John Carlos Baez on Nostr: There are 30 different ways to put the numbers 1,2,3,4,5,6 on the faces of an ...

There are 30 different ways to put the numbers 1,2,3,4,5,6 on the faces of an ordinary 6-sided die, one number on each face, if you count two rotated dice as the same.

Why? Well, there are 6! = 720 ways to put numbers on the faces, but there are 24 ways to rotate a cube, and 720/24 = 30.

So make 30 dice, one of each kind. Now here's where things get spooky. You can put them in a 6 × 6 square array with no dice on the diagonal, since 6 × 6 - 6 = 30. But what's *really* spooky is this. You can arrange them so that:

• If you pick up the five dice in any row or column of the array, then each pair of the numbers 1,2,3,4,5,6 shows up on opposite faces of exactly one of these five dice.

• The die in position (j,i) is the mirror image of the die in position (i,j).

This stuff relies on some amazing properties of the number 6, described by Peter Cameron here:

https://cameroncounts.wordpress.com/2010/05/11/the-symmetric-group-3/

For the mathematicians out there: it's connected to the outer automorphism of S₆.