What dice to use in a game that revolves around triangles?

Question

I am currently in the process of prototyping a game that revolves around triangles. I want my players to roll 3d4 dice to move, but am worried about the effects it might have vs. a traditional 2d6 setup.

The reasons I want to use d4s are:

1. They are triangular prisms.

and

2. I can use three of them and still only have a maximum of 12 rolled at any one time.

I am worried about the rolls always being just a few numbers. For 2d6, the odds of rolling a 12 (or the lowest #) are 1 in 36. For 3d4, the odds are 1 in 64, making my rolls less diverse. I seem to roll 7, 8 or 9 every time I roll 3d4.

I need to know if this would pose any problem to the movement factor in my game (for now, assume it has 40 squares).

Answer 1

Doing some research shows you will still 5-9 range but it will be slightly more when you are using d4. Now if that difference causes a problem that would be up to you to decide. One other thing to note is you will no longer be able to roll a 2.

This comes from https://anydice.com/program/51b6

#    |  % for 2d6   |  % for 3d4   |
2    |   2.777778   |     0.0000   |
3    |   5.555556   |     1.5625   |
4    |   8.333333   |     4.6875   |
5    |   11.11111   |     9.3750   |
6    |   13.88889   |     15.625   |
7    |   16.66667   |     18.750   |
8    |   13.88889   |     18.750   |
9    |   11.11111   |     15.625   |
10   |   8.333333   |     9.3750   |
11   |   5.555556   |     4.6875   |
12   |   2.777778   |     1.5625   |

Answer 2

In general, rolling more, smaller dice will give a narrower range of possible answers and a distribution that's more concentrated around the average value, compared to rolling fewer, larger dice. This is easy to see by looking at the extremes: rolling a single d12 will give every value between 1 and 12 with equal probability (broad range, no peak at all), whereas rolling twelve d1s will give you 12 every time (narrow range, nothing but the peak).

This is, essentially, the central limit theorem in probability theory, which says that adding up more and more independent random numbers (e.g., dice rolls) gives a better and better approximation to a normal distribution. The width of the "bell curve" depends on the variance of the random numbers that are being added, and the variance of a single dice roll decreases as you decrease the number of faces it has.

Joe W's answer gives the specific stats for the two cases you're considering, so I'll not repeat those.

Answer 3

General opinion is that D4s are unsatisfying to roll.

D8s have triangular sides and roll well. You can get D8s with different values to bias your distribution.

3 D8s with distribution: 1, 2, 2, 2, 3, 3, 3, 4 will still give you outputs ranging from 3-12, but your bell curve will be more extreme.

link with distribution: https://anydice.com/program/14e90

Answer 4

Of interest is the number of rolls to finish. For a 40 square game, rolling 2d6, it takes an average of 40/7 or 5.7 rolls to finish.

The Var(2d6) is 70/12. Variance of "5.7" rolls of 2d6 is 33.25, SD is then about 5.76, and 95% confidence is then around +/- 11.3 or about +/-1.6 rolls. So naively, it will take 4.1-7.3 rolls to finish.

For 3d4, average is 7.5, so 5.333 rolls to finish. Var(3d4) is 45/12, over 5 and a third rolls that is 20 variance, so 4.47 SD and +/- 8.76 confidence, or +/- 1.17 rolls; so 4.16 to 6.5 rolls to finish.

(This uses napkin math based distributions; the actual statistics are a bit more annoying, but will be close.)

Feeling the difference between those game lengths is going to be hard.

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I'd argue you should start upside down. How many rounds do you want the game to be? How should the on-board stuff (say, move forward) have impact compared to dice rolls?

I'd also argue that "rolling a bunch of dice and adding them up" isn't a great mechanic. There is so much more you can do with dice -- pairs, runs, etc.

Answer 5

If you want to have something very similar to 2d6 but using d4's, do 4d4 - 1d4. This can be implemented by having five d4's, four of which are the same color and one is a different color. This is not exact, and it does add a 2% chance of getting numbers higher than 12 or less than 2, but it's closer than 3d4.

https://anydice.com/program/14e8a

If you want to go heavier on the flavor, use 3-sided dice rather than 4-sided (these can also just be 6-sided dice numbered 1, 1, 2, 2, 3, 3):

3d3 will give you a much lower average roll than 2d6, but you could adjust for this by making the map smaller or just putting things closer together on the map.

https://anydice.com/program/8667