation of the NPC, and the minimum level of competence indicated by the interaction mode, select personality traits that are compatible with the result of step 5 and that justify/reflect that minimum level of competence.
7. In any aspects of the character not already defined, select one or two traits that are going to be memorable, expressive, and distinctive, even if they act to limit the future advancement of the character within their career.
8. Generate any other required parameters or traits accordingly to create a cohesive character.

Analyzing the process

Okay, let’s break this process down.

1. Previous interaction experienced by this PC

There are two things that you ideally want this encounter to contrast with, and the first is the last part of the story that this particular PC was involved in.

What you don’t want is for one particular player to be able to complain that every time he encounters an NPC, they are unhelpful, but other players do not encounter the same resistance. Or for one of the other players to complain that the NPCs always cooperate with the character belonging to Player X. You don’t want those opinions to manifest even if they are never spoken out loud!

But even beyond that sort of reaction, justified or not, ensuring such a contrast helps simulate the ups and downs of life, and makes the game feel more ‘real’ to the players in a broader sense, so it makes good sense, anyway.

2. Mandated interaction mode short-cutting the process

It might seem that this exclusion question would make more sense as item number one in the process, but there’s good reason not to do that; even though the interaction mode is being pre-specified by the nature of the encounter, having that previous interaction from step one in the back of your mind permits it to influence and nuance the dictated interaction.

For example, the NPCs employer might be allied with or otherwise supporting the PCs enemies, or simply be hostile toward the PCs for some reason; as a result, he has given instructions that his employees are not to cooperate. Or it might be basic corporate policy for this employer.

So the encounter is one in which the PC is to be frustrated and not to obtain whatever he or she is looking for – goods, information, cooperation, money, whatever.

That does not preclude the NPC having a different opinion to that of his boss, it simply limits what he can do about it. I had a similar situation arise a while back in the Zenith-3 campaign; during the encounter, the NPC was extremely regretful, but had to refuse the PC’s request even though he was not legally permitted to do so. The NPC then met the PC “by complete chance” shortly thereafter and ‘accidentally’ left the information that the PC wanted, and which was supposed to be publicly available, ‘lying around’ afterwards.

The encounter previous to this for the PC involved was one where the same ‘boss’ (a corrupt politician) had called in favors which resulted in a flat refusal to cooperate with another reasonable request from the PC. Clearly, there was a significant difference between the encounter described above and an outright refusal – but, nevertheless, it was a refusal to cooperate, as mandated by the situation.

Nuance can make all the difference in the world.

3. Preceding PC-NPC interaction within the narrative flow

The other thing that you want to contrast with is whatever was happening just before this encounter takes place. Except, of course, when you deliberately don’t want to contrast the two, but that tends to be an exceptional circumstance.

The motivation for ensuring a contrast here can be summed up, “Dice have no memory, but players do”.

Most of the time, the participants in one scene will have no knowledge of what happened in the scene immediately prior to this one, and the encounter should start from a neutral position (influenced by the interaction intended to occur, of course).

But, at the same time, you have to tell a cohesive story, woven around the PCs and their interactions with the game world, and the players know what has happened even if their characters don’t. It’s the players who are both audience and stars of the show, and their characters who are their roles in that show, and that can never be forgotten.

Contrast between scenes helps keep that story narrative alive and fresh and interesting, with ups and downs and highs and lows.

4. Contrasting interaction mode

My example earlier in the analysis of the process has probably taken most of the air out of this step of the process. Contrasting with one interaction is easy; contrasting with two separate interactions is a little harder, but still leaves innumerable possibilities for you to choose from.

But there’s one additional requirement to be met, and there are likely to be relatively few of those innumerable possibilities that survive that consideration: your choice has to advance the plot, or at least to enable the plot to advance. It has to fit into the overall story, in other words, and that can be the most constraining requirement of them all.

Life is so much easier in this respect if you have no pre-planned plotline at all, but there are such serious drawbacks to that methodology that I can’t recommend it.

5. Base personality profile derives from PC and interaction mode

Once you know how the NPC and PC are to interact, you can start to design a character that will have that particular interaction with the PC. Essentially, this involves giving the NPC a motivation for having that particular reaction toward the PC or for causing the PC to have that particular reaction to the NPC, or both. Nothing that does not contribute to this set of reactions should be considered fixed, not yet.



Image by Gerd Altmann from Pixabay

6. Justify the minimum level of competence required

It will happen regularly that a character with the personality traits that create the specific interaction you have chosen will find their career path hindered by those traits. That’s true about half the time, in my experience.

Sometimes, this hindrance will be so severe that you have to wonder how the character actually rose to the position they are now to occupy.

The nature of that position makes a significant difference to this factor; a research scientist is different to a lawyer, who is different to a hot dog vendor, who is different to a wizard, who is different to a cop. So you have to start by thinking about the actual requirements for holding that office, and sketching out the beginnings of an implied personal history that ends with the character occupying his current position – perhaps securely, perhaps precariously.

The other half of the time, the personality traits can or will make the NPC more ideally suited to their current role. Most of the time, this does not pose a problem, but occasionally they can be so suited to the role that you have to wonder why they have not been promoted out of it; in such cases, you need to weaken the character’s suitability in other ways, just enough to justify the situation as the character finds it.

There are other solutions that can be employed occasionally – an employee recently fired or suddenly retired, or even suddenly promoted, forcing their supervisor/boss to act in both roles until they find a replacement, for example.

In almost all cases, this will sketch in additional character personality traits. Once again, though, don’t incorporate anything that doesn’t directly contribute to this requirement – not yet, anyway.

7. Make the character uniquely memorable, expressive, and distinctive

The next step is to add in a quirk or distinctive personality trait or two that will make the NPC stand out and be memorable. Care must be taken to ensure that these traits do not upset the careful balance that you achieved in the previous step – you may need to strengthen or weaken some of the traits that you added, or even replace them entirely with the quirk. There are too many combinations of traits and quirks and occupational roles to try to get more specific than that.

As an example, however, at one point a PC needed to consult with a representative of the New Orleans Historical Society at their offices – they were trying to track down some extremely specific and obscure information, the details of which don’t matter. Following some of the additional advice that will follow later in this article, I decided that it would be interesting and memorable to have the individual occupying a position as a historian (unqualified) to be fascinated by a particular vision of the future. So I quite deliberately made them a Trekkie.

This meant that the NPC could be quite knowledgeable and helpful, but also totally memorable. Especially once I threw in some blonde dreadlocks, peace symbols and other badges demonstrating activism and idealism to go along with that primary quirk.

8. Complete a cohesive character

By now, you probably have a fairly good idea of the personality of the NPC, and in most cases, that’s all that you need – see Creating Partial NPCs To Speed Game Prep.

Well, almost all. If you don’t have one in mind already, it’s time for one of the most critical decisions of the character construction – the character name. You may also need names for their employer, and/or for their boss.

I’ve written a LOT of articles on the subject of choosing a good name, why it’s so important, and how to go about it; you can find most of them listed in the Blogdex on this page.

Refinements

There are a number of further hints that can be applied to enhance and refine the process and its results. In fact, some can be relevant general advice even if you don’t employ the process described. I’ve already hinted at one of them – the second one on my list.

Interaction Mode flow, not stark contrast

Complete reversals of fortune are less common and strain credibility more than gradual morphing from one extreme to the other, with the occasional mountain or valley along the way.

But there are two flows possible, and you only need to accommodate one to tick this box – the overall flow of the plot is probably the easiest and potentially the most useful, but the flow of the plot from the perception of the one character helps to maintain a consistent narrative flow through the narrative thread, and is probably better in the long term.

There is, however, no need to be consistent in this respect. You can switch from one continuity flow to the other at the drop of a hat; this is a tool in service of the plot, and not a chain to bind you.

If there can be one rule of thumb in this respect, it’s that early in an adventure, it’s better to make the plot threads flow, and later in an adventure, it’s more useful to focus on the overall flow of the adventure rather than on the individual plot threads experienced by any particular character. But the differences aren’t absolute, and this guideline can be ignored at the drop of a hat if you find it warranted.

Occasionally, play against type

Playing against type happens so frequently that it has become something of a cliche, almost as ubiquitous as the cardboard cut-out. There was a time when this wasn’t the case, and playing against type was great characterization advice, but those days have passed.

Unless you get clever about it, of course. A character who is striving to overcome an innate lack of ability for some very good reason, where that reason is supported by the character traits that make them tick, is perfectly acceptable – if not overused.

A character whose personal focus is the diametric opposite of their professional focus in some respect – like the historian whose personal philosophy and ethos and ideals are grounded in a vision of the future – is perfectly acceptable – if not overused.

I think you can see the trend…

Subvert every cliche that you don’t embrace

I’ve given this advice before, I think. But it still remains excellent. There will be times when you want or need to embrace a cliche, in which case, go all the way with it, totally over the top – and then put a layer of characterization beneath the surface that doesn’t quite fit the cliche. This only works well if the specifics of the encounter give an opportunity for that subsurface layer to find expression in actual play or dialogue, however. If you can’t do that, then you can’t undercut the cliche and give the character depth.

If you can’t embrace, subvert. That goes beyond making the character the exact opposite of the cliche indication, it demands character traits that make the cliche absolutely impossible in this particular case – or that redirect it to support those it would normally victimize – the Tax Collector who supports the poor and vulnerable, the sleazy lawyer who hates corruption in public office and is willing to do whatever it takes, the rural cop who passionately supports minorities, the research scientist who can’t balance his checkbook because he gets distracted writing complex formulas on the stubs…

Organization traits, contrasts, and confluences

There’s a lot to be said for treating organizations as characters with their own personality traits, ambitions, lines they will not cross, quirks, and so on. These don’t mandate the personality of those who work for or serve the organization, only the ways they require the employee to behave. The character’s personality determines how they feel about and react to those mandates.

Characters whose personality traits permit them to act as required without qualms or conscience problems are likely to get retained, trained, and promoted. Characters who see ways to take advantage of the required actions to their personal benefit will often be capable of suppressing any such qualms or conflicts, and are also likely to do well – at least for a while. Characters who encounter difficulty in following the rules and policies laid down will not last and are unlikely to be promoted beyond the bottom rungs of the ladder. But sometimes these characters can suppress their qualms by doing ‘good things’ in their off-hours, rationalizing their ‘professional’ activities as the means of doing those ‘good things’.

For any given profile of an organization and a give character profile, there is an ongoing interaction – a relationship – between the two, just as there would be a relationship between two characters, one that can be as conflicted and complex as any other.

Use these facts to your advantage – when creating an employee and an organization, start with the relationship between the two; this will be directly related to the interaction mode between a PC and the Employee. So start with the defined facts and find the combination of traits that supports that, and the organization will define itself.

Fixed Points in a maelstrom

You can extend this principle to work backwards from known elements within the adventure to define unknowns, either before that point or subsequent to it. Using the principle of interaction mode flows, you can work both backwards and forwards to define the personalities of virtually every encounter.

This confers a sub-current of inevitability to the adventure in which each encounter feels like it perfectly belongs there, creating an internal logic and storytelling momentum that, even though it may not intrude upon conscious awareness, is nevertheless felt by the participants in the story.

Let it Flow

Humans are fairly good at perceiving trends. The momentum and internal logic that I just described are the results of an awareness of trends, flows, and sub-currents within the events experienced by their characters.

But that’s a side-benefit. The real payoff for this approach to character design and placement is that it is faster, easier, and provides greater internal consistency, which in turn creates greater verisimilitude – especially when coupled with the Partial Characters concept, which translates the principles of sandboxing to character construction.

So, let it flow!

