Image by Free-Photos from Pixabay

I originally started writing this as a contribution to the May 2020 Blog Carnival, hosted by Moebius Adventures, but when I wasn’t able to finish it in time, I set it aside for later completion. It is now “later”…

Travel

The theme of the May carnival was “Are we there yet”, and the subject matter was the journey.

Travel is all about three things: time, space (usually a distance and direction), and things observed or encountered along the way.

There’s a hierarchy to these elements as most GMs and game systems apply them. Time is subordinated to Space and Events – or, more precisely, the potential for events – is subordinated to time.

“It takes so long to get to here” subordinates the time element to the spacial element – the destination to be reached, relative to where you are, defines the time that the journey takes.

Most systems then assign a flat percentage chance of an encounter transpiring to the passage of time – one, two, four (or whatever) chances per day of travel. More sophisticated systems may amend the chance according to the terrain, or to the total elapsed time since the last encounter.

This implicitly defines “departure” and “arrival at destination” as encounters – which is fair enough; I will frequently employ “last sight of where you were” as a psychological full-stop to whatever transpired where the PCs were, and “first glimpse of destination” as a prelude to whatever is going to happen at that destination.

It also implies that a change of terrain is, in itself, an “encounter”.

All well and good; that all makes a certain level of common sense. But it’s all been written about before; I started to wonder if there were other approaches or perspectives that might enable journeys to be elevated to “the next level,” and immediately made an association with a couple of philosophical musings that have occupied my mind from time to time over the last couple of years.

Time

To start with, I thought about the nature of time as it would be observed by PCs. Instead of the relentless simplicity of marking time by the clock – a device that commoners didn’t really have access to until the latter parts of the 19th century – time would be marked by a succession of mundane but implicitly variable events – first light, dawn, morning, noon, afternoon, sunset, twilight, night.

On top of that, there may be the completely independent but also variable events of moon-rise and moon-set. If your environment has multiple moons, or a binary star, all this may become more complicated.

But the time that these events transpire doesn’t matter. Instead, these are simply markers that are used to label “when” something did or will (perhaps “should” is better) occur.

In reality, Time is the period of existence between significant events – milestones, if you will.

This is both the physical reality and the subjective reality of the PCs, when you think about it. All our clever timepieces transform some change into a measure of time by means of defining a degree of such change as a specific time interval. We use hours, minutes, and seconds for those intervals. This is obviously true of such things as water-clocks, where a fixed quantity of water can drip from a container in a given period of time, so that the quantity of water remaining provides a ‘reading’ on the time interval since the ‘clock’ was last filled; it is equally obviously true of sundials, which uses shadows and the motion of the sun through the sky to mark time. It doesn’t matter if it’s the flow of electricity from a battery in an electrical clock, or from an external supply, or the mechanical release of energy from a pendulum’s swing, or the release of energy contained in the contraction of a wound-up spring – all objective measurements of time operate on this fundamental principle. The only difference between these measurements and the subjective reality observed by PCs would be in the definition of “significant”.

So, let’s start by discarding the ‘regularity’ of modern time-keeping devices, at least in terms of the fantasy milieu. Steampunk and modern and sci-fi campaigns are a completely different kettle of fish due to the ubiquity of timepieces and the psychological and social impact of being able to keep a precise schedule.

Rainwater coverage

Bear with me, it’s time for the conversation to take a left-field turn.

I was watching the first drops of rain fall on one of my windows at one point…

Assume a surface of area A. This surface starts dry but one raindrop after another falls until total coverage is achieved. It’s informative to break this process down and look at the events between these two milestones.

To simplify, let’s divide the span into tenths, and assume constant rainfall in both number and size of droplets. Let’s also assume that in each of these tenths, twenty droplets fall.

At the 0.1 mark, 20 droplets fall, covering an unknown percentage of the surface. Call this unknown percentage A1, and the total area that is wet at this point, W1. Obviously, W1 = A1 × A / 100.

At the 0.2 mark, 20 more droplets fall, covering another unknown percentage of the surface – but there is a chance that some of these will land on part of the surface that is already wet. This chance – assuming even distribution overall – is going to be 100 × A1 / A (%). So A2 = A1 × (1 – [A1 / A]), and W2 = A1 + A2.

