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

$\text{Die }1$ $=\{1,2,2,3,3,4\}$

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

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
Canonical name	CrazyDice
Date of creation	2013-03-22 15:01:43
Last modified on	2013-03-22 15:01:43
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 05A15