The Twists haven’t stopped yet!

rpg blog carnival logo
This month’s Blog Carnival, hosted by Campaign Mastery, isn’t finished yet!. The subject is still “With A Twist” and it covers anything about Surprises, the Unexpected, etc.

I started with an article on the rules interpretation of Surprise, and followed that with a two-part article looking at types of Plot Twist that would work in RPGs after discovering that the literary types all had problems when applied to a communal format (Part One, Part Two). After a mid-carnival break, I came back to the subject to look at the plot potential of the unexpected gift.

Next week, I have one final salvo to fire in the Blog Carnival department, plus the wrap-up at the start of January, but for now: By definition, the one thing that is supposed to be a surprise, by definition, is the result of a die roll…

dice-in-blue-vortex

I thought I’d throw out a post for everyone from absolute beginners to experienced GMs today, about die rolls, and a few little tricks that I use regularly.

In particular, I want to look at all the exotic dice that are out there, and what they can be used for.

This article is divided into three sections. First some basics, then some practical solutions for random-rolling of values that are frequently needed, and finally the dice roster.

With well over 40 sections and sub-sections to get through, I can’t spend much time on any one section (and there should be something for everybody), so let’s get busy…

Flat Probability

When you roll one die, and each of the sides have the same chance of coming up, you have what’s called a flat probability when you graph the chances of getting each result. It doesn’t matter too much what that chance actually is, just that it’s the same for every result.

2 dice probability

As soon as you add a second dice to the mix, this changes. Instead of a flat probability line, you now have something often described as a curve but which is, in reality, a stepped triangle. That’s because dice don’t roll completely randomly, they only roll integers, In other words, we’re talking about rounding error.

If you count out the number of chances of getting each result on two dice, do a quick table with one die roll across the top and another down the side. Fill in the possible results. When you do this, you find that there’s 1 chance in whatever of getting the minimum result, 2 chances in whatever of getting the next highest result, 3 of the one after that, and so on up to the integer of the average result. Then it starts back down, until you get to 1 chance in whatever of the maximum result.

Since this always happens, once you understand it, you will never go through the tedium of calculating the table again – you’ll just write it out.

The average result of two dice

Most dice I’ve ever seen that aren’t designed for cheating, or rolling averages, have results that run from 1 to maximum without gaps. That means that the average on each is half the maximum result, plus one-half.

  • The average of d4 is 2 + 0.5 = 2.5.
  • The average of d5 is 2.5 + 0.5 = 3.
  • The average of d6 is 3 + 0.5 = 3.5
    …and so on.
Outcomes

The number of outcomes is often something you need to know. You can work it out by multiplying all the maximums from each dice together. So there are 16 possible results from 2d4, 25 from 2d5, and 36 from 2d6. The difference between the maximum and the minimum, plus one, tells you how many results these are spread amongst. So the 16 possible outcomes on 2d4 are spread over 8-2+1=7 results; the 25 possible outcomes on 2d5 are spread over 10-2+1=9 results; and the 36 possible outcomes on 2d6 are spread over 12-2+1=11 results.

Multiple Dice averages

To average multiple dice, simply add the averages of the individual dice together. So:

  • The average of 2d4 is 2.5 + 2.5 = 5.
  • The average of 2d5 is 3 + 3 = 6.
  • The average of 2d6 is 3.5 + 3.5 = 7.
  • The average of d4 and d6 is 2.5 + 3.5 = 6.
  • The average of 3d6 is 7 + 3.5 = 10.5.

Which brings me to:

Roll_of_3d6-sm

3 dice probability

Now we’re getting a proper curve, as you can see. In fact, what we have here is commonly known as a bell curve, or even a normal distribution – which is to say that there’s a section in the middle where results are far more likely, and where the average lives, and flatter lower sides where the extremes may be found. The shape is symmetrical, ie the part that’s above the average (right side of the diagram) is the mirror image of the part below it (left side of the diagram).

To work out what the chances are of getting any individual result, list one dice down the left and the tally from the rest across the top. Then, starting with the first row, copy the tally. With each subsequent row, start from one further to the right. When you’ve finished add them all up and that’s the number of ways that you can get the result indicated by the tally line. See the (partial) example below for 3d6.

The patterns should be fairly obvious.

The patterns should be fairly obvious.

So if you want to know what the chance is of getting, say, exactly 8 on 3d6, you take the tally, 21, divide it by the number of possible outcomes (6 x 6 x 6 = 216) and multiply by 100 to convert to a percentage – 9.7222222222%. Call it 10%.

Or, if you want to know what the chance is of getting eight or less, add up the tallies from the 8 result down (1 + 3 + 6 + 10 + 15 + 21 = 56) and divide by 216, and multiply by 100 for the percentage: 25.9259259259%. Near enough to 26%.

The Middle Third

I always find it useful, now and then, to know the middle third of a frequently-used die roll. That’s the result that discards the lowest 1/3 of the outcomes and the highest 1/3 and tells me which results are most likely to occur.

This sort of analysis makes that fairly easy. 33 1/3% of 216 is 72, so I simply need to count and exclude the bottom 72 of the accumulated tally. I already know that 8/- is 56 of that 72. The next tally is 25, for 9/-, which brings the total to 81, well in excess of the 72 target. So the middle third starts at 9 and runs through to the number on the far side that also receives a tally of 25 outcomes, 12.

The middle third of 3d6 is in the relatively narrow range of 8-12. If your target for success (needing to roll low) is less than 8, you have a worse than 1-in-3 chance of success, and so will probably fail any given check. If the target is 13 or better (needing to roll low), you have a better than 2-in-3 chance of success, and so will probably succeed on any given check. If you need to roll above the target number, these are reversed in sequence but the numbers still apply. Only in that middle third are chances so even that you can’t predict with any reliability what is going to be the result of any given check, success or failure.

I was going to include a table of common “Middle Third” results but decided not to, for two reasons:

  • First, there are too many combinations for one to be practical without being overly lengthy;
  • Second, they are so easy to work out using Anydice – use 3d6 and the data given above, have a play around, and you will soon work out how; and,
  • I ran out of time – which is probably the most important reason.

