WOLFRAM|DEMONSTRATIONS PROJECT

Sicherman Dice

​
faces on die 1
4
5
6
7
8
9
10
11
12
13
14
15
faces on die 2
4
5
6
7
8
9
10
11
12
13
14
15
pick dice set
1
2
+
1
3
4
5
6
8
1
2
4
5
6
7
9
2
3
5
6
7
8
10
2
3
5
6
7
8
10
3
4
6
7
8
9
11
3
4
6
7
8
9
11
4
5
7
8
9
10
12
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
(
1
x
+
2
x
+
3
x
+
4
x
+
5
x
+
6
x
)
=
2
x
+2
3
x
+3
4
x
+4
5
x
+5
6
x
+6
7
x
+5
8
x
+4
9
x
+3
10
x
+2
11
x
+
12
x
. The sum of the dice and products of the polynomials are equivalent, due to the addition property of exponents.
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
(
1
x
+2
2
x
+2
3
x
+
4
x
)
(
1
x
+
3
x
+
4
x
+
5
x
+
6
x
+
8
x
)
. In this Demonstration, polynomial factorization is used to find all similarly-faced sets of dice with a distribution matching the chosen set of dice.