At each successive time mark, the chance of a droplet falling onto a dry part of the surface diminishes. By the 9th time mark, there is far less than A1 still dry, but most of the droplets will land on an area that’s already wet.

Even at the 10th time mark, there is (in theory) still a small part of the surface that is still dry, but that part is so small that it has fallen below the threshold of detection; to all intents and purposes, the surface is completely wet.

Some people may find all this easier to visualize if each time period makes a simple percentage of the surface wet – say, 25%, or 1/4.

At time mark 0.1, that would mean 1/4 of the window was wet and 3/4 was not.

At time mark 0.2, 1/4 of the raindrops fall on already-wet surface, leaving 3/4 to add to the total that is wet: 3/4 of 1/4 is 3/16ths. So the total that’s wet is now 7/16ths.

At time mark 0.3, 7/16ths of the raindrops fall on the wet surface, so 9/16ths do not. 9/16ths of 1/4 is 9/64ths, so that’s how much more surface becomes wet. 28/64ths is already wet, so the total is now 37/64ths. More than half the surface is now wet.

At time mark 0.4, 37/64ths will land on the wet surface, leaving 27/64ths. That means 27/64ths of 1/4 get added to the wet area, or 27/264ths. The area that’s already wet is 37/64ths, or (37 × 4) 264ths, so the wet area is now 148/264th – so the wet area grows to (148+27) = 175/264ths. About now, it become easier to work with decimals than fractions: 0.66288 is wet, 0.33712 is dry.

At time mark 0.5, the rainfall is divided 0.66288 (wasted) to 0.33712 (useful), so 0.08428 lands on the dry surface (1/4 × 0.33712) – so the wet area increases by 0.08428 to 0.74716, leaving 0.25284 dry.

At time mark 0.6, the rainfall is divided into 0.74716 waste and 0.25284 useful, so there is a 0.06321 increase in the wet area, to 0.81037 – in other words, 81% of the surface is now wet. The dry part is now 0.18963, or 18.963%.

Enough rain has fallen at this point to cover the entire surface one-and-a-half times over, but almost one-fifth of the surface is still dry!

Time mark 0.7: 81% of the fall is wasted, 18.963% is not. The wet area increases by 0.0474075 to 0.8577775, so 14.22225% is still dry.

Time mark 0.8: Increase in wet area 0.035555625, wet total is 0.893333125, so dry remaining is 0.106666875, just over 10%.

Time mark 0.9: Increase in wet area 0.02666671875, wet total is 0.91999984375 (close enough to 0.92), and 8% (or 0.08) of the surface is still dry.

Time mark 1 (final): increase in wet area is 0.02, total wet is 0.94, and 6% of the surface is dry. Which shows that by our time definitions, 25% coverage isn’t quite enough. The actual number needed doesn’t matter; what matters is the progression – each time the dry area decreases, more of the raindrops are wasted, and the increase of wet area gets smaller.

Generated using fooplot.com

This is an example of Xeno’s Paradox – which we have avoided with the concept of a threshold of detection, and the assumption that if we can’t detect anything above that limit, the amount is actually zero.

The mathematical expression that describes this is y = 1 – (1 / x), and if you graph that, you get the graph to the left/right:

My Coffee’s Gone Cold

Another practical problem that I’ve thought about from time to time is this: my kettle is essentially a cylinder of water surrounded at the base and sides by plastic, and with a capacity of a bit under four liters (7 pints in American terms). If I boil the water, will it cool faster as coffee in a mug that is a smaller cylinder 2.5″ tall and 3″ across, or am I better off leaving the hot water in one big mass until ready to drink the coffee? (Of course, I could simply measure it, but where’s the fun in that?)

Heat is list to a mass of boiling water in two significant ways. The first is by evaporation – the carriage of kinetic energy by water molecules as steam away from the central mass. Because all the other sides are enclosed, the top is the only part of the mass where this can occur – so the smaller that horizontal surface is, the slower the water will cool. Since the diameter of the coffee mug is half that of the kettle, the surface area of the mug is 1/4 that of the kettle. Which argues strongly that I should make my coffee as soon as the water is hot.