Whew! It feels good to be back on my regular schedule! I expected the “post little posts as quickly as possible” approach to be liberating, but I found that it came with a lot of pressure to post something regularly – and while the approach made room for the medical testing that I needed, it simply wasn’t possible to post some parts as quickly as I wanted to, producing pressure to perform. So, while it got me through a difficult period, it is not an experiment that I will be rushing to repeat anytime soon…

The story so far…

This is the last in a set of mini-posts that I have written and published as quickly as possible (given a number of health-related interruptions), something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each of the posts so far has examined one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality (if you’re looking for a specific article in the series, there’s a list at the end of this article).

The goal has been to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination. Along the way, there have been a number of tips and tricks to enhance the productivity of specific image searches.

This final part of the series will look at more than a dozen example images from my different campaigns, sharing a few war stories along the way, and giving me a last chance to offer some final hints and tips. In a few cases, concerns about copyright restrictions, or about associating real people with in-game identities of which those real people might not approve, have led me to drop in a couple of ringers of more impeccable credentials.

1. Dr Who Materialization

I always open a Dr Who adventure with the materialization scene, in which the TARDIS appears from nowhere. These are “unique” images that are hand-crafted from a location image; I’ve previously isolated a TARDIS image and partially faded it’s lower half, giving the impression that it’s fading in – in fact, I have three different ones showing slightly different angles and fades.

These composite images are the very literal essence of establishing shots, giving a graphic depiction of the locale in which the adventure is to take part. Sometimes, it will show something critical to the adventure, but more often, the location will be used to set a tone for the location, permitting more important location images to be placed within a context and interpreted correctly.

In direct terms, the game value of these illustrations is minimal, but with these added benefits, considerable effort is easily justified. And, on top of that, they serve as punctuation, signaling that the adventure is moving forward.

In order to preserve this function, I am careful to avoid depicting materialization scenes mid-adventure – I’ll show the Tardis already in place, instead.

I discovered the power of the materialization scene by accident in the course of adventure two of the current Dr Who campaign. So far, materialization has been shown on the planets Oa and Escar, an alien monastery, a cargo hold aboard the colony ship Carthage, the dry stores compartment of the nuclear submarine USS Ardent, the TARDIS docking bay on Gallifrey, An alien wasteland on Cornova-III, and the hanger of a Gas Giant mining operation.

2. Towns in Zenith-3

In my superhero campaign, the PCs are on a journey of exploration through Arkansas, looking for a suitable location for a base of operations. There are hundreds of these, which poses a different challenge – trying to distinguish one from another..

Overall, I employ a structured approach to representing these towns. I start by writing the history and demographics of the place, as delivered by the guidebooks being used as primary reference material by the PCs. That gives me a sense of the character of the location.

Next, I search for key images to convey that character – my first preference is to use actual images from the location, my second preference is to use a high-quality similar image, my third preference is a screen capture from Google streetview.

There is a focus on the economy of the location (banks and shops); then a focus on the culture (parks, churches, government), then anything unusual or distinctive.

Where a town has something that distinguishes it, I’ll often front-load that to appear in advance of anything else.

Next, I need to convey a sense of the housing commonly available in the location. Real Estate websites and Google Streetview are my primary sources for these images.

After that come any images of potential bases, described in-game as Contenders. There are several sources for these – to start with, I have a list that I generated before starting; if any image found for the town matches one of those, it might well find itself situated within the town in question. Next, there are any images of the town that show buildings that are obviously suitable. And sometimes there are notable buildings identified in the research back at the start of the process.

Increasingly, as planned, these travels are being used as a delivery mechanism for interesting encounters and mini-adventures – for example, two of the PCs are currently attempting to rescue some kayakers from some giant (sentient) spiders. These collectively are telling a broader story about the game setting, a foundation that will be used for later adventures. These need to be illustrated, as well; and all these illustrations then have to be integrated into a single cohesive and coherent narrative.

Occasionally adding to all this are maps, which are employed only sparingly, but sometimes nothing less will suffice.

To represent these, I have chosen a trio of images.

Oh, all right – one more, just because I’m very pleased with it! This image, which brings a whole new meaning to the term ‘ghost town’, has not yet appeared in-game. It has been built around a screen capture from Google Street View, as you can tell by the map inset in the lower left corner. Each of the buildings was individually sourced and composited, and then I used the same sort of ‘fade effect’ that was used with the TARDIS Materialization image (shown earlier) to create the Ghost effect. (ADDENDUM: I wasn’t sure that readers would get the full effect at the reduced size, so you can now click on the image to get the full-sized image in another tab).

3. Locations in Pulp

A Pulp setting brings a different set of problems and opportunities. These adventures are set in a time when there were cameras, so for every image search, you have to choose between searching for a modern image and searching for one appropriate to the era of the setting.

There are so many considerations that go into that decision that it’s almost instinctive. How likely is it that the location has changed since the 1930s? In many cases, the answer will be, massively; in other cases, the answer is “not at all’. There is also an element of practicality, of ‘this is what we can find’.

A third consideration is how picturesque any period images might be. Part of the remit of such campaigns is to bring the era to life as a game setting, and “color” from back then helps achieve that.

For example, from the current adventure, we have the San Juan police having a parade…

….and then there is this image of Rio:

But here, for contrast, is a modern image of the Carpathian Mountains of Ukraine from our previous adventure (which was written before the current invasion started):

And, of course, sometimes there’s nothing for it but to create your own. In the part of this series that covered locations, I displayed the image of the Valley that I manufactured for the current adventure, for example, and for the next adventure, here’s the airfield hanger from Twin Bridges, Montana – winterized.

4. NPCs In Pulp 1: specific NPCs

The same considerations are in play when it comes to NPCs, especially real people from the era – usually politicians, occasionally actors and businessmen, but also thrown into the mix are official portraits. But we’ve used everything from toys to propaganda posters as sources.

We’ve found that if we can get a name, we can usually get an image. It’s when no name can be turned up in our research that images become hard to find.

And yet, there are strange anomalies. There are relatively few images available of New York mayor LaGuadia, for example, so we’ve had to recycle and reuse the same two or three images multiple times.

5. NPCs In Pulp 2: important NPCs

When we can’t identify a real person, usually because it’s a character created specifically for the adventure, we have to use someone else’s image to represent them.

Our primary criteria is always to select images that show a lot of personality, first because they tend to be more interesting to look at, and secondly because we can use the personality content as inspiration. Generally, I will perform the initial selection and present a set of 4 or 5 possible choices to my co-GM, who narrows the choice down to a couple; I then make the final selection from that pair.

There are five general searches that we use to generate the short list, and we only move on to the next one on the list if the current search hasn’t produced enough results that fit whatever criteria we have applied.

Those criteria usually include image size as well as image content, and the presence of any anachronisms that can’t get painted out – sunglasses were uncommon except by prescription, and T-shirts had not yet been invented except as undershirts, for example. Digital watches are a definite no-no, as are mobile phones!

Telephone styles in general are often a problem – no pushbuttons allowed! But there are innumerable ways to trip up if you aren’t careful – we once found what we thought was the perfect “look” for one of our NPCs, only to notice at the last minute that a computer monitor was reflected in the lenses of his glasses. It was only faint, but once you saw it, you couldn’t unsee it.

Tie styles are another trap – especially when it comes to images from the 70s! Too wide or too colorful or with anything other than a simple pattern – they are all incorrect for the time period.

The five criteria are Emotion/Style, Occupation, Descriptive Terms, Descriptive Synonyms, and Antonyms.

Emotion/Style

The first choice of search is always to try and match the overall emotion or style that we want the character to possess. “Angry Man”, “Suave Man”, “Determined Woman”, etc. It is also common to add “1930s” as an additional search term.

Occupation

It’s also normal for us to replace “Man” or “Woman” with an occupation. “Angry Lawyer” or “Angry Lawyer 1930s” are far more likely to give useful results.

Descriptive terms

If that finds too many images, or doesn’t find enough, we add some descriptive terms to the search. “Blonde” or “Tall” or “Scarred”, for example. Due to the way search engines work, this works in either situation.

Descriptive Synonyms

But, sometimes, even this fails to find a suitable image, or enough suitable images for a full short list. When these searches fail, it’s time to replace the occupation with another descriptive term; we will often have to try several synonyms because this search is less likely to produce satisfactory results.

Antonyms

Being forced to compromise is sometimes not enough. Our last resort is usually to replace some of the search terms with antonyms, because a character who “doesn’t look the type” can often be used in place of a character who does.

I decided not to use any of the archived images I have stored away as examples because I couldn’t be sure which ones were in the public domain and which were not. Instead, I went hunting and found the image below.

6. NPCs In Pulp 3: generic NPCs

I’ll let you in on a little secret: there are no generic NPCs per se in our pulp campaign. We treat every NPC as though they were Important; the only difference is that these NPCs tend to have fewer preconceptions. That way, if we – or the players – decide to elevate the character in importance, the image is good enough to support that role within the campaign – though sometimes we won’t name a generic NPC.

An illustrative example took place in Adventure #30 in the campaign, “The Locked Door”. As usual, before the main plot started, we embroiled each of the PCs in a mini-plot that told them where they were and what they were doing when the main plot begins. One of those mini-plots was a restaurant sequence in which a couple of over-excited children were going to interact with a generic mob boss.

After discussing our options, we decided that we wanted to make it clear from the visuals alone that this was a mob boss, and that we would simply describe him as a ‘businessman’. That meant that we needed a really iconic representation; we soon decided that none of the photographic alternatives that we found were quite generically iconic enough, though they might have been fine for a specific crime-boss; that meant using character artwork.

I have the impression that we found the image that we ended up using for ‘Crime Boss At The Restaurant’ on DeviantArt, but a reverse image search doesn’t show it; instead, the image is all over Pinterest in multiple categories.

While we didn’t name this particular character at the time, we should have done, as it became necessary for some of the restaurant staff to address him by his surname and the mobster’s moll, by his christian name. I think we invented a name on the spot – “Reggie Romano” or something along those lines – but we should have anticipated the need.

7. NPCs In Pulp 4: Undefined NPCs

The less you know about a character, the more inspiration you can draw from a good image – and the more important it is for that image to contain inspiration for you to draw upon.

Perhaps a more typical example comes from Adventure #27, “The Fate Of The Golden King”. We needed a super for a flophouse and after tossing the question of what tone he should project around for a while, settled on “creepy” from memory. Or maybe it was “old man”. In any event, we somehow found an image of Australian author Patrick White, taken in Kings Cross in 1980 by William Yang. I’d love to show it to you, but it’s clearly copyrighted, even though it has appeared on many sites quoting White. You can look at it by clicking on this link.

What we really wanted was someone world-weary, who was tired of fighting for his prosperity every day, bowed down by the burdens casually visited upon him by the transient ‘tenants’ of the flophouse, and who was skirting the edges of sanity without actually crossing that line. The image found doesn’t quote capture all of that, but it comes close enough and looks vaguely unsettling when shorn of its literary context.

8. Priorities In Fantasy

Let’s be honest and clear – there are probably less than 1/10th as broad a subject matter available when it comes to Fantasy as there is for a more modern game setting such as something in the Pulp genre. The consequence is that most of the time, Fantasy images will be much more work to find and the game value of such images will need to be considerably higher in order to justify that effort.

That doesn’t mean that it’s not worth the effort, just that you need to be a little more selective at times. There are plenty of landscapes to use out there, both of exotic locations in our world and the work of a great many talented digital artists. Using the Pulp techniques described, you will often find something suitable for most of the important NPCs and many of the common ones.

The chances of success when searching for the latter are generally enhanced if you add a search term describing what the NPC should be doing in the image – “medieval money counting”, for example. Or “blacksmith forging horseshoes”.