As a player, my goal in any encounter is to – at minimum – get my chances of success better than the low chance through manipulating circumstances into my favor. If possible, I also aim to get my enemy’s chances below the high by taking away advantages that he might have.

As a GM, my goal is to arrange circumstances so that the players are on the wrong ends of these numbers, but have the capacity to swing things the other way. They start out dealing with overwhelming opposition and, one-by-one, strip away the enemy’s advantages while adding to their own, until they end up with either a fair fight, or better yet, one in which they have the advantage.

At lower PC levels, this doesn’t make so much of a difference; characters have so few hit points that the fight is over before these subtleties really have an effect. With increasing levels, combat becomes more and more tactical in nature (at least in theory), and nuances become very important.

Sidebar: Average Blows To Death
Another value that I use in conjunction with the middle third on a frequent basis is “Average Blows To Death”. There are two scales to this: PC and Normal.

The PC value takes the weapon that the attacking character most commonly employs, determines its average damage, and adjusts for the chance of a critical hit; it then divides the total HP of the opposition within the encounter by this amount. The result is the average number of successful blows that the attacker has to succeed with in order to kill the opposition. Dividing by the chance of a successful hit gives an indicator of the number of combat rounds a battle is likely to take. Allow an extra 25% on the top for rounds spent maneuvering, and the results are usually pretty close, and a vital planning tool.

The normal value takes a typical NPC (1st level, if your game uses levels) and performs a similar calculation. This gives a clue as to the fearsomeness and general impression of the creature being attacked, another vital tool in the planning process.

more dice probability

You can keep adding more dice to the total using exactly the same technique. 4d6, 5d8, 17d4 – whatever you want to know. What you will find is that the more dice you add, the steeper the sides of the central curve get – though it’s not always obvious because the number of results contained within that central section of the curve also increases.

There’s a wonderful table that I found at Dragonsfoot which charts as percentages the shape of the curves for 2d6, 3d6, 4d6, and so on, all the way up to 9d6. It’s about half-way down this page.

Of course, if you need to calculate an exact result’s chances, curves like this aren’t all that useful; you need tables. There is a shortcut that may be of a great deal of value to you when this happens.

If you tally the results for a d6 across the top of a table, and b d6 down the side, you can quickly work out the chances of any given result from a and b, and therefore for a+b, by multiplying the respective tallies. To save table space, it’s a lot easier to write the totals for a+b in the same cell in a different color. Here’s a partial example, showing 3d6 by 2d6:

Calculating the outcomes of 5d6

Calculating the outcomes of 5d6


Tallying the 5d6 results

Tallying the 5d6 results

Look at these tables closely and the patterns should become fairly obvious. You use the top table to generate the entries for the bottom table. Any box in the top table that gives a red 5 result goes next to the red 5 in the bottom table, and the same for a result of 6, and 7, and so on, all the way up to the highest result possible (18+12=30). Once you understand the principles, you can work this trick with any combination of dice – you might have 3d8 across the top and 2d12 down the left, or anything else you can come up with.

One or two hints:

  • Always show your working. The number of times that I have gone “…4,5,6,7,8…” when I meant “…4,5,6,5,4…” beggars belief – and I know what I’m doing. Seriously, its almost impossible to find an error if you don’t. And there WILL be errors.
  • It’s often a lot easier to use every 2nd column instead of single columns as I’m doing here, but I wanted to clearly show both the result and the tally contribution in the same cell.

If you look to the right, you will see how to tally the results from the table above. There’s only one cell with an outcome of 5 in it, so that cell’s results stand alone for the 5 line. There are two cells with outcomes of 6, so the total of the tallies of each is the number of outcomes that gets you a six. There are three cells with outcomes of 7, so the total of those three is the number of outcomes on 5d6 that equal 7, and so on. The maximum number of results on a line is the smaller of the two table axes – in this case, the 2d6.

And, if you plot out the results of the 5d6 tallies, you end up with the curve below:

5d6 bell curve

5d6, generated using anydice.com, click graph to visit the site

More and ignore highest (or lowest)

These techniques don’t work when trying to calculate NdX and ignore the highest (or lowest). To calculate this, you can’t simply accumulate the results, you need to work out every possible combination. There are still shortcuts, but they are nowhere near as short or as pretty.

Fortunately, there are several sites who have done this sort of maths and graphing for you.

For 4d6 and drop the lowest I recommend this page.

Next best is the graph below, also generated using anydice. Unfortunately, to make it fit, I’ve had to reduce it in size.

4d6plusminus
4d7plusminus2

The first thing you should notice about the above is that the probability curve of “4d6 drop lowest” is the mirror image of “4d6 drop highest”, reflected about the average result of 3d6. When you think about it, this is exactly what you would expect to be the case, but it’s a great confirmation that I’ve done it right!

I’m a great believer in having the numbers to go with a graph – you never know when you’ll need them, as I’ve learned the hard way on a number of occasions. So, to wrap this section up, to the left are the actual results, obtained from anydice once again and fed into a spreadsheet – then pimped to look pretty:

Mixing Dice

These principles and techniques work regardless of the combinations of dice that you need. If d8+2d6 is what you think you need, the three-dice technique gives you numbers.

So why might you want to mix dice?

Answer: to get the probability curve that you want, with the maximum, minimum, and average result that you want.

For example: 3d6 gives a standard curve. Replacing one of those 3d6 with a d8 and tacking on a -1 gives you exactly the same shape of curve over the 3-18 range. but also extends the range of results down to 2 and up to 19. Replacing a second one and tacking on another -1 gives you a 1-20 roll that is very different in probability to a d20:

bell curve 1-20

Alternatively, you might want to replace one of the d6 of with d4+2 – which gives a 3-18 range, but boosts the average result by 1. Or even all three, to get 3d4+6 – which boosts the minimum result to 9 without changing the maximum.

Any time you want a range of results that bias toward the average, you’re talking about using multiple dice – and that requires understanding them.