The other method is convection cooling the container, which in turn cools the water inside. The most significant factor here is exposure to the air, so the area of the top and the sides dictates, in relative terms, which container is more effective at retaining heat. We already know that the coffee-cup is at a significant advantage in terms of the surface area of the top, but the surface of the sides of the jug is certain to be many times as large as that of the mug. In the case of the mug, 47.124 square inches to the side and 7.07 sq in at the top. The kettle is taller, too; about 10 inches or so. That gives it’s sides a surface area of 377 square inches and the top an area of 28.274 sq in. The totals are, mug 54.194 sq in and kettle 405.274 sq in.

It doesn’t really matter how significant one method is, relative to the other; in both cases, the mug wins, easily. Or does it?

There’s one other factor to contemplate. Because it is a much larger mass of boiling water, the kettle has a lot more energy to lose. This is a question (essentially) of the VOLUME of the two shapes. The kettle holds ten cups worth – so it has ten times the energy stored within the water. Is the kettle ten times larger in either measure? In terms of the convection, the ratio is close to that (405.274 / 47.124 = 8.600) but close doesn’t cut it; it’s not ten times. And for the method that I consider likely to be the more significant, evaporation, the kettle is only 4 times the mug, which isn’t even close to a factor of 10.

Conclusion: I’m better off leaving the hot water in the kettle until the time comes to use it – it will retain more heat for longer.

But that gets you thinking about the evaporation process. The time when there is the most energy to be carried off by steam is when the water is hottest. If you divide the passage of time up, and measure the heat loss through evaporation alone, the greatest amount of loss will be at the start, when the water is hottest, and the least will be at the end, when the water has cooled somewhat.

Already, the parallels between the rainfall question and this phenomenon are clear. In the former, the largest loss of dry area was in the first time interval (0.25+something); by the tenth time interval, the loss wasn’t even 1/10th of this. The numbers might vary, but the same mathematical principle applies.

Fixed Effect Milestone, Relative Time Intervals

What if, instead of marking how much took place in a specific time-frame, we divided the total change into intervals and measured how long it took for each such % change?

Obviously, the shortest such intervals would occur at the start of the process, while the longest would be at the end of it, by a significant margin. Each milepost would be a constant multiple of the temporal ‘distance’ from the preceding milepost. Again, it doesn’t matter what the actual numbers are.

Let’s illustrate this with a 20% gain:

1
1.2
1.44
1.728
2.0736
2.48832
2.985984
3.5831808
4.29981696
5.159780352

That’s ten intervals worth. The numbers grow even more spectacularly with greater gains, as shown by a 50% gain:

1
1.5
2.25
3.375
5.0625
7.59375
11.390625
17.0859375
25.62890625
38.443359375

Again, ten intervals worth. This is an exponential relationship (n to the power of x).

If we add those together to get the total time-span in “units”, we get 113.33 and a fraction. So, if the total is complete at the tenth interval, the first time measurement is 1/113.33, and the last is 38.44359375/113.33 – or 0.88% from the start and 33.92% from the end, respectively.

But let’s go back to those 1.2 factor results, which aren’t so extreme, and convert them to percentages of the total (trust me, there’s a good reason):

1 + 1.2 + 1.44 + 1.728 + 2.0736 + 2.48832 + 2.985984 + 3.5831808 + 4.29981696 + 5.159780352 = 25.958682112

100 × 1 / 25.958682112 = 3.85%
× 1.2 = 4.623%
× 1.2 = 5.547%
× 1.2 = 6.657%
× 1.2 = 7.988%
× 1.2 = 9.5857%
× 1.2 = 11.503%
× 1.2 = 13.803%
× 1.2 = 16.564%
× 1.2 = 19.877%

We can test the accuracy of these numbers by adding them up and seeing how close the total comes to the ideal 100%. For reasons that will become clear, I’m going to show each subtotal along the way:

3.85%
+ 4.623% = 8.473%
+ 5.547% = 14.02%
+ 6.657% = 20.677%
+ 7.988% = 27.334%
+ 9.5857% = 36.9197%
+ 11.503% = 48.4227%
+ 13.803% = 62.2257%
+ 16.564% = 78.7897%
+ 19.877% = 98.6667%

That’s pretty close. In fact, let’s round those off (and tweak them just a little) to something a little more convenient:

5%
10%
15%
20%
30%
40%
50%
60%
80%
100%

Those are the intervals of the total time that approximate milestones in the progression.

