Brain teaser: RPG Dice

You’ll need the classic d4, d6, d8, d10, d12 and d20. For those of you who don’t know, these are dice mapped out on regular polyhedrons, except d10, which is mapped out on… something else. You might want some in front of you, or if you’re a criminal mastermind, you could visualize them.

First, take the four sided die. As long as it ‘works’ (eg. has all the equal numbers in the same corner, so that it’s possible to see what you have rolled), it doesn’t matter how the different numbers are related, all you have to do to see a given combination (1 over 2 and 3, 1 over 2 and 4), is turn the die.
Now, if you’re with me, this means that you don’t have to work out a pattern for mapping out the die, as any working die would include all possible relations. (Actually, you could have two different sequences for each of the lower numbers (1 over 2 and 3 differs from 1 over 3 and 2), but no different sequence will be more harmonic than any other (One could say that 2 and 3 is more harmonic than 3 and 2 because they are then in order, but no reading direction is inherently more harmonic)

Then, take the six sided die. As most would know, it is generally agreed that the sum of all pairs of opposing sides should be the same. Let’s adopt this as a rule. Now, lets look at the die with 1 as the side facing up. 6 is facing down, but how do 2, 3, 4 and 5 relate to 1? 5 can’t be next to 2, and 4 can’t be next to 3. Thus 5 is next to 4 and 3, with 2 opposing. It wouldn’t really matter if 4 is left or right to 5, as again, no reading direction would be inherently more harmonic.

So it seems quite simple rules apply to d4 and d6. How would you map out the remainig dice?
A friend and me thought we had beat the dicemakers to it a couple of years ago, but we examined them again tonight, and found even cooler patterns as they are.

Are you saying in other words, how many different ways can you arrange the numbers on dice, and are some arrangements more symmetrical than others?

For a start, I agree opposite faces should all add up to the same number. (d20= 1-20, 2-19, … 10-11) Using that restriction, there are only two ways to arrange numbers on a d6: one where the corner where 1,2,3 meets is arranged clockwise and the other counterclockwise. I believe there’s even a convention that “official” dice should always be one way, but I’d have to look it up. I don’t know how many permutations are possible for d8, d12 and d20.

For d10 it’s even more confused because that geometric figure (technically, the “pentagonal trapezohedron”) doesn’t have opposing faces and isn’t regular.Usually d10 will have the odd numbers 1,3,5.7,9 in one group and even numbers 0,2,4,6,8 in the other, so that going around clockwise (or counterclockwise), you go sequentially 0-9.

Why? Assuming the weight of the die is evenly distributed, the odds of turning up one number are the exact same as any other number, regardless of the distribution of the numbers among die faces.

Today’s dice can be perfectly fair without having any regard for the placement of the numerals around the die faces.


You’re partly right about the sequence, but d10, like any rollable die, do have opposing faces: When one side is up, another one and only one has to be down.

Mtgman: As the borderline schizofrenic I like to consider myself, I can’t accept the numbers simply being randomly distributed on the die. I want some kind of compelling pattern, and it’s generally accepted that opposing sides always giving the same sum is a harmonic thing for dice.
Besides, figuring out the best possible pattern, or why the one on the dice got chosen (they seem to all be the same, and we’ve seen a couple of rpg dice in our time.) is a rather good mental exercise.

There’s no point to it, it’s just been done that way since time immemorial.

Actually there is a point. Essentially is all the “high” or “low” numbers are grouped, it’s easier to toss the die in such a way as to skew the probability of getting a desired “high” or “low” result. For example, if one wants a high result on a D20, and 20, 19, 18 are on one side, a dishonest person could attempt to slide the die across the table increasing the probablyility of a higher number.

If 1 and 20 are near each other, it gets difficult cause if you miss the 20, you could get a strongly undesireable result.

clnilsen has the right of it. Wizards of the Coast release a “Spindwn life counter” a while back: a d20, but with numbers arranged so as to be easier to find. It’s spectacularly unsuited as a RNG, though, due to how easy it is to consistently roll a 15 or higher.

An unscrupulous player could also shave the edges slightly around all the high numbers so that if the die were about to stop with that number down, it’d keep rolling. If all the high numbers on a d20 were on one side, you could rig it so it would pretty much never roll low. Average to high, but hardly ever low.

I once was the proud owner of a d100 – looked like a golfball, complete with the little divits in it. You had to look straight down from the top to read which number had come up. No real good purpose for it, but a little fun to play with.

I think that the number of ways to uniquely distribute the numbers around the face of an n-face die (other than the d10) is (n-1)! divided by the number of sides of the face of the die. (That’s with no restriction of opposite-faces adding to n+1)

a dodecahedron with the “top” and “bottom” in points rather than flat.

Agreed. My d12 has opposite faces summing to 13, as is the convention, but it has 2,3,4, 5, and 6 arranged around the 1 face, and likewise (of course) the high numbers arranged around 12. Even with my nerdishly low dex score, I can consistently (about 80%) roll the high half, if I try.

If we’re requiring the convention that opposite faces add up to the same number (except for the d4), I get 2 possibilities each for the d4 and d6, 16 for the d8, 384 for the d10 (it’s so much higher because the d10 is less symmetric than the others), 768 for the d12, and 61931520 for the d20. This is counting all three-dimensional rotations as equivalent, but parity inversions as distinct (since real dice can and do roll, but they rarely invert). If you don’t want to count parity inversions as distinct, halve all those numbers.