Percentage Conversions

Some people convert a 3d6 result to a percentage by simply dividing by the maximum result, and then wonder why it doesn’t add up to 100%. They have neglected the effect of the minimum.

A slightly more sophisticated group simply subtract the minimum from the maximum and wonder why that doesn’t work, either. This ignores the fact that the minimum result is also a valid result.

To convert a range to a percentage, you have to spread the 100% evenly over a range equal to MAXIMUM-MINIMUM, Plus 1.

Why is this important?

There are all sorts of occasions when what you want is a flat roll, and others where you want a “normal” probability that clusters around the mean. On still others, you need still more complicated results. For atmospheric temperatures, for example, where you need two different normal probability curves on different scales – and, occasionally, a d% to fill in the gaps. But I’ll get to that a little later.

Which brings me to part 2 of this article…

Creating Small Custom Tables

So you have a list with an odd number of entries – 11, say – and you want to turn it into a random table. This is easy to do when the number of entries is exactly the same as a standard die size, but that’s why I made it an odd size. The easy answer is to make it a d12 table, and if you can’t come up with a twelfth entry, you can simply make it roll again.

But there’s more than one way to skin a cat. The d12 solution works if you want a flat probability curve – but what if there are some results that seem more likely to you than others?

There are two obvious ways to handle this with a flat-probability die: minimum-and-add, and maximum, distribute, reduce, distribute. And then there’s a trick with multiple dice that can sometimes be simpler.

Minimum-and-add

You need a die size that’s bigger than the number of table entries required. Divide the size by the target and that gives you the “minimum” per entry. Then distribute whatever’s left as you see fit. The bigger the die relative to the target, the more precision and flexibility you can get.

  • 11 on d12 gives a minimum of one and leaves one. So you can make one entry twice as likely as any other. That’s a fairly blunt weapon.
  • 11 on d20 gives a minimum of one and leaves seven, giving you lots of entries to spread around. So that shows rather more finesse.
  • 11 on d% gives a minimum of 9 and leaves 1 – not a lot of room – but reducing the minimum to 8 leaves 12, or reducing the minimum to 7 leaves 23 – and that’s a lot of wiggle room.
Maximum, Distribute, Reduce, Distribute

The flat-roll alternative is to take a large die like d%, decide how big the most common result will be, divide the remainder over the rest, reduce by one each, and distribute the excess.

  • Most common result set at 25%: leaves 75 to distribute over the remaining 10, so 7% each, and leaving 5 remainder. Reducing by one to 6% baseline gives an additional 10 remainder for a total of 15 to distribute. So you could have a 25-17-11-6-6-6-6-6-6-6-6 pattern.
  • Most common result set at 20%: leaves 80 to distribute over the remaining 10, so 8% each. Reducing by one gives a remainder of ten to split up and spread around – probably adding 5% to the likelihood of the next two most likely results, giving 20-12-12-7-7-7-7-7-7-7-7.
  • Most common result set at 15%: leaves 85 to distribute over the remaining 10, so 8% each, with a remainder of 5. Reducing by one gives a baseline of 7% and leaves 15% to distribute. This can give a 15-12-10-9-9-8-8-8-7-7-7 pattern, a reasonably smooth curve.

In fact, there’s a shortcut that I often use: With a baseline of 9 under the minimum-and-add, I double it and round both up and down to get some idea of where the optimum pattern is likely to lie. More often than not, rounding down will be the better choice.

From this starting point, it’s easy to further tweak the resulting table. Taking the 15% maximum pattern and dropping the least likely to a 5 would permit +1 to the 10 and +1 to the first 9, giving an even smoother 15-12-11-10-9-8-7-7-5 pattern. Taking another 1 from the second seven to add to the first, or to the 5, further smooths the curve.

exotic solution

An exotic multidice solution

Here’s a solution that I’ve found useful a couple of times. The image to the right explains it – you start with a d12 or whatever die is larger than the list. If you go beyond the bounds of the solutions listed, you get “the roll to the right” result, dropping a die size. Keep going until you get to d4.

In the example offered, that means that there’s a one in twelve chance that is divided amongst 1 to 9 by the d10, and a one in 120 chance that is divided amongst 1-7 by the d8, and so on.

To save time, you can even roll all the dice at once, since they are all of different sizes.

Usually, when I employ this technique, I also save space by using different colors of text instead of explicitly including a column for each die type.

You can also spread the probabilities out a lot more by going up an additional die size without extending the table – so the first roll is on a d20 instead of a d12, the next is a d12 instead of a d10, and so on. That would mean that it’s 45% being divided up by the d12, and 15% being divided up by the d10, and so on.

I don’t use this very often, but it can be a useful trick to have in your back pocket.

Creating Big Custom Tables

How big is big?

Using a d6 x d6 array, two rolls give you a table for 36 random choices. Make it d10 x d10 and you have 100 – though why you wouldn’t simply use d100 is beyond me.

I once created (and will one day publish as an e-book) a table which used d20 to select from 20 subsequent tables of 20 personality traits and guidelines for integrating the results. An additional roll specified the number of rolls to be made on the table for any individual. That’s effectively a table with 400 entries, 20 across and 20 down. (The problem is that this doesn’t seem to go far enough these days – I have a lot more entries to add – but I can’t assume access to a d30. So I need to rethink the structure a little). And table weighting is another serious consideration. Anyway…)

These are all flat tables – each different result on the die rolls yields a different result on the table. They are relatively trivial problems. A rather more complex problem comes when you construct Encounter tables

…which I intended to demonstrate at this point. I soon realized that it was too big a topic and would have completely overshadowed everything else in this article, while not being granted the recognition that such a big topic warrants. So I made the decision to spin that off into a separate article, which I’ll publish sometime in early January. It probably won’t be in time for the Blog Carnival, unfortunately, but you can’t have everything.

AM/PM

Probably the easiest thing of all to determine. Roll a die with an even number of faces – if it comes up high, it’s PM, if low, it’s AM.

