Fun mathematical facts about dice

On a standard six-sided die, opposite faces add up to seven, and there are only two ways to arrange the numbers following that rule: one is the mirror image of the other. There are also ‘non-transitive’ dice, where A beats B, B beats C and C beats A.

Why opposite faces add up to seven

1 opposite 6, 2 opposite 5, 3 opposite 4. This arrangement became widespread centuries ago and helps balance the weight. We mention it in are all dice fair?

Dice in other shapes

Besides the cube, there are dice shaped like the other Platonic solids, with 4, 8, 12 and 20 faces. These are the only regular polyhedra in which all faces are identical and have the same probability of coming up.

Non-transitive dice

By putting specially chosen numbers on the faces, you can make three dice where each beats the next more than half the time, in a cycle. It makes a brilliant after-dinner game to surprise your friends.

The most likely total

With two dice, 7 is the most likely total. We explain this in how dice probabilities work.