Travel

Any trip can be divided up into intervals and will therefore follow these approximate milestones.

The most significant in terms of what has happened previously and reactions to it will be when it is freshest, i.e. close to departure. The gap between these minor milestones is therefore going to be an exponential relationship.

Which means that if we look at the total distance to the destination, or the total traveling time in hours not spent camping, we can use the percentages derived above to get the intervals between the events – 5%, 10%, and so on.

What’s more, as a general rule, the more immediately-significant an event is, the sooner it is likely to occur – so we can rank these events in relative immediate significance as 100%, 80%, 60%, and so on. Of course, events are rarely so predictable, so there would be some random noise to that assessment – maybe plus-or-minus 10% or 20% or whatever.

This also does not factor in some agency deliberately waiting until a threshold of time or distance is achieved – waiting until characters are far enough out of town for an ambush, for example. So there are going to be limits.

Nevertheless, there is a ruthless kind of usefulness to this sequencing – especially if you also add in a little randomness to these approximate times.

Travel. II

Things get flipped around if the relevance is to the destination. The intervals are from the destination – so 4% away, 8.5% away, and so on – while the significance is 100%, 78.8%, 62.25%, and so on.

The terrain and environment that’s most significantly different from that at either end of the journey is obviously going to be the bit in the middle. The greatest likelihood of an encounter that would be suppressed by proximity to civilization is, likewise, going to be in the middle.

In Practice:

That means that you can map out your encounters.

ENCOUNTER INTERVAL MAPPING & SUBSTANCE
Interval Mark	% Chance Encounter Relates To Departure Point	% Chance Encounter Relates To Terrain	% Chance Encounter Relates To Destination	Total %
5%	80	5	5	90
10%	60	15	10	85
15%	50	30	15	95
20%	40	50	20	110
30%	30	80	30	140
40%	20	80	40	140
50%	20	50	40	110
60%	10	30	50	90
80%	5	30	60	95
100%	5	15	80	100

You may have noticed that these totals often don’t sum to exactly 100%. Some only come to 95%, others reach a whopping 140%.

There are two ways to interpret this:

1. The individual percentages need to be adjusted to get a correct percentage breakdown of the encounters at each stage of the journey; or
2. The combination of the various factors (i.e. the total) is the adjustment to the base chance of an encounter according to where in the journey the PCs are.

Adjustment One:

This is done by multiplying each chance shown by 100/Total. For example, the 20% distance line contains the following chances:

• % Chance Encounter Relates To Departure Point = 40
• % Chance Encounter Relates To Terrain = 50
• % Chance Encounter Relates To Destination = 20
• Total = 110

So these relative values become absolute values (suitable for a die roll) as follows:

• % Chance Encounter Relates To Departure Point = 40 × 100/110 = 36%
• % Chance Encounter Relates To Terrain = 50 × 100/110 = 45%
• % Chance Encounter Relates To Destination = 20 × 100/110 = 18%
• Total (included to check the logic) = 110 × 100/110 = 100%
• Cross-check: 36+45+18=99, so one of these needs adjustment to accommodate the rounding error.

There are two schools of thought regarding these rounding errors:

• Add or subtract an equal share any adjustment to the largest values, effectively “swamping” the error in the biggest percentage; or
• Add or subtract an equal share of any adjustment to the smallest values, effectively highlighting the rarest encounter by a smidgen.

The actual approach used is up to the individual. Note that if there is only one “lowest value” or “highest value”, you don’t need to worry about the “equal share” part of the prescription.

More Epic Journeys

This approach is fully scalable – you can apply it to a long journey through several townships, or to individual legs of such a journey.

More epic journeys result from doing both, and combining the results.