Hour of day

In conjunction with AM/PM, this is easy to determine – just roll a d12. If you would rather get more technical, you could ignore the AM/PM and roll a d24 (they do make those, don’t they?) but this seems an unnecessary complication.

Morning/Afternoon/Evening/Night

The obvious way of determining this is by rolling for the hour. But sometimes that’s more information than you need; a d4 does the job just fine.

But it’s not always that simple. In summer, morning and afternoon are both longer than Evening and Night; in winter, it’s the other way around. Daylight savings may rob afternoon of an hour and give it to morning. And you might want a separate indication of Sunrise and Sunset as well. When you list out all these little bits and pieces that you might want to know, you end up with 16 slices of time of differing lengths.

There’s no easy way of expressing all this complexity in a die roll with any accuracy. So, a long time ago, I came up with a table to simulate it as simply as possible. It’s a little imperfect, as close examination will reveal, but it’s not too bad.

morningafternooneveningnight

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Also available as a PDF for your convenience, click icon to download

Spring/Summer/Autumn/Winter

There are times when you don’t even know what season it is – and need an answer. As with the period-of-day question, this can be simplified to a d4-roll-and-get-on-with-it approach, or you can dig into the quagmire of complexity. Northern Hemisphere? Southern? Latitude? Altitude? Climate? Variation? Sunspots? El Niño? Trade Winds? And on, and on, and on.

The easiest way to avoid all this complexity – well, to build it in under the skin of your die rolls – is to look at the topology of your location, then map it to an equivalent point on Earth – a point for which reliable and detailed climatic information can be accessed. Look for average daily maximums for each month of the year, then simply roll a d12. As a bonus, this is likely to give you information on average minimums, records maximums and minimums, precipitation, and so on. Next to talking to a local about the weather – at great length – this is as good as it’s going to get.

Of course, the more complex your game reality, and the more it differs from our terrestrial experience, the more difficulty you’re going to get yourself into. What is the relationship between climate and axial tilt, to name just one question that cannot be fully answered. We’re still getting to grips with Earth’s climate, and finding that it’s always more complex a question than we thought – never mind trying to take values that are fixed and turning them into variables!

Month Of The Year

A bit of an anticlimax, this – until you wonder how difficult it would be to take into account the different lengths of the months, and then you are suddenly in deep, deep, kumpcha (unless you use some sort of software die roller that lets you specify die size).

In order to work out the month of the year by days since January 1st, you’re looking at a d365 or d366, and knowing when to use which, and all sorts of other complications. Frankly, the easiest method that I’ve come across is d20 x 20 + d20 -20. This is the equivalent of a d400. Then just re-roll anything over 365/366, which will happen roughly 9% of the time. That can be cut back by realizing that a 20 on the first d20 is an automatic re-roll, and a 19 is almost certain to be a re-roll.

Random Month – bias winter

So you don’t know what time of year it is, but you know that it’s more likely to be winter than summer? There are two ways of handling this. The easier way is to roll 2d6 to get the month, with “1” being the middle month of summer. The first alternative is to roll 3d4-1, with “1” meaning the same thing; and the second alternative is to roll 2d4+2, again with “1” being the middle of summer.

The 2d6 still gives a chance of getting a summer result; it’s just not very likely, about 16 2/3 %. The 2d4 approach rules summer out of the question entirely, ruling 4 months of the year out of bounds on the results. That leaves the 3d4-1 approach; technically, it has to be “3d4-0.5” in order to get the same average as the other results. Making it 3d4 biases results towards the end of winter, leaving it as 3d4-1 biases results toward the start of winter.

Which may make it astonishing to readers who don’t know their math when I suggest that the 3d4-1 approach is actually the most technically accurate. That’s because it isn’t 3d4-1; it’s INT[3d4-0.5], and it identifies the month that the die roll places you in the middle of. In other words, if it says October, the actual result is read as “the middle of October”. With the others, they are indicating the start of the given month, which only qualifies as being that month by the skin of it’s teeth and an accident of the calendar. Really, what’s the difference between August 30th and September 1st, in practical terms? A day in the middle of, say, October, is technically as “October” as it is possible to get.

So, which one should you pick? The answer is to forget all the technicalities and decide how much bias you want toward winter. If you list them in order of increasing severity of bias, the order is 2d6, 3d4-1, 2d4+2.

Personally, I’m lazy when I can be; I’ll use 2d6 unless I have darned good reason not to.

Random Month – bias summer

The same results work perfectly to give a summer bias; just set “1” as the middle of winter.

Month of the season

I’ve sometimes needed to know whether something was happening, or going to happen, at the start, middle, or end of a season. I use a d6 for this: 1-2 is early, 3-4 is middle, and 5-6 is late.

This is actually more convenient than it seems. Most months have 30 or 31 days, which is close enough to 30 for our purposes; so a “1” indicates the first 5 days, “2” indicates the second 5 days, and so on. If you have an extra day, tack it onto the end – so a “6” can indicate the last 5 or 6 days of the month. The only month where this doesn’t work is February, and even that is close enough for practical purposes – and closer on leap years.

Minute of the hour

d10 x 6 + d6 -6 gives you the exact minute of the hour.

Usually, it’s close enough to use d12 x 5 – 5, which gives you 5 minute intervals – H:00, H:05, H:10, and so on.

Seconds of the minute

If you need to, do this in exactly the same way.

Latitude and Longitude

Okay, now we need d360-180, where results <0 are South or West, and results >0 are North or East.

The best way to get d360 is to use d18 x 20 – 20 + d20. But most people don’t have a d18 handy, even though they exist.

d12 x 30 – 30 + d6 x 5 – 6 + d6 is a d360 with errors. You can simplify it to (d12 x 30) + (d6 x 5) + d6 – 36.

A better method, because it reduces the frequency of errors is (d20 – 1 ) * 18 + d20. It’s still not perfect; you can still get results of 361 or 362, and some results like 19, 20, 37, 38, and more, seem to come up more frequently than they should. But it’s only two dice being rolled and a bit of calculation.

Still better is a way that does away with that calculation altogether, even though it involves slightly more die rolls – a d4, to be specific.

