# Sicherman Dice

Sicherman Dice

Somewhere at a casino, a person is rolling two six-sided dice and adding the values of the top faces. Out of 36 outcomes, both snake-eyes (1+1=2) and boxcars (6+6=12) can occur in exactly one way, while 7 can occur in six ways. Note that the distribution for the sums 2 to 12 is 1 2 3 4 5 6 5 4 3 2 1. Next, note that = +2+3+4+5+6+5+4+3+2+. The sum of the dice and products of the polynomials are equivalent, due to the addition property of exponents.

2

(+++++)

1

x

2

x

3

x

4

x

5

x

6

x

2

x

3

x

4

x

5

x

6

x

7

x

8

x

9

x

10

x

11

x

12

x

Does a different set of dice produce the same distribution? Yes: dice labeled (1 2 2 3 3 4) and (1 3 4 5 6 8) correspond to the factorization . In this Demonstration, polynomial factorization is used to find all similarly-faced sets of dice with a distribution matching the chosen set of dice.

(+2+2+)

1

x

2

x

3

x

4

x

(+++++)

1

x

3

x

4

x

5

x

6

x

8

x