You will often struggle to find illustrations of the more exotic creatures from the many sources both official and unofficial, but casting your net a little wider and being prepared to adapt the creature description and stat block to what you find from an appropriate image source can both broaden the encounters in your game and stir your creativity; that said, though, since 3.0, the artwork in the various official sourcebooks has been excellent and quite suitable as an illustration.

The more outlandish a vehicle or object, the greater the struggle to find a good image, but objects can be surprisingly tricky at times as well. Good photographs of wooden barrels, for example, were hard to find the last time I looked – I actually needed them to insert into a scene for Pulp Ultimately, I ended up making our own image from multiple parts of the one source and adding a barrel-maker to conceal some of the imperfections and prevent the image from being totally static. Again, I don’t think it’s an image that I can share.

To illustrate this section, I thought that I’d offer up a pair of images. The first is of a Bavarian Castle, and the second, a Fantasy Knight in an enchanted Forest..

9. Sources Of Sci-Fi

There are some subjects that have very few illustrations in Fantasy but are relatively well-represented in Sci-Fi, and vice versa.

There are hundreds of aliens to pick from, for example, so you have no excuse for not choosing one that exactly matches what you want to convey. There are almost as many spacecraft to choose from. But try searching for a cryogenic chamber! Or a steel coffin…

There are four terms that can be added to searches to increase the chances of finding what you need. The first is the fairly obvious, “SciFi”. The second is a variation on that, “Sci-Fi” – and it will often find results that the first search doesn’t. Third, “Futuristic”, and finally, “Concept Art”.

If none of those work, then it’s time to go without the additional terms and look for a contemporary representation that can be edited.

When it comes to critters and creatures, additional search terms that can be useful are “Fantasy” and “Horror” – plus those listed earlier.

For interiors, it’s not uncommon to have to rename / repurpose depictions of one type of room to another, perhaps adding some window dressing. But it’s worth searching thoroughly because sometimes there can be the perfect image lurking in the results, even if initial attempts are fruitless.

It’s fair to say that greater patience is needed for sci-fi than for anything more contemporary like Pulp. Whether or not it’s worse than fantasy depends on what you are searching for!

“Gadgets” is notable, by the way, as a particularly difficult search term. Try “Machine” or “Device” instead!

10. Sources Of Superheroics

When it comes to superhero campaigns, 25% of the images you need will be modern photographs, 20% can derive from Fantasy sources, 30% from Sci-fi sources, and the last 25% are the most problematic, because they are genre-specific.

In general, that means that the first 25% are fairly easy to find, the next 50% are a little more work, and the last 25% are the most difficult.

When it comes to depicting superheros and villains, you have two choices: use what you can find (possibly editing the colors) or use a service like the Hero Machine. A third choice available only to relatively expert digital artists is to use nude photographs and convert the skin tones into costumes. Be prepared to spend 10-20 hours on each such image, so I reserve it for only the most essential images.

The Power Of Images

An image, it is said, is worth a thousand words. In the case of the right image, I would tend to agree. But few groups will have the patience to listen to the GM for 1000 words of description and narrative; RPGs are supposed to be interactive.

But that’s only the start of assessing the power of images. The mere fact that you are slicing chunks of narrative out of your delivery and replacing them with an image that can be absorbed and appreciated in a fraction of the time means that images often accelerate the process of play.

And, on top of that, images can conjure emotional reactions, something that prose can sometimes struggle to achieve when delivered orally (which is not the same as reading it on a page).

And, on top of that, there’s the benefit of getting everyone on the same page, as noted in earlier parts of this series.

That’s a powerful weapon to have in your arsenal – but that’s only if the image is right. Those advantages can quickly drain away if you are forced to compromise because you can’t find that “right” image – which may in fact, not even exist.

If you had unlimited time at your disposal, it would be easy to achieve the maximum possible benefits. The reality is that this is a luxury that is a rare event, and that demands that you maximize your efficiency in searching for illustrations, spending your time where you get the biggest return on your investment of time.

Achieving that requires understanding the value to your game of each image relative to the degree of effort required to achieve it, hopefully restricting your compromises and corner-cutting to those illustrations where it doesn’t really matter.

This series has aimed to give the reader that understanding, and a bunch of tips along the way to enhance your prospects of success. Hopefully, it has achieved that purpose.

This series has taken more out of me than I expected, especially with interruptions and delays caused by medical issues. I’m not sure that I’ll have a post ready to go for the usual publishing schedule, though I’ll try.

So this was all ready to post – and then my internet connection went out. Thankfully that was only a problem for a day or two. But it has delayed the series slightly.

The story so far…

This is the sixth in a set of mini-posts that I’m writing and publishing as quickly as possible, something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each post examines one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

So far, the series has looked at Objects, People, Monsters (and other encounters), Vehicles, and Locations. In this penultimate post, I turn my attention to Events that may need depiction.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

There are all sorts of things that occur in an RPG which the GM might wish to depict for various reasons. Most of those reasons would contain no surprise for those who have read other parts of this series; a common frame of reference, for example. Others speak to the integration of a location within an environment, or bringing a dramatic presence to an event that would otherwise be unremarkable..

Most of these fall into the general category of weather effects, but there are others – avalanches, flash floods, lava flows, spell effects – that lie beyond this simple classification. “Effects & Events” is very much a catch-all label for anything that doesn’t fit the earlier categories.

In general, the contents of this category are all things that can happen to, or be encountered by, the PCs, but that don’t fit one of the earlier categories.

Mundane

The most common effects are mundane interactions that are essentially the same everywhere and every time they are encountered – campfires, for example.

There functionality of such illustrations is analogous to some that have already been considered – they act as punctuation within the game-play, for example, and the darkness can be a location in its own right. Like a vehicle, there is also an implied passage of time inherent in the depiction of such phenomena. So there are lots of good reasons for presenting a graphic depiction of such phenomena at appropriate times..

This game value is even greater in proportion to the difficulty in locating a suitable image. As a general rule, you will be spoilt for choice. This is a good thing, because it permits a different image to be used each time, useful in a repeating and regular event.

I actually suggest gathering a dozen or so examples and rotating between them, perhaps in a random sequence. Once these become familiar, you can add to them.

Common

Common images are weather phenomena. A group of adventurers will encounter weather every day, but sunny days and good weather tend to be incorporated into location images and taken for granted. It’s exceptions to this that generally need to be depicted, and those exceptions will be different on a regular basis.

Consider a rainy day, for example – the rain can be light, heavy, or monsoonal. It can and will be affected by wind, and what can be seen through the rain is always going to be different – rain in a forest is quite different to rain on a plain.

This sub-category can also include other temporal markers like dawn and sunset.

All this means that generic representations have only a limited utility, and greater specificity is required – and that means that ‘common’ images can actually be specific in their requirements, and rare and hard to find.

It can be tempting to attempt to create your own images, adding weather effects to preexisting location images. Unfortunately, that’s not as easy as it sounds; weather effects beyond fog and mist are extremely difficult to do well, at least with Krita. Other packages might offer better options.

It also has to be noted that such specificity is often a task that often far exceeds any reasonable game value (there can be exceptions), but – again – to some extent that’s a limitation of the software that I have available, and hence, subject to change.

In the meantime, it can be necessary to use generic images; these can be enhanced by structuring the narrative accordingly.

Specific

Specific images tend to refer to unusual events that are nevertheless natural phenomena. Volcanic eruptions, aroauras, comets in the sky, icebergs at sea, even tsunamis…

Paradoxically, the drama inherent in such images permits the event/effect that they feature to fully occupy the attention, permitting generic images of the phenomena to be utilized in many cases. That means that these are often easier to find than many “common” images are!.

Unique

As usual, unique images are those so specific in their required content that you either have to make them yourself, or find the image you’re going to use first and then write to it. Spell effects, rainbow bridges, and – perhaps surprisingly – futuristic gadgets – are all included. Steel cryochambers.Supernovae. Galaxies. Planets. In fact, most astronomical phenomena fall into this category.

So do most spell effects. And that includes illusions in which you can tell that you’re looking at an illusion.

Finally, things like floodwaters affecting a specific location fall into this category.

Unlike most unique images, however, I have to question the game value of most such presentations. It’s also worth pointing out that most images are static, and you may be better served (when it comes to spell effects) with a narrative that emphasizes the dynamics of the spell.

In fact, as a general rule of thumb, the more easily you can find a ‘unique’ image, the greater its game value. But there are exceptions!

The final post in this series will look at some actual experiences from my campaigns, some war stories if you will. These were significant enough that they have remained in memory.

The story so far…

This is the fifth in a set of mini-posts that I’m writing and publishing as quickly as possible, something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each post examines one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

So far, the series has looked at Objects, People, Monsters (and other encounters), and Vehicles. Which must mean that it’s now time to turn my attention to Locations.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

Locations

At it’s most elementary, a location is someplace for something to happen.

That something could be roleplaying, or a skill test, or a combat, or a narrative passage that conveys essential information to both PCs and players.

But when you start digging a little deeper, complications start to emerge. For example, let’s say that you have a map and you have an image; the first describes the tactical situation, the second gives a sense of the atmosphere and trappings, the look-and-feel of the location.

Which came first, the map or the image? If the map came first, the image is a mere representation of what the location is like, not what it actually is. If the other way around, then the map is an estimation, an approximation, of what is depicted in the image.

Either way, what do you do if there’s a discrepancy that you didn’t notice and take into account during prep? Which do you regard as canonical?

Most GMs will choose the map, in the process undermining the credibility of not just this image but all images presented to the players. Counter-intuitive though it may be, I would sooner adjust the map.

The even better answer is to tell the players in advance that the image is just to convey an impression. By taking the implied promise of accuracy off the table, you can keep the map as accurate without the resulting price-tag.

Regardless, that elementary definition tells us what we need to know, because it establishes a direct correlation between the basic commonality of the image and its game value. The more important the image to the plot, the more specific it has to be, and its Game Value is commensurate to that plot functionality.

Locations as punctuation

A location image does more than set the scene for something to happen; it signals the end of the previous sequence of events, serving as punctuation within the adventure. New locations almost always work well as break points within an adventure, a good place to end for the day or take a five-minute rest break.

Locations as time

There is also something of a psychological ‘reset’ that occurs when you present a new location. It’s as though we subconsciously associate the change in scenery with an associated passage of time, and with everything that such a passage implies.

The GM needs to explicitly connect the new scene with the one just past if there is not such a passage of time, or the players will experience a discontinuity between what their head and their instincts are telling them which is distracting and undermines the verisimilitude in both areas.

It might say something about human nature that unless explicit mention is made of the arduous nature of the transition, most players will usually respond to a new location as though they were refreshed – “a change is as good as a holiday”, so that’s something else that the GM has to explicitly mention. The only exception to that is when the previous location, or the events that follow it, deliberately embed awareness of the difficulty in reaching the next location into the narrative.

Tonal Shift

Finally, there is a natural expectation and receptiveness to a shift in tone or intensity with a new location. That makes them serve as ‘mile markers’ of the journey through the adventure from start to finish.

Game Value

All of these imbue a location image with Game Value regardless of the commonality of the image. The rarity of an image still provides increased value, however, but it does so with subtext, tone, and other intangibles.

The greater the ‘rarity’ of the image, the more these intangibles are communicated to the viewer – that is the determinant of ‘image content quality’, the ambition that you hope your image search will satisfy.

This is an important point to understand. For any given basic search, there will be dozens of location images to choose between. To some extent, temporal content will restrict your image choices, but when the ability to edit images is taken into account, that ceases to be a definitive factor; it’s a consideration and a constraint, but nothing more.