Better yet is ((d4-1) * 90) + ((d10-1) * 10) + (d10-1).

But the best answer is (d36-1) x 10 + d10. You get a d36 with (d6-1) x 6 + d6.

Step by step:

  • Roll d6-1.
  • Multiply by 6.
  • Roll d6 and add it to the result.
  • That’s the 10’s place. For the digits, roll a d10.

That’s fairly straightforward. Then just subtract 180 – or (far easier) subtract add 20 and subtract 200.

Day of the week

If I didn’t know the date or couldn’t consult a universal calendar, I used to use d8 and re-roll 8’s. But these days you can get 7-sided dice marked with the days of the week. I don’t know when I’ll need it, but when I do, I have it handy.

Day of the month

I’ve already hinted at the trouble that odd lengths of months can cause. Well, here they are again. There are three different solutions to this particular problem, and a variation or two on those answers to consider, as well.

d30 method

The simplest technique is to roll a d30, assuming you have one. Re-roll the result if it’s too high, and ignore the possibility that a month might have 31 days.

d16 method

If you have a d16, roll d anything; on high, add 16 to the result of a d16 roll. For practicality, roll both at once. Re-roll if you get a date that doesn’t exist, like February 30th.

If you don’t have a d16, you can simulate one with a d8 by rolling a separate d-anything and adding 8 to the result if the d-anything is high.

For example, using d6 for the d8-to-d16 roll and d10 for the d16-to-d32 roll:

  • Rolls: d6:4 d8:5 d10:3
  • d6 is high, so d16 result is 8+d8=13
  • d10 is low, so d32 result is d16+0=13.
  • If the d10 result was a 7, the d32 result would be d16+16=29.
d7 & d4/d5 method

If you have a calendar, you can take advantage of the fact that there are 7 days in a week and never more than 5 of any given days in a month. d7 gives day of the week, and d4 or d5 (depending on how many of that day there are in the month) gives the occurrence of that day, i.e. the exact date.

For example, in December 2014, the first was a Monday and there are 31 days, so there are 5 Mondays, Tuesdays, and Wednesdays, and 4 of everything else.

If you don’t have a d7, use a d8 and re-roll 8’s.
If you don’t have a d5, you can either use half a d10, or a d6 and re-roll 6’s.

My Choice

Because it gives extra info and is so much easier than anything else, my first choice would be the d7/d4/d5 method. If I didn’t have a calendar, my second choice would be the d30 technique – but I don’t have a d30, so I would fall back on the d16 method.

Quick Temperature

How hot is it outside? How cold? Temperature is one of THE big questions that needs to be answered regularly in any RPG. It’s so big a topic that it might have to be excerpted out into an article of it’s own, just as I did with Encounter Tables, but I’m hopeful that I can trim it to size enough for it to be one this article’s centerpieces.

There are two key facts that you need to know for the simplest solution: What is the average daily high or low, for this location, at this time of year, and how much variation can be expected from that?

Average Daily High & Low

These are relatively easy to find for real places. For example, if the climate is the same as Southern Italy, fire up Google Maps, zoom out, find Italy, and zoom in until you find a town of reasonable size (more likely to have weather information online) that looks like it has something close to the right Geography – ocean to the correct side, mountains in the right places.

Next stop is Wikipedia. Enter the town name and see if there’s weather info – occasionally the answer will be yes, more often no. If you don’t find it, go to Google and search for “[Town Name]” +weather the inverted commas and plus sign are very important. If that gets you nowhere, look for a neighboring town – I leave Google Maps up in a separate tab just in case – or an alternative. And sometimes it can be useful to zoom out one or two steps, when the town is simply too small – wait for it to disappear and look at the places that are left.

For example, Ravenna is located towards the north of Italy, is on the coast of a narrow sea, has a mountain range some distance away to the west and a bigger one some distance to the north. The city has a Wikipedia page but no climate info. Google pulls up a number of sites offering forecasts, but I ignore those; I want less current information and more long-term statistics. A “Trip Advisor” website comes to eye, as does another entry listing “average temperatures” – opening both of these gives a gold mine. The Trip Advisor site includes a chart showing average monthly maximum and minimum temperatures for six months of the year, and a great narrative description of the climate. This would be immediately snaffled! The Weather site proves to be “Knowitall.com” and contains a chart of daily temperature averages, both maximum and minimum, over the last four years – perfect! This would also be grabbed, if possible, for future reference – in this case, I can, but quite often this isn’t possible.

The problem you are likely to face is that you will often get winter low and summer high, and not a month-by-month value. That’s good enough – a little experience and one key fact is close enough for gaming purposes. But more specific information is better.

Variability

You either have this information, and need to analyze it, or you don’t, and need to make assumptions.

Frankly, you shouldn’t get too hung up on the analysis side of things; weather is so variable that it will make no real difference in the long run.

Let’s take a look at the weather chart from Ravenna, continuing the example from the previous section.

Text

A simplified version of the Ravenna Weather Chart from Knowitall.com.

When I first glanced at the original, five facts leapt out, which I have carefully preserved in the simplified version to the right.

  • Peak temperatures are fairly flat over the peak of summer, which is July/August.
  • In midwinter there is a bump in temperature averages – December and February are both colder than January.
  • After summer, the temperature range between day (maximum) and night (minimum) gets much smaller very quickly and the drops in both averages are very consistent through into December.
  • In comparison, the spring range widens suddenly in May and then narrows again, due more to the nights staying cool while days continue to rise in average temperature.
  • The minimum day-to-night range is about 6º C / 15º F; the maximum is about 11ºC / 21ºF. These ranges seem relatively narrow, until you realize that we’re talking AVERAGE temperatures. Any given day can be hotter than the maximum indicated, or colder; any given night can be hotter or colder than the minimum.

Once I know the time of year, I can determine the day’s minimum and maximum temperatures; once I know the time of day, I can estimate where in that range the temperature currently sits.

What you want is a “normal” die roll, not a flat one, and a modifier so that the average result matches the appropriate average temperature.