To map out the encounter patterns for such an epic journey, we need to first divide the epic journey into intervals at the same milestone markers – 5, 10, 15, 20, 30, 40, 50, 60, 80, 100 – and then relate them to physical milestones on the geographic map. These define the stages of the overall journey.

Next, for each interval, we need to define a ratio of global vs local significance. I would use 80-60-50-40-30-30-40-50-60-80 – meaning that in the first leg of the greater journey, 80% of encounters will relate to the overall journey and only 20% will be local.

These values then scale the contributions to the encounter table of the greater journey to each leg of the trip.

Example:

• Leg 1:
  • Global:
    • % Chance Global Encounter Relating To Departure Point = 80 × 80% = 64%
    • % Chance Global Encounter Relating To Terrain = 5 × 80% = 4%
    • % Chance Global Encounter Relating To Destination = 5 × 80% = 4%
    • Subtotal = 90 × 80% = 72%
  • Local:
    • % Chance Local Encounter Relating To Departure Point = 80 × 20% = 16%
    • % Chance Local Encounter Relating To Terrain = 5 × 20% = 1%
    • % Chance Local Encounter Relating To Destination = 5 × 20% = 1%
    • Subtotal = 90 × 20% = 18%
• Leg 2:
  • Global:
    • % Chance Global Encounter Relating To Departure Point = 60 × 60% = 36%
    • % Chance Global Encounter Relating To Terrain = 15 × 60% = 9%
    • % Chance Global Encounter Relating To Destination = 10 × 60% = 6%
    • Subtotal = 85 × 60% = 51%
  • Local:
    • % Chance Local Encounter Relating To Departure Point = 60 × 40% = 24%
    • % Chance Local Encounter Relating To Terrain = 15 × 40% = 6%
    • % Chance Local Encounter Relating To Destination = 10 × 40% = 4%
    • Subtotal = 85 × 40% = 34%

… and so on

Narrative Key-points

But there’s an even more powerful approach available to GMs – the use of these to flag narrative key-points, which are then used as guidelines in formulating specific encounters

For example:

• 5% – Departure, Attempt to delay the journey (possibly indefinitely)
• 10% – Attempt to turn the journey back
• 15% – The journey encounters a setback
• 20% – Attempt to redirect the journey / entangle them in a side-quest
• 30% – Progress is blocked by an independent force manipulated into engaging by an enemy – enemy loses track of the journey
• 40% – A temptation is put before the party
• 50% – The motivations for the journey are misinterpreted
• 60% – The journey receives unexpected aid after a setback
• 80% – Enemy reacquires the journey and mounts a significant attack to prevent it’s successful conclusion
• 100% – Reach Destination (campaign milestone), begin destination Adventure

This transforms the journey into a story with ten chapters, an Adventure in its’ own right.. This will usually be a smaller-scale adventure than the Adventure that takes place at the journey’s end, but so long as that Adventure is larger than any individual chapter, the story will “feel” right.

(NB: I kept the example fairly generic; in actual usage, specific individuals and groups would be named).

Ten Intervals Is Too Many?

Ten intervals is easy to work with, mathematically, but may not be the most useful breakdown. I would actually start with the narrative breakdown, and count up the number of intervals required on the basis of that narrative. That’s why I’ve been careful to show my working in this article – to give you the flexibility to do it differently. So if you really want to apply a basic three- or four-act narrative structure to your encounter planning, you can.

Conclusion

Each entry on a journey can be a single line mentioning the weather, the ecology, the society, the terrain, or supplies. Or it can be something more substantial, but still inconsequential in terms of the main plot. Or it can represent a development or milestone in an ongoing plotline.

Throwaway passages reinforce the sense of traveling, and “anchor” the journey. They make it feel like something that’s actually happening. Everything on top of those throwaway narrative passages – which may be a single sentence in length – adds to the “reality” of the journey. Even ‘wandering monster’ encounters become a milestone along the journey.

But, by providing a guideline as to the content of any encounters, this process anchors the journey with respect to departure point and destination, giving them an added reality that both enhances the journey between them but also enhances the locations themselves. That’s a lot of reward for very little effort.

Make your journeys a memorable story in their own right. It may take you more game time to get to your destination, but it will feel more real when you do.