That permits other factors to come to the fore, the intangibles mentioned earlier foremost amongst them. It’s more important for an image to convey the right intangible messages than for it to match precisely other restrictions because they can be edited in or out in many cases.

I’ve lost count of the number of air conditioners that I have painted out of windows to make a modern-day image reflective of the way a building might have looked in the 1930s for the Adventurer’s Club campaign, for example. Each time, the building in question had the right look and the right context – but one or two too-modern features. Massed phone lines are often another item that needs to be redacted. And don’t get me started on modern cars….

A valuable tip: what you lack the time or skill to paint out can sometimes be conveniently covered over by importing some relevant image. An old car in place of a new, for example. Or a convenient wall. A snowbank (doesn’t work in a summer scene, of course). Even a convenient tree!

Mundane

Mundane images are the most easily found with an image search, it says so on the tin. Unfortunately, there can be a gulf separating expectation from reality, and the breadth and depth of that gulf is largely a function of genre.

For example, search for “Science Fiction Corridor” and you’ll get fewer usable results than if you search for “Lunar Colony” even though you would expect corridors to be more ubiquitous. The reason is because Lunar Colonies are “sexy” subjects while corridors are boring.

“Castle Corridors” on the other hand, are reasonably well-represented, at least in comparison. Why? Because there are any number of actual castles out there with corridors, and some of them will have been photographed.

Mundane locations are therefore fairly generic in nature. A snowy field is more about the snow than about anything underneath it. Drop a genre element or two on top (and anchor it with a suitable shadow) and it can be anything from a sci-fi setting to a fantasy wonderland with minimal effort.

Generic scenes, by their nature, add little to the setting in terms of intangibles, but they provide the same basic benefits as all location images.

Common

Common images can be a little trickier, because you are generally forced to deal with anachronistic elements. I’ve already mentioned air conditioners for 1930s locations; for one particular image used in my Dr Who campaign, I had to paint out modern rubbish bins, and two modern cars, replacing the latter with snow and road in one instance and road, sidewalk, and wall in the other. It wasn’t difficult but there was a lot of fiddly detail required.

You are more often forced to compromise with common images, in my experience. The more specific your description of a village, the less likely you are to find an exact match – unless you do the image search first and then base your description around a chosen image.

The problem with that approach is that there are only a certain number of usable ‘village” images available, and once you have used them, you are inevitably looking at photo-editing something, and some time thereafter, will inevitably be faced with choosing between a more accurate depiction of a village that is going to involve more work to prepare, or a less accurate depiction that is also going to be less work.

If you have the time in hand, the first one would be your choice every time – but GMs rarely have excess prep time to devote to projects of such limited value. There’s almost always something you could better spend that excess time on than a common scene – so almost certainly, you will tend to choose the quick-and-easy option and compromise the fidelity of the image’s relationship with the narrative.

All this makes ‘common’ images much harder to deal with than most people expect. There are few easy answers to the problems described; you simply have to do the best that you can in the time available.

Specific

Many of the problems with Common images go away when you turn to more specific locations, because there will often be either real images of the location to choose from, or because it’s a more interesting subject, and so has been depicted by artists more frequently.

Either way, it’s far more common to find usable images, even of small and obscure locations – with the same caveats regarding anachronisms.

With the greater range of search results, a different problem manifests itself, however. What you find is very dependent on what search terms you feed into the search engine, and getting the most out of them involves a deeper understanding of how the search engine works than most people ever need.

When you search for an image, it’s commonplace to employ search terms that describe the image content, and sometimes that can be successful. But most search engines work by finding the search terms within the text, offering up all the images found on a relevant web page, and then sorting and weighting the results based on the image information itself. It’s also not uncommon for more recent images to be weighted to appear closer to the top of the results.

With most image searches, that approach is more likely to find what you are looking for, and it’s relatively easy to do, which is why the search engines do it that way. We, on the other hand, are using image searches for a purpose for which they were never intended, and since we can’t change the search algorithms, it’s up to us to adapt by using different search terms, i.e. image searches that are more likely to produce viable results. And this can be harder than it looks.

The best approach is always to be flexible, and think around the problem. Use synonyms – if you are searching for a snowy village, “winter hamlet” might find the perfect result. Creativity in selecting search terms, and making multiple attempts, can be the difference between ‘easy’ and ‘hard’, and between greater compromises and better results.

Unique

As usual, it’s when you get extremely precise about what you need that it becomes unlikely that you will find it with an image search. That leaves you with three choices: compromise with a less precise image, create your own, or adopt a half-way position between these extremes, creating something that is close but still not perfect.

It can actually be quite surprising how little specificity you need to reach these levels of restriction; some subjects are surprisingly hard to find represented. For example searching for cave entrances will find you many images taken from inside the cave looking out, but images of cave mouths from the outside tend to be rarer than hen’s teeth; if you need such an image (and they occur in almost every genre, at least occasionally, and often regularly), you will almost certainly need to make it for yourself. Fortunately, that isn’t all that hard to do.

Which of the three alternatives you employ in any given case will vary with the subject matter, as the above example implies. It all depends on how closely what you find matches with what you wanted.

Of course, you can always ‘cheat’ and do the image search first, based on a generic label or description, choosing an image that looks good and describing it in your narrative and other planning. This compromises the adventure content to match the image results available, but if you aren’t locked into a specific need, it can often be the easiest solution here, as it has been in other categories.

I’m going to close this mini-post with a surprising real-world example. For the current Adventurer’s Club adventure, we needed a jungle valley with a certain grandeur and scale, completely surrounded by mountains or better yet cliffs. The valley needed to be large enough to have a flat floor through it’s center, and it could contain nothing man-made – all such objects were specified by the plot and unchangeable; they would need to be constructed and inserted behind layers of vegetation.

This is a very specific set of requirements, but a valley seen from a mountaintop sounds picturesque enough that I was hopeful. Unfortunately, nothing was quite right; there were alpine valleys and Canadian valleys that had the right shape and size, but were full of the wrong trees, and there were jungle valleys that had the wrong shape and often the wrong size, but the right vegetation.

In the end, I created my own, disassembling 14 source images into more than 70 components and layering them (with some paint-work) Some components were blurred and some sharpened, some were color-shifted or otherwise manipulated.

I shared the unfinished image on social media as a work in progress. Below, I have not only shown that unfinished image but the final version without the all-important details.

As usual, the more important an image’s content is to the plot, the more easily you can justify taking the time to do something like this. It’s a perpetual balancing act between the cost (in time, which is a function of your skill with your chosen photo-editing application) and the benefit to your adventure. If it’s a quick and easy edit, it’s relatively easy to justify; if it’s more involved, it needs to be sufficiently important.

Two mini-posts remain in this series – Events and Effects is up next!

The story so far…

This is the fourth in a set of mini-posts that I’m writing and publishing as quickly as possible, something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each post examines one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

So far, the series has looked at Objects, People, Monsters (and other encounters), and now it will turn its attention to vehicles. I’ve expanded this post more than most because it’s both useful to do so and the post will be published in the usual window for CM.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

Vehicles

There are three, no four, attributes that combine to yield the Game Value of a vehicle.

Distinctiveness

The more unique the vehicle, the greater the imaginative leap required to visualize it from description alone. With some players, that doesn’t matter too much, but with others it can make all the difference in the world. Either way, it is something that they need to concentrate on in addition to playing, and that means that they can benefit from a graphic representation doing that part of ‘the work’ for them. Hence, the greater the distinctiveness of the vehicle, the greater the game value of an illustration.



Image by Jean photosstock from Pixabay, sky by Mike.
Try describing this vehicle in a couple of hundred words – whatever the impression your text creates will be inadequate next to the graphic visual image.

Plot Impact

The more important the vehicle is to the plot, including as a setting, the greater the Game Value because important pieces of roleplay will take place there. Making an RPG immersive is one of the greatest challenges faced by a GM, and illustrating a vehicle with significant plot impact punches above its weight in this area.

It doesn’t even matter too much if the vehicle itself has low plot impact, being nothing more than a place where things happen while characters travel from A to B. If significant events take place there, then it can be considered an important location, and that earns it the same plot impact as though the vehicle were plot-significant.



Both of the spacecraft images below would be suitable as either a base of operations for a group of PCs or as a vehicle for the regular use of a group of PCs, but they convey very different impressions and subtexts. The first image is by Thomas Budach, while the second image was shared by Thomas Budach, both through Pixabay. Images rotated and cropped by Mike.

Campaign Penetration

I struggled to find the best terminology for this aspect of a vehicle. Simply put, if it’s going to appear in multiple adventures, if it greatly expands the choices open to the PCs, if it functions as a mobile headquarters for a recurring character of any kind, then the vehicle achieves a higher level of campaign penetration than it’s relevance to any specific adventure or encounter, and that gives it greater game value than might meet the eye.

Metaphor, Metagame, and Implication

These three elements, even in combination, are not as significant as those described before them. That’s why I have grouped them together into this one banner headline rather than counting each separately.



Image by Eduardo Davad from Pixabay

Proposal One: Metaphor

Telling players that their characters have traveled from A to B is not as effective as telling them this while showing an image of the vehicle that conveyed them. The vehicle itself is a metaphor for the act of traveling. One picture is worth at least 500 words in this case, maybe more – even if absolutely nothing of interest is going to transpire on board and you intend to hand-wave the entirety of the passage.

Proposal Two: Metagame

A vehicle can contain a cultural context that holds significance beyond it’s simple existence. Think back to the first appearances of the other schools in Harry Potter and the Goblet of Fire (the Durmstrang Institute and the Beauxbatons Academy) – the first thing you see is their mode of transport (the carriage drawn by pegasi and the ship rising from the sea floor). Both of these begin the process of establishing these schools and their styles even before they are introduced by Dumbledore.

This is foreshadowing, which readers might not have realized is inherently metagaming – it’s the GM using his foreknowledge of the plot to hint at what is to come, and (in this case), manifesting it in an image. The image is therefore a critical piece of the plot, designed and intended to subtly prod the thoughts of the players in the “right” direction.

And that gives such images a level of game value beyond any that meets the eye.



This image places two representations of the same train from two different sources side-by-side. Which image you choose depends on the impression that you intend to convey. The mostly grayscale image was provided by Brigitte Werner, the color image is from blizniak, both from Pixabay, cropping and compositing by Mike.

Proposal Three: Implication

The players see an alien spaceship land. Even if the pilot is about to appear in its doorway and steal the spotlight, the first hint that the PCs will have as to the nature and intentions of the occupant is that spaceship. Depicting the ship lets the players process exactly what their characters are seeing, and that gives it a higher game value because it conveys implications about the contents.

The analysis below, as usual, is based around the first of the four attributes just discussed. That means that the primary source of nuance and differentiation within each resulting sub-category are the other ‘three’ attributes – Plot Impact, Campaign Penetration, and the three-legged Metaphor / Metagame / Implication bundle. Collectively, these comprise the Importance of the vehicle. Even the most common types of vehicle can have enough Importance to justify an extensive search; rising rarity simply elevates the “minimum importance”.

Mundane

Mundane vehicles are widely available in the campaign. Depending on the genre and time period, availability of images can range from the hard-to-find to the routine. For example, if you search for “1970s car”, you’ll be spoiled for choice; you will usually be better of with a more specific search – “1970s Ford”, “1970s sports car”, “1970s pickup”, and so on.