The only unanswered question is how much variability to have. In winter, the nighttime temperature doesn’t seem to vary very much, while the daytime temperature can be a bit more variable, in my experience; in summer, daytime temperatures can be very variable, while night-time temperatures vary to about the same degree as winter daytime temperatures.

A good rule of thumb is that the extreme results from the daytime average must never be colder than the average minimum, and the extreme results from the night-time temperature must never be hotter than the lower 1/3 probability mark of the average maximum. Yes, there can be exceptions on rare days, but this gives a guideline that’s close enough.

Forecast Daily Maximum

So the place to start is with the daily maximum, because we’re using the results of that to get the daily minimum.

It’s easier to explain with an example, so let’s pick a month – April – and see what happens in Ravonna.

  • Average Maximum is about 18ºC / 65ºF, Average Minimum is about 10ºC / 49ºF.
  • The difference in temperature is about 2/3 of the the die difference (maximum-minimum) that we want. In this case, 18-10=8ºC / 65-49=16ºF; multiply by 1.5 (round up if necessary) to get 12ºC / 24ºF.
  • With 2 dice, 2 is always the minimum result, so the range is maximum-1. With 3 dice, 3 is the minimum, so the range is maximum-2.
  • One half of (Range-1) tells us the X in 2dX that is needed.
  • One-third of (Range-2) tells us the X in 3dX that is needed.
  • For Ravonna in April, half of 12ºC-1 = 5.5 – and there’s no such thing as a d5.5. But 2d6 should be close enough. Half of 24ºF-1=11.5; 2d12 is indicated.
  • Also for Ravonna in April, 1/3 of 12ºC-1 = 3.67. So 3d4 is acceptable. 1/3 of 24ºF-1=7.7, so 3d8 will work.
  • April is in spring, but a glance at the Ravonna chart says it’s closer to Winter than summer. So I would use the fewer dice option – ie the 2d4 for ºC / 2d6 for ºF – because there is less range for variation.
  • Work out the average results of these die rolls, rounding down if necessary. Average of 2d4 is 5; Average of 2d6 is 7, no rounding necessary.
  • We need a modifier that turns these averages into our average maximum temps – a simple subtraction. 18ºC-5=13; 65ºF-7=58.
  • So the Daily Maximum temperature for Ravonna in April is 2d4+13ºC or 2d6+58ºF.
  • The Middle Third is always useful in these circumstances, and we need it for the Forecast Daily Minimum. These show that the most likely maximum temperatures are 17-19ºC / 64-66ºF.
Forecast Daily Minimum

This process is very similar, but instead of using the daily average maximum, we are using the lowest of the most likely maximums.

  • Lowest Probable Maximum is 17ºC / 64ºF, Average Minimum is about 10ºC / 49ºF.
  • The difference in temperature is about 3/2 of the the die difference (maximum-minimum) that we want. In this case, 17-10=7ºC / 64-49=15ºF; multiply by 2/3 (round up if necessary) to get 5ºC / 10ºF.
  • Half of 5ºC-1=4. So 2d4 works. Half of 10ºF-1=4.5, so 2d5 are indicated.
  • For the same reasons as before, I’ll be using the 2dX choice. I already know that, so there’s no need to calculate the three dice version.
  • Work out the average results of these die rolls, rounding down if necessary. Average of 2d4 is 5; Average of 2d5 is 6, no rounding necessary.
  • We need a modifier that turns these averages into our average minimum temps – a simple subtraction. 10ºC-5=5; 49ºF-6=43.
  • So the Daily Minimum temperatures for Ravonna in April is 2d4+5ºC or 2d5+43ºF.
  • The Middle Third shows that the most likely minimum temperatures are 9-11ºC / 48-50ºF.
Current Temperature

Current temperature is best worked as a flat roll between the two rolled extremes. But why bother rolling, or even working out a die roll? Use your experience and the time of day to work out where you’re at in that range, and estimate the current temperature.

Using ºC:

  • Roll 2d4+5 for last night’s temperature: 12ºC.
  • Roll 2d4+5 for tonight’s forecast maximum: 9ºC.
  • Roll 2d4+13 for today’s forecast maximum: 18ºC.
  • Determine time of day. I get mid-morning, which in late winter or early spring means that the temperature is about 2/3 of the way between last night’s low and as hot as it’s going to get. So I estimate the current temp to be about 16ºC.

Using ºF:
This is done exactly the same way, only the die rolls change.

Quick Weather

So you know the temperature. There are a lot of complicated ways to determine weather, but here’s the simplest I’ve come up with:

  • Roll d5 for the amount of cloud cover. 1 = clear, 2 = scattered cloud, 3 = sunny breaks, 4 = cloud cover, 5 = threatening, heavy clouds. Make allowances for desert environments, etc.
  • Roll d6+1. If the result is less than or equal to the cloud cover, it’s precipitating.
  • Roll d4 for the intensity of the rain IF it’s raining. 1 = light showers/snowfalls, 2 = hvy showers/medium snowfalls, 3 = solid rain/snow, 4 = heavy precipitation/blizzard/hail. Use temperatures to determine the nature of the precipitation (rain/hail/snow). If you roll a 1, it may be fog instead, depending on temperature and time of day.
  • IF it’s hailing, roll d8/2 for the size of most of the hail, in 1/2 cm or 1/4 inches. Exceptions can be 2-3 times this size.
  • Roll d8-1 for the wind strength, in 5kph / 3 mph units. If you get a natural 8, roll again and add 7. If you get a second natural 8, roll again and add 14. If you roll a third natural 8, roll d5 and add 21. If the total is 25 or more, roll d5 for the category rating of the hurricane.
  • Roll 2d4-1 for the strength of gusts, in units of +10%. If you roll a double-four, roll and add d4-1. Average result = +40%, 6.25% chance of needing the 3rd die, 1.56% chance of a result of +100% ie double wind speed.
  • Roll d8 for direction – it’s up to you whether this is the direction it’s coming from or the direction it’s going to. Override the results if they don’t seem to fit the terrain. I usually count clockwise with “1” being North.

