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Recovered: 7d6, split between two attributes

On 20 Feb 2020, PlanetNiles (Ey/Em) NB on the workshop channel of the OSR Discord asked:1

7d6, split between two attributes

What values are even possible?

One way to think about this is not in terms of the values themselves, but in terms of the difference between the two paired attributes. The maximum difference is when all dice are allocated to one attribute, leaving the other empty. This is then equivalent to the sum of the dice.

The minimum difference between the two is a form of the well-studied partition problem: given a set of numbers, can it be split into two sub-sets with equal sums? A heuristic method that can usually2 find the smallest difference between our two subsets (and therefore if they can be equal or not) is called the Karmarkar-Karp method, which works as follows:

  1. Take the two largest numbers in the set. Assume that these will go in opposite partitions.
  2. Because they will go in opposite partitions, we subtract them from each other.
  3. Instead of now deciding which partitions they will go in, we return the difference between them to the set. This is effectively deferring that decision until later.
  4. Repeat steps 1-3 until there are only two numbers left. The difference of these last two numbers is the smallest difference we can make with this set.

Note that we didn’t actually find what the partitions are, only what the difference between them is here. I hope this is clear enough, but I’m also sure that clearer explanations can be found with minimal searching.

With some wrangling, we can produce this lovely chart, where a heavier hexagon is a more likely pairing.

A hexagonal grid, with axes labeled “Minimum Difference” from 0 to 6 on the vertical and “Maximum Difference (Sum) to around 40 on the horizontal. The hexagons are shaded blue, very lightly along minimum 0-2 until around maximum 15, then darker centering on maximum 25, then eventually lighter past maximum 35. Along minimum 3-4 the hexagons are very lightly shaded from around maximum 20 to 40, and minimum 5-6 are even more lightly shaded from around 27-40.Figure 1. Hexagons are elegant, as odd/even combinations that cannot occur are skipped naturally.

From this, it looks like the system gives a lot of flexibility in assigning scores to your attributes.

What Values are Likely?

I have a concern about this mechanic though: if you give someone a range of numbers, and tell them to pick one, they will tend to pick in the middle.3 If a player is dead-set on being boring, how boring can they be?

It might not be exactly true, but we can show the distribution of all the high stats and all the low stats together, and then compare them to the distribution of 7d6 literally divided in half (which is approximately normal).

“Individual Scores” along the X axis are plotted for “7d6 Halved” and “7d6 Partitioned.” The partitioned scored very nearly align with the halved scores, in a roughly normal curve.Figure 2.

This doesn’t look great, but PlanetNiles has actually already got us covered here:

Of course. I’d further consider including subsystems where the difference between attributes had some sort of effect. So favoring one over the other would prove beneficial in some way, or at least open up different options.

The strength of this mechanic then will rely on the strength of the system. Given the range of the first figure though, I have confidence that an interesting system could be built here.

What about other mechanics?

Suppose we were looking for a similar mechanic, except that it would force a difference between the attributes where possible. We might expect intuitively, that fewer dice and larger dice are harder to partition effectively. This table gives the probability of a forced difference (although does not consider the size of that difference).

A grid with die sizes from d4 to d20 along the top and quantity from 3 to 8 along the side. “8d4” is shaded the lightest gray and marked “0.51” this progresses relatively smoothly to “3d20” which is almost black and labeled “0.93.”Figure 3.

From this table, I thought I would look at 3d20, because it forces a difference the most often. The figures below have the same interpretations as the similar ones above, but for 3d20 instead of 7d6.

Another hexagonal plot of minimum (0 to 20) and maximum (1 to 60) difference. It is uniformly darker, making a double-triangle shape with one peak around 20 maximum and one peak around 60 maximum (both peaks around 18 or 20 minimum). From 40 maximum onward, there is no shading along the bottom, creating a “leaning” triangle shape for the second peak.Figure 4. Another plot of individual scores, but where “3d20 Halved” is still roughly normal, “3d20 Partitioned” increases almost linearly from 0 to 20, and then has a discontinuous tail above that which is much lower.Figure 5.

Python

This was my first project using Python, and I think it’s an all right language. It’ll probably displace Octave in my repertoire, but I’m sure I’ll be right back at Perl if I start doing string stuff again. The code’s a bit janky, but you can take a look here.

This post was first shared on March 24, 2020.

  1. “Asked” a little more directly this time, if only because I asked first.↩︎

  2. According to Wikipedia, this method is “bad for instances where the numbers are exponential in the size of the set,” so like, probably fine?↩︎

  3. I only know this anecdotally: if you tell a plant operator to keep some process temperature between a high limit and a low limit, without fail they will control it to the middle of the two. It makes sense to a person, but the optimal temperature is provably at one of the two extremes. Possibly, this is an extension of the anchoring effect.↩︎

Tags
Recovered Dice Mechanics Programming Math
Date
July 26, 2024
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