Analyzing Alternative Resolution Mechanics: Different Dice

Found this here.

GURPS uses the six-sided die. It uses three of them. And that is how it is done. Every once in a while, someone might complain that the six-sided die is boring, mundane, pedestrian, etc. I don't have a problem with it. I like the bell curve of 3d6. However, for this article, I explore a rather cumbersome method of substituting my 3 cubic friends for dice with more fun-like shapes.

Overcoming the Bell Curve

The big thing that makes using other dice more difficult to translate exactly is that GURPS works off of a discrete normal distribution, which is hacker talk for "numbers in the middle are easier to roll than high or low numbers." Note that if anyone seriously wanted to use these rules and wanted to be fair, they would require a fairly tedious, but not difficult time of cross-referencing tables and dividing and multiplying non-integer rational numbers. Below, I break down what a roll against each number would potentially be equivalent to using polyhedrals, and sometimes coins.

3 or 18

These numbers at the poles of the possible results have very tiny odds of registering on 3d6, below the resolution of anything that could be fairly represented on a d20. Rounding the value, the odds of rolling either number are 1/200. This is rarer than the Sword of Kings in EarthBound; but we roll dice faster than we fight Starman Supers, so it still has a higher rate of occurrence. To simulate trying to roll a 3, throw a d100. If you roll a 1, flip a coin. If you got heads, you just rolled the equivalent of a 3 in GURPS.

18 similarly is flipped, you need to roll a 100 and get a tails. But! That's just rolling an 18. GURPS being a roll under system, typically, we want to roll less than 18. Additionally, we might care what the margin of success is in terms that can be converted to 3d6 margins of success. Take a look at this table.

3d6 Roll	Odds Of Number	Cumulative Odds	d100 (1%)	d20 (5%)	d12 (8.3%)	d10 (10%)	d8 (12.5%)	d6 (16.6%)	d4 (25%)
3	0.46	0.46	1	x	x	x	x	x	x
4	1.39	1.85	2	x	x	x	x	x	x
5	2.78	4.63	5	1	x	x	x	x	x
6	4.63	9.26	9	2	1	x	x	x	x
7	6.94	16.2	16	3	2	1	1	x	x
8	9.72	25.92	26	5	3	2	2	1	x
9	11.57	37.49	37	8	4	3	3	2	1
10	12.5	49.99	50	10	6	5	4	3	x
11	12.5	62.49	63	12	7	6	5	x	2
12	11.57	74.06	75	15	9	7	x	4	x
13	9.72	83.78	84	16	10	8	6	x	x
14	6.94	90.72	91	18	x	x	x	x	x
15	4.63	95.35	95	x	x	x	x	x	x
16	2.78	98.13	98	x	x	x	x	x	x
17	1.39	99.52	99	19	11	9	7	5	3
18	0.46	99.98	100	20	12	10	8	6	4

This table is the face of the insanity of converting 3d6 to other dice rolls, and the inherent unfairness of a "naive" interpolation. Looking at the d100 column, look for the number you rolled, or the next highest number if the number you rolled doesn't appear. Go over to the 3d6 column. This is the number you can use for margin of success. This table will be referred to multiple times in this same document.

4 or 17

To roll a 4, one should use a d100 and attempt to roll a 2 or better. To beat 17, one should roll a d100 and attempt to roll a 99 or better.

5 or 16

To roll a 5 or better, roll a 1 on a d20. To roll better than 16, roll a 98 or better on a d100.

6 or 15

A 6 is a 2 or better on a d20. a 15 is a 95 or better on a d100.

7 or 14

A 7 is a 3 or better on a d20. a 14 is an 18 or better on a d20.

8 or 13

An 8 is 1 on a d4. a 13 is a 6 on a d8.

9 or 12

A 9 is a 3 or better on a d8. A 12 is 3 or better on a d4.

10 or 11

A 10 is a 4 or better on a d8, an 11 is a 5 or better on a d8.

Wrapping Up

This was something of a "failure". That's not saying it wasn't valuable. This article illustrates that translating polyhedral dice to GURPS in a fair way is a non-trivial mathematical exercise and is not as simple as just using a different die. I think the table also gives a decent graphical illustration to how nuanced the curve is for a pool of dice versus a flat distribution. It also shows that even though a d20 has more degrees of freedom, it somehow has a lower resolution regardless; to use an analogy, the d20 is a huge, but slightly blurry photograph; 3d6 is a tiny but inordinately detailed postage stamp that has details that can't be realistically seen without some mechanical aid. Realistically, if designing a system from the ground up that only has binary tests (pass or fail), mathematically a curved distribution or a flat distribution is usually not that important. But in a system like GURPS, where one uses resolution mechanics like degrees of success or failure, it can be impactful, as larger relative success is more difficult to achieve with a curve than with a flat distribution.

Precis - A "failed" exploration of using dice rolls besides 3d6.