This doesn’t account for all weather phenomena, but it covers the most common. And it’s very quick.

Random Choice off a list of unusual length

And that brings me to the piece of ordinary randomness that I use most often of all – where I have a list, and I want to select a random entry from it.

There are two approaches that I use to this process: One Flat Roll and Divide and Conquer

One Flat Roll

For this, you always want a flat roll, and the most important thing is to know the length of the list.

That length divided by 10 and rounded up, becomes the X in my dX +10 -1.

That always leaves a few entries left over; the difference between list length and remainder get excluded if necessary, to leave a short-list of entries to get the “extra chance”.

It’s very quick and relatively straightforward.

Example: there are 39 first-level Sorcerer/Wizard spells listed in the PFRPG Core Rulebook (it took me about three seconds to count them). So I want a d4 for the tens place on the roll and a d10 for the digits. I have one entry left over; I’ll decide what to do with it if I need it. I roll a two on the d4 and a 6 on the d10 for a result of 26. Another count (I was a bit slower this time, it took about 5 seconds) gets me to “Magic Aura”.

This might be the spell that has been cast on an area, a spell on a scroll, a spell that is nullified by a particular magic item – whatever I needed it for. Counting 1-2 seconds to pick up the dice, a second to roll them, and another 1-2 seconds to read them and get my answer, it took me less than 15 seconds to pick the spell at random.

Divide and Conquer

This technique employs 3d6, though if the list is really big, I may substitute one or more larger dice.

Roll all the dice you think you need, and line them up in the order they lie on the table after the roll, left to right. Then shift dice to the right according to die size.

The first d6 tells me top, middle, or bottom of the list. I do this quickly and roughly, by eye.

The second d6 tells me top, middle, or bottom of the selected part of the list. Again, this is done quickly and by eye.

There will be one entry in the middle of the selected range. I count up from that roughly half the size of the dice – well, when I say count, it doesn’t have to be exact. I then use the final dice to determine where in that range the actual result that’s been selected is. It only takes a second or two to nail down a result in quite a long list.

I’m looking at the double-page list of weapons in the Pathfinder Core Rules (table 6-4). I have no idea how many entries there are, but I want to pick a weapon off the list. First, I need some idea of how big my third die should be – I estimate by eye a tenth of the total list and then estimate by eye the number of entries in that span. I get a result of about 10 entries. So I’m actually using either 4d6 or 2d6 and d10. It’s always easier to go “start, middle, end” than to count to 10, so I’ll choose the 4d6 method.

I roll 3, 5, 1, and 4. The Three says the middle third of the list, which is roughly the last 1/3 of the first page and the top 1/3 of the second page. The Five says the last third of that range, which starts a couple of entries down from the top of page two – from somewhere in the vicinity of “Greatclub” to “Shortbow”, say. The one says to focus on the first part of that list, from “Greatclub” to “Guisarme”. The final roll puts me in the middle of that range – either Heavy Flail or Greatsword – and since it was on the high end of that middle, I choose Greatsword. Total time: about 6 seconds, 1/3 of which was spent deciding between 4d6 and 2d6,d10.

I can estimate by eye, taking into account entries that span two lines, very quickly. But others aren’t as adept at it. So your mileage may vary.

Rolling impractical numbers of dice

Have you ever rolled 860d6? I’ve had to, once, or more specifically, one of the players in the Zenith-3 campaign had to, a matter of various spell-amplifying circumstances, and a runaway chain reaction. (the rules that led to this are now a lot more constrained).

Here’s how to roll a ridiculous number of dice.

Once you get over 20 dice, there’s not going to be a lot of difference between four times the result of five dice and rolling all 20 dice. However, the final point or two can be crucial if we’re looking at thresholds and game mechanics that subtract from the total. So:

  1. pick a convenient factor, something that the number of dice will more-or-less be evenly divisible by. I’ll usually use 10 as the factor, but there have been times when something else has been more convenient.
  2. Do the division. Round the results down to get the Multiplier.
  3. Multiplier times factor will be less than the total number of dice required, which should always be a reasonable number – more than 6, say. Call this the remainder. Reduce multiplier if remainder isn’t a reasonable number.
  4. You can either roll factor-number-of-dice and multiply by multiplier, or roll multiplier number of dice and multiply by factor. Pick the one that’s most convenient.
  5. Make the roll, and multiply the result by the other part of the factor-multiplier pair. Then roll the remainder dice, and add them to the total.
  6. Job done!

It actually takes a lot more time to explain it than to do it. Here are a couple of examples:

  • 40d6. Roll 10d6, multiply by 4, and add a roll of 10d6. Note that the other way around, Rolling 4d6 and multiplying by 10, is too random it’s outcome. Rolling 8d6 and multiplying by 5 would be OK.
  • 63d8. Roll 10d8, multiply by 5, and add 13d8.
  • 124d6. Roll 11d6, multiply by 10, and add 14d6. Or add two rolls of 7d6, if you prefer.
  • 394d10. Roll 19d10, multiply by 20, and add 14d10.
  • 860d6. Roll 8d6 and multiply by 100. Roll 5d6, multiply by 10, and add to the result. Roll 10d6 and add to the result.
  • 1024d6. Roll 10d6 and multiply by 100. Roll 4d6 and double the result, then double it again; add to the total. Roll 10d6 and add to the total.
  • 1,120d6. Roll 10d6. Multiply by 100. Roll 6d6. Double it, then multiply by 10, and add. Subtract 35 (the average of 10d6). Roll 10d6 and add to the result.

It really is that quick.

Bonus Tip: quickly adding up lots of dice
This seems really obvious to me, but I’ve seen players who don’t know it, even after months or years of gaming experience.

Count in groups if you have to. Four people can total 20 dice as fast if not faster than one person can total 5 dice – if they are all using the technique outlined below.

Create 10s. Put 6’s and 4’s together, and pairs of 5’s and so on. Keep these in a line. You will be left with a few. Try using three dice to make tens – 4,4,2. 4,3,3. 6,2,2. It’s a lot easier to count 10, 20, 30… or 20, 40, 60, 80… and then have only a couple of oddballs left at the end.

