# crazy dice

It is a standard exercise in elementary combinatorics to the number of ways of rolling any given value with 2 fair 6-sided dice (by taking the of the two rolls). The below table the number of such ways of rolling a given value $n$:

$n$ # of ways
2 1
3 2
4 3
5 4
6 5
7 6
8 5
9 4
10 3
11 2
12 1

A somewhat (un?)natural question is to ask whether or not there are any other ways of re-labeling the faces of the dice with positive integers that these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a re-labeling, via the labeling

 $\displaystyle\mbox{Die }1$ $\displaystyle=\{1,2,2,3,3,4\}$ $\displaystyle\mbox{Die }2$ $\displaystyle=\{1,3,4,5,6,8\}$

and a pair of dice with this labeling are called a set of crazy dice. It is straight-forward to verify that the various possible occur with the same frequencies as given by the above table.

Title crazy dice CrazyDice 2013-03-22 15:01:43 2013-03-22 15:01:43 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 05A15