This fact provides the solution to the problem of the hard-to-find categories. For example, searching for “Carriage” brings up a lot of railroad images but not too many of the horse-drawn variety; to actually get results you are better off ditching the term “carriage” completely and substituting the specific type of carriage that you want, something that I’ve learned the hard way. Your starting point should therefore be the Wikipedia page dedicated to the type of vehicle desired – for example, this page for Carriages, and this page for Boats (hint: hover your mouse over each of the types to get a pop-up preview of the page, which generally includes a brief description).

Things become a little more problematic again when the vehicle is a type that doesn’t actually exist yet (and might never exist). For example, “Space Freighter”. Once again, adding “concept art” can find images that would otherwise be missed. In general, you are forced to apply functional descriptions as search terms because any other kind of specific yields few or no images at all.

Being well-read in the genre can make a huge difference. If you know that, for example, Space: 1999 contained exactly the right “look” for the vehicle you’re after, searching for “Space 1999 Vehicle Concept” will usually find you choices that are more useful because they have the right appearance for your intended purposes. Knowing many of the different sci-fi television shows, novels, authors, and artists can be a lifesaver. For example, do an image search for “Chris Foss Spaceships” – here, I’ll make it easy for you:

DuckDuckGo Image Search: Chris Foss Spaceships.

Foss is a quite famous sci-fi artist whose work features on a number of sci-fi novel covers; his style tends to be instantly recognizable.

Nevertheless, as a general rule of thumb, image searches for spacecraft and the tend to be either too broad, or too specific; there doesn’t seem to be much middle ground. Be prepared to aggregate the best results from multiple searches in order to find enough choices.

Other additions to the list of search terms that can be useful are “Primitive”, “futuristic”, “Sci-fi”, “Scifi”, “Interplanetary”, “Interstellar”, “Intergalactic”, and so on, both singly and in combination. The sheer number of possible search terms that result can quickly become overwhelming; it’s almost unheard-of to search for all the possibilities. That means that a significant fraction of the possible terms never get searched for, and a significant number of the possible results will never get found. To combat this, I try to perform subsequent searches in alternating sequence – first to last, then last to first.

Further expanding the scope of potential results (and I’m sure I’ve mentioned this before), the algorithms used by the different search engines are all different and frequently find different results. Learn to use them all; have one primary go-to (mine is DuckDuckGo because I find it more convenient, my fallback is Google (which used to be my #1 choice, but they have made it progressively more annoying and hard to use), after that Qwant, then Bing – except that sometimes I’ll scramble the order to avoid over-reliance on the same sources..

The devil’s in the inconsistent detail

Two of the biggest problems that you will face is technology that looks too dissimilar and technology that looks too similar.

Two alien races should have differing engineering philosophies, and that should translate into a distinctive appearance for their respective spacecraft (or sea-craft, for that matter). At the same time, the same physics will usually apply to both, and so there should be some consistency, too. And that can be a very difficult duality to achieve.

Star Wars doesn’t quite pull it off; their designs are too variegated for consistency. Star Trek does a better job – Vulcan, Romulan, Klingon and Federation ships all operate on similar physical principles, and so there’s a consistency of general design principles, but each race also has its own style. If a ship shows up in an episode that looks different, it will prove to operate on different physical principles, often with benefits and disadvantages that the rest don’t share.

For contrast, study the different ships of Babylon-5, in which the technologies and design priorities of each race are distinctly different and so each race’s ships are similarly distinctive (Earth ships look like they were designed by Chris Foss…). Each race’s design ethos also translates into many other design manifestations – from homes to diplomatic quarters to weapons to… well, you get the point.

Decide where your dividing line has to be and stick to it like glue; that is your only pathway through the complexities of inconsistent consistency and consistent inconsistency!

Game value of images can range from the trivial to the monumental, but even the trivial ones tend to still offer some reward to the GM who seeks them out.

Common

Common vehicles are less widespread but still easily obtained if you go to the right place. They are vehicles that most travelers would see regularly if not with great frequency. For example, while sailing the Mediterranean, it would not be all that surprising to see a Spanish Galleon in the appropriate era, or a Dutch Trader, but a Barque showing the colors of Turkey, or one of the Scandinavian countries? Much less likely.

Or, to take a D&D-relevant example, Carriages would be mundane, Royal Carriages would not – but there are enough nobles that you would probably see one every month if you were traveling regularly. Or carriages might even be ‘common’ and wagons ‘mundane’ in your game world.

The same search tips and techniques apply to common vehicles, as do the same problems.

Specific

Specific vehicles are where things often start to get sticky for the fantasy GM and easier for the sci-fi GM, because – as with all searches – the more specific you can be, the more targeted your search results. You may get fewer matches, but the average result will be a better match. But specific for the Fantasy GM means something that matches a specific style, and they can be very hard to match.

At one point, for the Pulp campaign, we needed to find an image of a longs hip being excavated. There were a number of viking long-ship images, but very few of them matched that specific criteria. What’s more, they were all photographed from above, we needed one that was at or above the eyeline of the observer. I ended up getting a ship from one image, the dragon-prow from another, some oars from a third, and some colorful shields in a row from a fourth, then compositing everything with a background image that was itself a composite of multiple images. This search was so specific that the full criteria approaches ‘unique’ status, but even discarding the additional requirements, there were so few matches that the general principle still holds.

Despite this, there is a ray of hope that gets bigger all the time. Artists keep creating images, and those images keep getting curated on the web. The pool of possible results is always growing. That doesn’t mean that obscure images will suddenly become commonplace – but it does mean that there is always hope that someone’s artistic product and your need will intersect!

Unique

Which brings me to the problem of unique vehicles. These are either something completely specific, like the Millennium Falcon, or Galactus’ spherical ship (from the Comics, not the movie), or the Battlestar Galactica (original or revised version). Or maybe Thor’s Chariot – which (surprisingly) has no unique name (the Norse named his goats, his hammer, his belt, his servants, his gloves, and his staff – but not the chariot drawn by those goats. Go figure).

Quite often in this category, though, ‘unique’ is a misnomer. You might need a ship with a particular figurehead or a specific name, for example, and have multiple other requirements, just as we did with the Viking ship; but we weren’t looking for a specific Viking ship, any ship that matched our needs would do.

Many of these are relatively simple editing jobs in Krita. It’s usually easy to paint out an existing ship name and replace it with a new one, for example.

What this adds up to is that the most unusual vehicle images are often less work than a highly specific one, because – ironically – you can compromise more regarding the image content when you know that you are going to be editing it anyway.

Another week of medical to-ing and fro-ing is in prospect, interfering with my ability to post. I’ll try to get the next mini-post, Locations, done for the usual posting time, but it may be delayed 24 hours or more.

This post has taken a lot longer than expected, delayed by medical testing that was like a black hole sucking in time. And there’s more of it to come, I’m afraid…

The story so far…

This is the third in a set of mini-posts that I’m writing and publishing as quickly as possible, something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each post will examine one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

Monsters and Encounters

‘Monsters’ have a commonality, in game terms, proportionate to the danger level they pose. In fact, in an RPG (regardless of genre), increasing rarity equals increasing deadliness.

There is even a level above the uppermost tier of commonality where you are dealing with named and discrete individuals, which are better treated as unique NPCs. In D&D that’s individuals like Beelzebub and Odin, Hercules and Tiamat.

Encounters are a little stranger, since monsters are already covered. Weather events are dealt with in the Effects category, NPCs are in the People category, environments are in locations – there’s not a lot left.

Or is there? There’s a lot of ground between the basic level of danger represented by the bottom tier of “Monsters” and nothing at all; in fact, in some genres, almost everything will fall into that category. Deer, birds, bees, sheep, cattle… So “Encounters” is being used as a label to reference “ordinary” animals (and plants), while “monsters” is being reserved for creatures that are the products of the collective human imagination.

Encounters

With that determined, I can look more closely at the commonality strata as they apply to these different categories.

Mundane Encounters

are

• the small creatures – birds, small lizards, rabbits, mice, and the like;
• small fish (did you know that properly-cared-for goldfish grow 1/2″ every year and can get to 14-16″ in length?); and
• insects.
• And most plants – grasses and bushes.

These creatures don’t add much Game Value in and of themselves. What they do is add a living, active, ingredient to the landscape in which they appear. If you need such an illustration, it should be judged as a Location image – at least under most circumstances.

There are a few exceptions to bear in mind. The first is where there is some additional plot relevance to the animal’s appearance. The second is where the animal is somehow out of place, which lends additional significance to the very fact that the animal is where it is.

For example, recently in the Zenith-3 campaign, I used a very picturesque Elk in the middle of ‘town’ to illustrate how small and decentralized the community in question really was. A second example is the (forthcoming) appearance of a toucan in flight in the Adventurer’s Club campaign, which (in combination with some other images) should suggest an environment teeming with life.

The examples show that there can be plot value in such images by virtue of the impressions that they suggest to the players, but outside of such purposes, they have very limited utility. If you can make valid use of an image of this type, they tend to be fairly easy to find, permitting you to select the image that adds the maximum value to the campaign.

Common Encounters

The contents of the Encounters category begin to diversify markedly in the Common sub-category.

• small creatures that are inherently dangerous – some frogs, snakes, and so on – and
• larger creatures that are not so dangerous – goats, sheep, deer, etc.

This also includes

• the smallest dinosaurs (up to, say, 1 foot in length);
• apes up to the monkeys,
• mid-sized fish – up to the size of trout, salmon, and barracudas;
• and any plants that are shorter than a person, including those smaller plants that can be considered exotic or dangerous (Venus fly traps, orchids, etc) – which complete the category.

These are a bit of a mixed bag in terms of Game Value – some can be quite high, others not so much. Aside from the dinosaurs, you should have no trouble finding what you need, so rarity is not a real consideration.

Let’s deal with the exception noted above. If you do an image search for “small dinosaur” you will be presented with a few options; opening the source page of one of the results should give you the scientific name of the species that has attracted your attention, and an image search for that specific variety of dinosaur will usually yield a better range of results from which to choose.

The impact of the creature on the plot should guide your decisions; if the creature is not significant, you can usually live without the illustration. Exceptions are the same as for the Mundane variety of encounters. As a general rule of thumb, and to state the obvious, the more dangerous the creature is, the more likely it is that the encounter will be significant to the plot.

Specific Encounters

• …start with the mid-sized hunters – eagles, hawks,
• jaguars and most of the big cats,
• wolves, wolverines, etc,
• all the way up to and including alligators and crocodiles;
• And the really big herbivores, including elks, moose, elephants, hippos, etc.
• Most dinosaurs also belong in this category, all the way up to multi-ton plant-eaters.
• The larger apes, excluding the really big ones (gorillas, orangutans) are also part of this category, as are
• fish up to the size of dolphins.
• and plants that are bigger than a person and shorter than a house. That includes virtually all the fruit trees, vines, etc.

Basically, anything that’s left that isn’t in the fourth subcategory, below.

As you can see from the listed contents, these creatures tend to be a lot more attention-getting and potentially quite dangerous (at least in most game systems). That makes them much harder to ignore, from the perspective of the players, and that translates into a high Game Value for an illustration (from the point of view of the GM).

In D&D, these are the sort of encounters that you use to let the PCs blow off a little steam without doing much to reduce the emotional intensity that has built up. Used in this way, the image has considerable Game Value.

I would be prepared to spend 5-10 minutes ferreting out an image for one of the encounters on that list. Maybe less for the fish, depending on the Story Value. Not that it should take that long, as most of these images will be broadly available. In fact, in many cases you may find yourself spoiled for choice.

Unique Encounters

Any animal that can’t be ignored, even if you are in a safe place.

• Raptors, T-Rexes, etc,
• Lions, Tigers,
• Gorillas, Bears, arguably Rhinos,
• Kangaroos and Cassowaries,
• Great Octopi and Giant Sea Squid,
• Whales, Sharks, etc.
• Plus trees taller than a house.