If there are too many oddballs, or you are rolling d12s or something, get a third party to select pairs of dice that add up to less than 20 – then reduce one until it reads 10 while the other gets increased by the same amount, or reduce it to whatever you need to complete another 10-combination. The adjusted “other” goes back into the dice pool remaining to be counted, the 10 goes into the 10s stack or 20s stack (whichever one you’re using). Note that the third party has to pick them up so that no-one else grabs them to make up a ten-combination.

Count the 10 or 20 combos going around the table as necessary – person one gets to say, 40-and-that’s-it and the next person then starts their count with 50 or 60 or whatever’s appropriate. Meanwhile, person one has added up the odd dice that remained.

Part 3 of this article is relatively short and looks at the properties and usefulness of dice, by size.

d1

Exotic, Beautiful, but absolutely worthless. Go to Awesome Dice if you want to look at one, and this page if you want one of your very own. They are even prettier in real life.

d2

Not so attractive – by a long shot – and not of huge value, since it’s so easy to get a high/low result on a d6.

d3

Similarly, there is not a lot of value in these when low/middle/high are so easy on a d6. However, those playing with younger children may find them useful.

d4

The smallest of the traditional polyhedra, and a die shape which I don’t like at all. It takes practice to be able to actually roll them, and effort. I usually cup my hands together to form a chamber in which I can’t see the die or dice and give it a shake to randomize the result before “rolling” them – just in case.

I’m far more in favor of some of the modern alternatives for reasons of practicality – the crystal shaped ones in particular.

d5

Most of these aren’t particularly attractive, either – but you can get some 10-sided dice marked only 1-5. The problem with the latter is that they are easy to confuse with a normal d10, and you don’t want to get the two mixed up. At best, it’s shooting yourself in the metaphoric foot, at worst it’s cheating, whether intentional or not. That said, Game Science make some very nice crystal ones.

That said, they are incredibly useful. There aren’t many dice out there with an odd number of faces, and that gives the d5 a niche that few other dice can fill.

However, it’s very easy to use a d10 and either divide by 2 (rounding up) or subtracting 5 from any result higher than 5, if you need a d5. Or you can simply roll a d6 and re-roll sixes. So they aren’t hard to simulate.

d6

The six-sided die has been the workhorse of dice games since forever. It’s such a simple shape, and only has one more side than we have fingers on a hand – and both are relevant factors. I have about 120 of these – enough that I can extract a couple of d6 that are visibly distinct (and all I think I’ll need) and toss the remainder onto a battlemap as boulders or the locations of mines or whatever.

d7

Except in exotic combination rolls, like some of the suggestions offered above, the d7 is useful only for days of the week. I have one for that purpose.

d8

This used to be almost equal in the workhorse department with the d6, simply because AD&D used d8’s for most monsters’ hit points. There’s more variety in use these days, but the d8 remains incredibly useful.

d10

The d10 was rarely useful in it’s own right; only when married to a partner to create a d% does it become almost irreplaceable.

d12

The d12 has always felt like the ugly duckling to me, never really getting the respect and attention it deserves. It’s most obvious use is for time – hours, minutes, seconds.

d14

I’ll be honest – I can’t think of a single reason for this dice to exist as anything but a curiosity. Days of the fortnight, perhaps? Either I’m overlooking something colossally obvious, or this is worthless in most practical senses most of the time. It’s probably just a showpiece, so if you buy a d14, make it a pretty one.

d16

Almost the same story as the d14, to be honest. Now if they marked one up with the results of a 2d4 roll, that would be valuable. However, the d16 can do one thing that the d14 can’t – and it justifies the existence of both of them. A d16 plus a high-low roll that triggers a +16 to the roll on a high result is just about as easy a way to get days of the month as I’ve seen.

d18

As a GM, I consider this a little more useful than you might think. Firstly, you can multiply the results by 5 to get a value of 5-90 – in combination with a d6-1, that gives a much simpler way to get latitude and longitude than I listed above.

Secondly, it’s great for a GM to use to cheat in the character’s favor, by making it impossible to roll a 20 in a d20-based game. You just slip your d18 in hand, and forget about inflicting critical hits – with most weapons, anyway. Use it when you find that you’ve overestimated how effective your monsters and tactics would be and don’t want to completely ruin your adventure by pounding the PCs into the pavement – and don’t want to ruin the fun (or your rep as GM) by admitting your mistake.

In any game where you need to roll low on a d20, the advantage swings the other way; 20 is impossible to reach, and you actually have a slightly higher chance of rolling a 1.

Use it with a GM Shield, for obvious reasons.

d20

So ubiquitous it has multiple game systems named after it. Not much more needs to be said.

d22

The other side of the coin from the d18 – so long as you remember that a 21 or 22 are to be treated as a “twenty”. Which makes twenties three times as likely (or thereabouts) as they are on a proper d20.

d24

I didn’t know these existed. Greatest use is for hours of the day.

d30

These are still treated as a novelty item by most gamers that I know, simply because it’s so easy to have three d10 tables selected by a d6 roll. But there are times when they can be quite useful.

d60

Wait, they make a d60 now? Since when? Obvious use is for minutes and seconds. Anything else is a bonus.

d100

I started this list with the sublime, and I end with the ridiculous. These are so hard to roll and keep within reasonable bounds that it’s not funny. And they can be hard to read – 100 results takes up a lot of space on a small die! I suspect that the d60 might suffer from similar problems, but I’ve never seen a real one. Use d10s, it really is MUCH easier.

Others

A footnote, this: According to Wikipedia, you can also find d26s, d32s, d34s, and d50s. I can see obvious utility for the d32; the rest I’m not so sure of.

The d26 might suggest a set of alphabetic markings, but ignores the fact that distribution of letters in English is not uniform – not even first letters of words.

Which brings me to the end of this particular article. While pitched mainly at inexperienced players and GMs, I hope that everyone out there got something from it!

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