Everything said in the last section holds true for this category except for the last paragraph.

Some of these have loads of images to choose between, but there are some that are surprisingly hard to find. You will either strike gold very quickly or you will need to get creative. And that won’t be as easy as it sounds in some of these cases.

Monsters

Attitude counts for a lot in this category, as does sentience, because it allows creatures that have more tools at their disposal than they are endowed with by their nature..

Mundane Monsters

Anything small than a man in weight that is not intelligent.

While a few examples, like Blink Dogs, may have sufficient plot value to justify an image search, these will be rare exceptions. More often, the depiction in your monster sourcebook will be as good as anything you can find online.

Common Monsters

Anything up to man-sized that is intelligent, and anything that can be considered inherently magical that isn’t in one of the two higher tiers. Like Unicorns.

It’s far easier to find interesting images online for most creatures in this category, and that (coupled with obvious plot relevance) yields obvious Game Value that justifies a search.

These include some of the favorite choices of subject matter for many artists. Which brings me to a useful tip: adding the words “fantasy art” to your search term can open up whole new worlds of results for your inspection, as can adding the words “concept art”.

Specific Monsters

Anything man-sized or larger that isn’t in the fourth category, excludes anything that is not considered sentient. That’s all your Ogres and Trolls, Ents and Elves, and a great deal more, besides.

Unfortunately, these are less popular as image subjects, so the increase in Game Value is matched by a decrease in image availability. Accordingly, you will often be in for an extended search of 30 minutes or more before you find anything useful. Fortunately, those image search tips offered in the previous section are still valid.

Unique Monsters

Beholders, Dragons, Mind Flayers, and anything of similar cache. Like the top tier of Encounters, these are the creatures that can’t be ignored, even if you think you’re in a safe place with respect to them. That would also include your higher-type Demons and Devils, of course, and Djinn, and Greater Elementals.

These are all creatures that should be treated as top-level NPCs, but often aren’t, even by experienced GMs.

On top of that, anything special brewed up by the GM as a featured monster – like the menacing Halloween creature used as an illustration at the start of this mini-post – also goes into this sub-category, which suddenly seams almost bursting at the seams.

Some of the most popular subjects for fantasy art occupy this category, but that can be a mixed blessing; two artists can have different visions for the same subject that are wildly incompatible. This can completely undermine the verisimilitude that justifies the image search in the first place, if you aren’t careful!

Sometimes it can be more valuable to do the image search first and create the encounter second, drawing upon what you find for inspiration.

Next: Vehicles!

The story so far…

This is the third article in this series, and the second of a set of mini-posts that I’m going to be writing and publishing as quickly as possible, something I’m calling a mini-blitz. My normal publication schedule will resume at the end of the series.

Each post will examine one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

The first mini-post dealt with Objects. Today, the subject is NPCs.

People (NPCs)

Compared to the category of Objects, People are relatively straightforward.

Mundane

Mundane NPCs are your generic crowd scene, or representatives from such a scene. These can be divided into four subcategories:

• Crowds that say nothing more than “the location is crowded.”
• Crowds that are important because of the common activity being depicted.
• Groups that are important because of some collective common feature.
• Individuals with whom no significant interaction is expected.

The Location is crowded

The crowd are superfluous window dressing in such images; the importance is actually dependent on, and attached to, the location itself, and (if anything) the crowd is likely to be obscuring that, though they would impart a greater realism to the scene. Such images should be evaluated as depictions of the location, which will get dealt with in a later mini-post.

Depictions of activity

Shoppers in a bazaar or marketplace, for example. This category is a half-way house in which the activity of the crowd and the location tend to be of equal significance, but an image of rioters would also fit into this category with no location value being evidenced.

Such activity depictions tend to be plot-significant, so these images are immediately possessed of Game Value. Depending on the nature of the crowd, they may be either easy to come by or extremely hard to find. “Medieval rioters” would be quite difficult, for example, although you might succeed by thinking outside the box – a still of villagers from Frankenstein, for example, might fit the bill.

Common features of a group

The King’s Guards. The palace courtiers. The Riot squad. An invading army of orcs, or androids, or martians. An ordinary group of people all suddenly wearing the same badge, or mask, or whatever. A demon horde.

No way any of these would ever be relevant, is there? Quite often, it’s the mere fact that there are a group of them that is the most important fact to convey; the actual appearance of the group is secondary in such cases.

This can get tricky, because you might be able to find an image of a representative member of the group but not of an assembled group. That puts you in the position of displaying the individual as a representative member, or of editing the image to insert additional copies of the individual which are then manipulated to create diversity to the required level. I’ve worked it both ways, depending on the Game Value of the group, but in general, the first would be the most acceptable compromise if an actual group shot can’t be easily obtained.

Individuals of no plot importance

The final subcategory deals with another form of window dressing, the movie-set extra. These exist for no other reason than to add verisimilitude to a scene, but they can be useful in hiding inappropriate content from the viewer in a visually-arresting way. It’s just as much work to insert a small dragon into an image as it is to paint out an air conditioner, but the scene-plus-dragon is likely to be the better result.

I’ve also used this technique to cover street signs, fire hydrants, and to replace inappropriate (modern) vehicles with something more era-specific; the general term I use for the procedure is ‘time-shifting’.

For example, you can take a modern image (without too many people) of a British village’s historic town center and use a few bits of window-dressing to set the image in a medieval era. All you need to do is recognize the possibility of the altered image when viewing the source, and plan what you are going to need to correctly “dress” the scene.

Beyond such purposes, though, rent-a-crowds have very limited Game Value.

So there are some functions of appreciable Game Value for images of mundane (in plot terms) people, and some of negligible value. This makes your intent in choosing to display such an image (assuming that you can find or make one) critical. You need to have a specific purpose, and you then need to select images or image elements that achieve that specific purpose. Anything more that you might get out of the image is a bonus.

Clearly, some of these will be more easily-obtained than others. As a general rule of thumb, the greater the Game Value of the resulting image, the more difficult it will be to find the right image. This simple relationship means that you should search until you find something suitable if a search is warranted at all.

Ideally, you will find two or three images to choose between, but there is often a degree of luck associated in finding anything at all. Therefore, when undertaking such a search, I don’t take the first result, but continue until either I have enough such options, or the search extends beyond what I consider the Game Value in prep time – at which point I choose between whatever I have found. That might be one image, or a pair of images, or even more – if the plot value was high enough that I had continued the search after achieving that ideal result (it has happened).

Common

Common People, from the standpoint of Game Value, are NPCs who may be named, but whose identity as individuals is subsumed to some other factor. That factor may be a personality trait (“angry young man”) or a social trait (“fop”) or a profession (scientist) or whatever.

The interaction with one or more PCs gives these individual NPCs significant levels of Game Value, and this was one of the drivers of the high Game Value score in my initial insight into the subject (the chart shown in the first post of the series)..

Naming such individuals gives the option of addressing them by name, having them introduce themselves, etc – it makes interaction easier. But it is often unnecessary. The rule of thumb I employ derives from the anticipated interaction between the NPC and the PCs – if this is such that it would be reasonable for the NPC to offer their name, or if the PC is likely to be directed to speak to the individual by name, then I name them.

Searching for such images is usually a case of searching for depictions of the “other factor”, possibly with additional qualifiers. Having a broad vocabulary helps. When searching for “Brazilian Deckhand 1930s”, the actual image chosen after a recent search was found using just “Tropical Deckhand”.

Because you will expect to make multiple searches, you need to be decisive. Use any tools on offer (image size especially) to narrow your search down. Any ‘contenders’ should be opened in new tabs and keep going until you have five or six of them, then winnow down to the most satisfactory image.

What’s more, amortized effort is again a consideration – the same NPC may reappear multiple times in the campaign. It’s worth spending a little extra time on such image searches because there will be virtually no effort required for subsequent appearances.

In terms of availability, people are one of the most commonly photographed subjects. That tends to produce a lot of images for you to choose from. Specific restrictions bite into that ubiquity – some more than others. Some searches will yield a lot of results, some few, and it can be unpredictable. So budget your time expecting trouble and take advantage of it when random factors align in your favor.

Specific

This sub-category generally indicates the need for an image with multiple specifics, but not a real, named, individual. Luftwaffe Captain with a scar, Riverboat Gambler with cane, Monstrous Hulking Porter, Beautiful Blonde Concierge… you get the idea.

The one thing that members of this sub-category always have in common is that you expect there to be considerable interaction between this NPC and one or more PCs, either now or in the future. That in turn means that the character that the image is to depict will play an important role in the plot, and that it is all the more important for the players to be clear about the character that’s doing the talking..

All the advice in the previous section still applies. Both the Game Value of the image and the difficulty of finding a match have increased, but in proportion, so the same standards of results apply. The increased plot value / interaction level goes on top of that, so this is a search in which it is worth taking your time and being a little more exhaustive.

I’m going to reference this sidebar again in the final part of the series, but it’s an important search tip for right now: you will often get better results, faster, if you do your image search and then write your descriptions, etc.

In the Adventurer’s Club campaign, we set a high standard for such images in terms of the amount of personality conveyed by the image. Sometimes, those results are achieved by searching for the personality profile we want (and being less selective with respect to other character traits), but more often, we will search for some specific desired quality and cherry-pick the results that show the most personality regardless of the makeup of that personality – then we can pick between the population of the resulting short-list.

The image above could be any government or business figure. The emotional content is subject to interpretation, but the image itself displays a lot of personality; it has impact.

Unique

There are two types of Unique individual. The first is someone real, often doing something specific – Stalin making a speech, Lincoln tipping his hat, Churchill looking stoic, etc. These are no more difficult to find than Specific images, but have much higher Game Value by virtue of the baggage and reputation that they carry. This is even true of images of individuals that the players won’t necessarily recognize – for example, James Buchanan – at least until you provide a relevant biography.

The other type is of an individual whose specifications are sufficiently distinctive that the likelihood of a successful search plummets. As noted earlier, the Game Value of such an image tends to rise proportionately, which means that you can justify spending quite a bit of time and effort generating a custom image, which is quite likely the only way that this sort of image search can succeed.

In both cases, the Game Value is about as high as it can get. These are always important characters in the adventures in which they appear (otherwise there would be no point in such distinctive characters appearing).

Depicting important characters is therefore about as important as it can get.

To close out this mini-post, I thought that I would repeat an image first shared about a year ago, of Brother Simon, the Pacifist Dalek, which is an example of the second sub-type of unique character.

Next: Monsters and Encounters

The story so far…

This is the second article in this series, and the first of a set of mini-posts that I’m going to be writing and publishing as quickly as possible, something I’m calling a mini-blitz.

My normal publication schedule will resume at the end of the mini-blitz.

Each post will examine one of the specific image categories nominated in the first post of the series, dividing them up into strata of commonality.

The goal is to define a set of policies, processes, and principles that use the game value of the result to define how and how hard it is worth searching for a particular image within each category / strata combination.

Objects

This first mini-post is likely to be one the larger ones of the series. It deals with objects – which is to say, things. This is a category full of nuance and shadings and exceptions, many of which will need explicit examination within.

Mundane

As a general rule, mundane objects are ubiquitous within the game environment and have virtually no Game Value. If you mention a lantern, the exact design doesn’t matter very much, and spending any time describing the details is counterproductive.

And if the exact design does matter, the mere fact that it mattered enough to search out an image instantly telegraphs that significance to the players, even if there is no reason for them to recognize it. If you wait until they do recognize the importance, they will already have formed a mental image of what the lantern looks like, so the image is at best superfluous, and at worst counterproductive again.

There are, however, a number of exceptions to this principle.

• Objects that are iconic to a culture that is different to that the PCs are used to can carry greater Story Value by symbolizing the whole ‘we aren’t in Kansas anymore’ message. The vase and other objects in the image above clearly illustrate this principle.
• If there is something else visually obvious about the object, it can carry greater Story Value by representing that the owner has unusual objects.
• Some objects are especially symbolic of culture, no matter how mundane they might be. Food is a great example.
• Some recurring objects can, by virtue of the repetition, have a greater Game Value. But these are usually a lot more specific than this category.
• Any Vehicle that does not have enough Plot Significance to warrant illustrating the interior is considered to be an object, and some of those will be ubiquitous enough to be a Mundane Object. You can quickly determine this to be the case when you don’t care what the ‘object’ looks like, just that it looks ‘good’ or ‘cool’.

Searching for such images is a lot easier with a touchstone term. The basic search term is the name of the type of object – “lantern”, to continue the example. A touchstone term defines a specific subcategory with a descriptive term. That term could be “oriental” or “Babylonian” or “gold” or, well, anything your can think of that will winnow through the chaff and get you the image you want.

Always remember what the search terms mean when you enter them in an image search – they are terms that appear on the web page where the image being displayed can be found. If you are very lucky, and depending on the subject of the search, that might be a description of the item. But it might not. “Babylonian Artifact Restored” might be an even more successful search term than “Babylonian Lantern”, even though it is more likely to throw up non-lantern results. You have to Design your search terms.

As a general rule, availability of these images is quite high, because – once again – you don’t particularly care (or shouldn’t) what the image result is, so long as it will look good. But even so, unless there is some specific Game Value that will be achieved, these images are not worth the time to pursue.

With one exception, in addition to those listed above: Immersion. This is always hard to achieve, but a succession of relevant images is one way of doing so. Such a sequence contains, collectively, greater game value. This is especially important when setting the stage for a mystery plotline.

Common

Common images are Mundane objects with one element of specificity about them. That specificity must have specific game value or it doesn’t count.

That usually means that you are looking to compile a consistent “look”, and (as explained earlier) that gives the image results a greater Game Value.

It also means that you are justified in making a quick search for images of the specific objects. You may not find everything you want, but a little creativity in your search will usually find most of your wish list.

The same exceptions listed above still apply, and now warrant something more than a cursory search. My personal standard is to try to find three choices for each image, enabling me to choose between them.

I also pay special attention to the background behind the object. There are three possible options:

• A plain background, which may be cut away digitally and replaced with a more relevant setting;
• A background that constitutes a reasonably relevant image already; or
• A background that constitutes an irrelevant image that will be difficult to redact and replace.

Clearly, the second option is the most efficient, the first is an acceptable outcome, and the third should only be considered if the proposed illustration has the highest possible Game Value, i.e. you consider it essential that you have something to show at that point. And that is especially rare when you’re still dealing with common objects.

Another way to look at these potential illustrations is as seasoning for your adventure. As with cooking, a little can go a long way, and it is very easy to use too much salt, pepper, or spice. Be selective and take the time to prioritize your desired results. The default option should still be not to add an illustration unless it carries extra Game Value.

In practical terms, the stricter requirements mean that only 10-40% of the images found under a “mundane” level search will still be relevant, possibly less. That’s another way of saying that it will take about 2½ times as long to do an adequate image search.

That in turn sets the threshold that you should apply – unless the object has at least 2½ times as much Game Value as some generic “flavor” object, it’s not worth the effort to search. That’s a threshold that only examples that are right on point can reasonably sustain. If the result isn’t at least some minimum shade of perfect, forget it; you’re better off without an illustration.

Specific

As the specifics of what you want increase, the difficulty of finding exactly what you want in an image search also increase, and it becomes increasingly practical as a solution (if you have the skill) to edit an image to get what you want.

The effect of the latter is to broaden the scope of a search that has become so narrow as to be difficult and time-consuming. The greatest probability is that it will still be so, even after that compromise.

In many (but not all) cases, the Game Value of the image as an illustration will also have increased, but has it increased enough to justify the time and effort involved?

Sometimes, the answer will be yes, and sometimes no.

The answer will be different from individual to individual, depending on how skilled they are at image editing and manipulation. But it will also, inevitably, be fuzzy in other ways.

You see, when an illustration is provided by the GM, the assumption is that the players will find it useful or even need it, but that’s something that will vary from player to player and from day-to-day. On one occasion, a given player might need the extra support to get a clear understanding of what’s happening in-game, and on a different occasion that same player might not.

It’s usually the best that you can do to play the odds, while allowing yourself a margin of safety.

Most of the time, you can ignore this and simply go with your gut; it’s when the effort required is close to the limits of what is acceptable that this assessment is most likely to be unreliable.

A “Specific” image is one in which most parameters of image content are fixed and unalterable, but the object is nevertheless sufficiently popular as an image subject that there is a reasonable likelihood of finding something that can be manipulated to be “close enough” for game purposes.

In the first post in this series, I discussed the cars that the PCs could choose between in my Zenith-3 campaign. One of those was a Sky Blue and Burgundy Sedan DeVille. The image that I found for this car was White and Burgundy. It had to be edited to provide the image that was ultimately used. In fact, about half of the cars on the list needed image alterations of this kind. Most of these changes took less than an hour, a couple took a full evening each. All told, the available prep time meant that this editing took a little less than two weeks.

The only reason they held enough Game Value to justify this was because we were in Covid Lockdown, and hence I had substantially more prep time than would normally be the case. In normal times, I might have edited the images representing the cars that the PCs chose after the fact, but not doing them all in advance.

There is also the possibility of ‘amortizing’ the effort over multiple adventures.

One of the ongoing ‘bits’ in the Zenith-3 campaign is the mysterious appearance on one character’s pillow each day that she is where she is expected to be of a set of perfectly-fresh gourmet muffins and an exotic, even magical, coffeepot. This always has two stalks, each of which dispenses a different beverage designed to ‘match’ the muffins.

Every time this coffeepot has appeared, it has been a different color. It took a while to isolate the coffeepot from its background the first time it appeared, but because I saved the file in Krita’s native format, it took only a handful of minutes to change its appearance the next time.

The image to the right is a compilation of the four colors that it has manifested so far in the campaign.

Oh yes, when the last cup of beverage has been drawn from within its depths and it is set down, it vanishes the first instant that no-one is looking. What’s more, the players have discovered that if they stay put in a strange place for long enough, the muffins and coffeepot will start showing up there – it just takes them a couple of days to find the PCs.

This involves a more sophisticated calculation – first use and ongoing use – because it can generally be assumed that ongoing use will require a lot less time than doing the initial image editing, while the recurrence increases the Game Value of the illustration.

Technically, because the examples are all supposed to be the “same” coffeepot, this would be a unique object; but because the images are different each time, it should be treated as four separate objects.

I use the same logic and technique when I have the TARDIS materialize in the Dr Who campaign – a different background each time, and I have three different “pre-faded” materialization to choose between. It takes very little time to marry a chosen background and a prepared Tardis Materializing (it usually takes a lot more time to find the right background, to be honest).

Because the baseline Game Value of these images is higher, anything on the exceptions list becomes a high priority.

Unique

Unique images have to match specifications so exact and so distinctive that the chance of finding one of them in an image search is essentially nil, and there is a near-certainty that a base image will have to be edited to get what you want.

Such images do not all have the same Game Value as illustrations, however. It depends on how central they are to the story. What can also be said is that even those of comparatively low Game Value – “Magic Sword,” for example – this value may be high enough to justify a search.

It’s usually a lot less work to find a suitable image and alter your description to fit, than to create a bespoke image. For that reason, it can be a good idea to do a lot of your image searches as you go.

There is a compromise approach that should also be borne in mind: curating a list of required images, with some sort of priority rating. That rating tells you how much effort to put into the image, taking everything discussed here into account. One-star might be “quick search, take anything remotely compatible”; two stars might mean “ten minutes search, accept only good images of a decent size”, and so on; that’s up to you.

As a general rule of thumb, I like to have two or three alternatives to pick from – sometimes more – as I may have said already. There are times when an image that is ‘good enough’ (or even ‘perfect’) pops out of the woodwork, and is accepted immediately, though.

Next: NPCs.

A False Start

Last week, I was discussing an image that I had composited for the pulp campaign with my co-GM.

I made an off-hand remark about how justified I felt about expending the time and effort on the image.

He agreed, positing that the greater the demands on the imagination of a scene, the more useful it was to relieve that load, enabling the players to focus on the game situation.

That sparked a thought that seemed like it would make a good post for Campaign Mastery, so here we are.

My initial reaction – the thought that I mentioned – was that there were two spikes in the utility of graphic depiction – one that focused on the specific but big-picture, and one that focused on isolating a specific from the innumerable possibilities that arose from a more generic subject matter.

The chart to the right describes this first impression, and it’s an insight all on its own that it would be worth exploring in a post.

Unfortunately, a second thought a moment after showed that it’s half-false and half-misleading. There was a critical factor that this alleged relationship was completely ignoring.

A Better Beginning

The missing factor: the potential importance to the plot of a clear representation. This could be said to multiply by the commonality of the subject to give a more reliable index to be charted against the game value of a graphic representation.

So, an image could be of something quite common (a low uniqueness) and even if very central to the plot, it would have a low game value. Or it could be of something quite unique, which means that even if not central to the plot, it has some game value – but if it is central to the plot, its value as an illustration achieves astronomical value.

And, I was all set to write today’s article on this subject – but the presence of one additional factor implied the possibility of more, so before I committed myself, I invested a bit more time in thinking about the subject, and sure enough….

The Final Attribute: Availability

…I found that there was an assumption buried within the analysis to date: that all images desired would be equally available. It took only a moment’s reflection thereafter to realize that this flawed assumption would completely invalidate any analysis that didn’t take it into account, and that there was no one-size fits all answer to accommodate the attribute of availability.

The more I thought about it, the more relevance seemed to attach to this question of availability. At first, I thought there were three types of impact, but further reflection suggested that two of the three were closely related.

Availability Manifestation 1 – The Likelihood Of Success

The first primary impact is on the likelihood that you will find an image matching whatever specifications you might have, or one that is at least close enough.

A secondary outcome would be an image that could be quickly and easily modified to transform it from unsuitable to ‘close enough’ – introducing a sub-variant factor to consideration, which also has to be taken into consideration. The easiest way of doing so is to expand the concept of quality of result to include the potential for modification to achieve suitability.

Availability Manifestation 2a – The Importance Of The Hunt

A problem that needs to be considered is that you can’t know the outcome of a given search until you actually undertake that search. While you can make educated guesses about the likelihood of success based on the specifics of the search, and your capacity to compromise, there are perpetual surprises of both the good and bad kind.

Going on an image hunt is not unlike rolling a skill check – all sorts of things can be reflected in the bottom line, but in the end, it all comes down to luck and the scope for the manifestation of that luck. You can never be completely sure of the outcome until you roll those dice.

‘Availability’ is a critical assessment that can never be perfect. That means that the greater you can refine your estimates of the likelihood of success before you start, and in the early phases of the search, the more accurately you can assess the likelihood of success – and whether or not it’s worth you continuing with the hunt, or should switch to an alternative approach.

Availability Manifestation 2b – The Value Of Image Generation

Not everyone is as adept at image generation as I am. That should surprise no-one. There are also artists out there who can generate works that leave me awestruck in the same time (or less) as it would take me to churn out something just barely adequate.

Also, to be completely honest, sometimes works seem to go a lot more smoothly and quickly than I expected before I started, and at other times, the simplest job seems to have unanticipated complications that add hour after hour to the project.

Again, we have a ‘best guess’ fuzziness attached to each project; you can’t predict in advance with complete precision where that project will fall.

What can be said is that if you honestly evaluate each project relative to an accurate perception of your skills, the actual difficulty and project time will fit more-or-less on a standard distribution, a dumbbell curve, centered on the ‘best educated guess’. In the long run, therefore, your estimates should average out if you are honest with yourself.

But it’s possible to bias the odds in your favor by choosing a base image that is more suited to being manipulated to meet your needs than one that needs a lot more work.

There are all sorts of skills involved, and you get better with all of them with practice – so the bar of what is achievable in a reasonable time-frame keeps rising, and the accuracy of your estimates improves constantly. This means that it doesn’t really matter how adept you are with a (digital) paintbrush; being unskilled simply means that you have more scope for improvement.

Generating an image yourself is always a compromise over finding the perfect image ready-made. Sometimes, that’s a compromise worth making; sometimes, it isn’t – and often, you can’t tell until you are neck-deep in the project.

That’s the point of the (ongoing Image Compositing for RPGs series – collected hints, tips, and tutorials to boost that learning curve into the stratosphere.

“Availability’ is not a simple linear measure of the chance of finding an image that can be modified to your needs; each possible answer also has to be evaluated with respect to the amount of work required, and the amount of prep-time available. It might be that a less-acceptable image with a lower overhead is going to be a better, more practical, choice than one that would yield a better result but would require three times as much work to complete.

Game Value vs Availability

It’s worth taking a moment to define exactly what the goal of the analysis was supposed to be. What I wanted was to develop a schema that identified the relative value attached to searching for, or manufacturing, an image. Such a schema would enable a master listing of priorities that would yield the maximum ‘bang for buck’ for image searches and define a threshold point at which the generation of images (if one could not be found) was worth the time it was likely to take.

I’m not entirely sure that this goal is still a viable one, given the complexities that have now been introduced – but any guideline is better than none. So it’s still worth making the effort. In order to yield a result from these efforts, though, I intend to simplify outrageously. I’m not looking for a definitive set of answers, at least not anymore. A process for making decisions, in which each of the considerations is taken into account, is going to be more universally useful.

Strata of Commonality

If you take another glance at the prototype chart reproduced above, you will notice that Commonality has been divided into four classifications. This still makes a great starting point, so let’s look at each of them.

Mundane

‘Mundane’ incorporates all sorts of everyday items. Most of these will have very limited game utility; what limited functionality they provide can also be achieved through more important images, at least most of the time.

In general, mundane objects should not be graphically represented. It is said that one picture is worth a thousand words – if you can’t see the need for at least 500 words being spent on the description, it’s just not worth searching for them, outside of some specific exceptions.

Vehicles are a special case, so don’t worry about them for the moment. We’re talking pots and pans and treasure chests and the like.

There is a huge contextual element here, however. Space suits may be mundane items in a sci-fi environment, but they would be exceptionally rare in a fantasy one – and, in a steampunk environment, would have a completely different look-and-feel. What’s more, even in a sci-fi campaign, the graphic depiction of a space-suit does so much to ‘sell’ the genre that they have a game value beyond the mere fact of their ubiquity.

Perhaps the greatest value of mundane objects is as ‘set dressing’ to enhance generated images of greater value. These can be transformative – take an image that could belong to any number of settings and toss in a mundane object of greater specificity, and you have suddenly nailed the scene to a far more specific setting.

Throw in an object that clearly doesn’t fit within that range instead, and you introduce a deliberate contradiction that serves as a visual metaphor for a far more complex situation – so much so that the results probably belong in the ‘unique’.range, below.

Common

There are two ways of interpreting ‘Common’. The first deals with generic backgrounds and setting illustrations – these can be useful as foundations of more specific images, or for the capture of genre / setting atmosphere. Or they can impart absolutely nothing of significance. But they tend to be easy to find, so the return on invested time can be relatively high.



Image by ELG21 from Pixabay

For example, a generic image of snow-topped mountains doesn’t add much specific information about a setting – but they carry a sense of grandeur, of sweeping epic scale, that even a thousand words might not be able to convey. Atmosphere alone can justify their use as an illustration.

The other is where the subject is so common that dropping the label is going to result in fifteen different interpretations in the minds of players. For example, “Innkeeper” – everyone will immediately have an image in their reminds as to what he (or she?) looks like, and everyone’s interpretation will be different.

Now, that might not matter – or it might be critical. It all depends on the role of this particular NPC within the adventure.

You can even take advantage of the multiplicity of impressions. For example, you tell the players that they are greeted by an innkeeper in a surly tone, then give each player a strip of paper and ask them to write a line describing what they imagine the innkeeper looking like – race, dress, features.

From that time forward, those descriptions are how each PC sees the NPC – until something happens to reshuffle the strips of paper.

But you don’t tel the players this – you simply incorporate visual elements that no-one else can see into the narrative as the scene proceeds. “He accepts your rebuke of his manners and tips his hat at you in apology.”

“His hat? I thought he wore a leather cap?”

“He does, with a broad rim and silver buckle”….

The players may not know What is going on, but will soon be in no doubt that something more than meets the eye (quite literally!) is transpiring!

Once again, the field can be subdivided, though into more distinct and nuanced classifications than just “generic” and “atmospheric”. I’ll have more to say about that when I get into specifics a little later in the article.

Specific

The more specific your search subject, the lower the likelihood of an exact match, and the more you either have to compromise or edit whatever you find to make it fit for purpose. In fact, the likelihood of any result is markedly lower.

That can be compensated for, to some extent, by making more exhaustive searches. You can also sometimes find results that won’t show up in Google Search results by trying a different search engine – I’ve noticed that they usually all give different overall results with a few images in common. My search priority these days is usually DuckDuckGo, Google, and Bing (in that order) and I’ve recently added a fourth string to my bow, Qwant.

After that, some of the big image repositories and clip art providers, like Pixabay sometimes have images that none of the search engines seem to find. I keep a large set of links to these for the purposes of searching out illustrations for Campaign Mastery, anyway. You might not have any equivalent justification, but don’t let that stop you. Avoid (as much as possible) sites that watermark their free images, though. I include Wikimedia in this category.

The deeper that your search has to proceed, though, the more you encroach upon the point of simply not finding what you are looking for. Being aware of this, I also open any images that could potentially be edited to give the desired result, preparing the ground for a potentially-necessary Plan B.

It is also important, when making such assessments, that the points made earlier are kept in mind – “is it actually worth the investment in time to edit this image into something with game value?” – the more time you spend in the search, the less time there is available for such editing; there comes a point where the answer to the italicized question becomes “no.”

Unique

This is really an extension and extrapolation of the trends identified in the last category; the only real change is that the Game Value increases massively by virtue of the “unique” label.

When it comes down to it, with very very few exceptions, “Unique” images follow one of two paths:

• You perform all searches with the expectation that whatever you find will require editing to convert potential game value into actual game value; or
• You choose the best image that you can find and modify the adventure to fit, not the other way around.

In terms of time, choice two is the obvious winner. So if I find an image that will “work” with this approach, that’s job done and move on to the next; if I don’t, then I’m looking at the ‘editable’ options that I have gathered in the course of the search. Pursue both options simultaniously, in other words.

Classification of Image Categories

Okay, let’s get down to brass tacks. I’ve divided image subjects into six categories. I’m then going to look at how each category intersects with the Strata of Commonality just discussed, permitting advice that is as specific as I can make it. The six categories are:

1. Objects
2. People (NPCs)
3. Monsters & Encounters
4. Vehicles
5. Locations
6. Events & Effects

There is some overlap – “Unique People” clearly overlaps with “Encounters”, for example, and “Objects” can include “Vehicles”. These overlaps were necessary to ensure comprehensive coverage.

Objects Vs Vehicles

If the PCs involvement / engagement with the vehicle is enough that it may require depiction of the interior, then the “Vehicle” category is the important one, with the “Object” treatment (exteriors only) a fallback position.

If there is not likely to be any Game Value in the interiors, then the Vehicle is an “Object”, and the advice attached to that category should be your guide.

Vehicles like motorcycles that don’t have an “interior” require a further exercise of mental gymnastics – pretend that the vehicle in question is actually one that has an interior and then assess the engagement as above.

People Vs Encounters

The “People” category has two non-exclusive objectives – implying the personality of the individual, and providing a common mental image to the players to aid in recognition of the NPC as an individual. If neither of those is the purpose of the image, it should be treated as an “Encounter”.

Encounters are more related to what the individual is doing in the image, which should match up with the plot purpose of the encounter. “Why is this encounter happening?” at a meta-plot level is the guiding principle.

I want to conclude this overview section with a comment about player agency. Giving players a choice is always a better choice than not, but it does make image searches more complicated. For example, in the Zenith-3 campaign, the plot had the players buying a couple of vehicles from amongst the options at a pair of used-car lots. What attributes the players chose to prioritize would dictate which options best matched; since I didn’t know this, I had to prepare for a large number of contingencies. I described the situation and the prep involved in How Good Is That Rust-bucket In The Showroom Window?

There were more than 200 vehicles in the spreadsheet that contained the results. After the players decided on their key parameters, from highest to lowest, I simply had to sort the results accordingly to determine which vehicles best matched. Then it was just a matter of playing around with the budgets to derive a set of choices. I could simplify this somewhat because these were to be bought from used-car lots that were no doubt trading in vehicles even as the PCs were searching – any option that yielded too many possible combinations or that didn’t look to be “fun” / interesting, I simply marked as “sold” while the PCs were looking around the lot.

Because I could perform “theoretical” sorts in advance, I could manifest a shortlist of the options that would represent the options to be put before the players. This ended up being a list of twelve vehicles; there were more, but another key metagame factor was whether or not I could find an appropriate or editable image of the vehicle.

For those who may be interested, the twelve second-hand cars offered to the PCs in post-Ragnarok 1986 were:

• 1983 Black Coupe DeVille Cabriolet d’Elegance
• 1982 Sky-Blue and Burgundy Sedan DeVille
• 1984 Black Escort LX 5-door Hatchback
• 1984 Beige Escort Series I Liftback Wagon
• 1984 Black Escort 3-door Hatchback
• 1984 Navy Blue Cadillac Sedan DeVille Automatic
• 1983 Lincoln Continental Mark VI Pucci Designer Edition – one of the two chosen
• 1984 Brown Chevrolet Cavalier Station Wagon
• 1983 Cherry Red Chevrolet Cavalier, badly faded
• 1982 Purple Buick Skylark Station Wagon – the second vehicle chosen, but it turned out to be a good-looking lemon
• 1983 Bright Red Chevrolet Cavalier – the vehicle chosen as a replacement for the Lemon above.

Each of these had their virtues from the PCs perspective. That meant that each had to be presented to the players when they were looking at their different purchase options – I had no idea which ones the players would eventually choose. To get two cars, I had to illustrate 12 (plus three interior views).

That’s a lot more work than shows in the ultimate results. Player Agency is the enemy of efficient prep – but that’s a necessary evil. The only real restrictions placed on the players choices were (1) that I could find / make a suitable illustration of the vehicle; and (2) that I wanted them to end up with two different makes and models so that I could compare and contrast the two, and so that the players choices had measurable impact on the game-play. Because if it makes no difference, it’s not really player agency, is it?

Where To From Here?

This article is now approaching an unmanageable length, given the available time. So I’ve decided to break the rest of it into smaller mini-posts, which I’ll deliver over the next week or so, a day or two apart. Each will examine one of the specific Image Categories listed above, breaking each down into the four levels of Commonality.