We all know the possible outcomes of throwing two usual six-sided dice. Have you ever wondered if there are other possible types of dice, i.e. still six-sided but with different face values, which gives the same outcome?
The answer is yes, and they are called Sicherman Dice after George Sicherman and again popularized by Gardner in Scientific American (1978).
The generating function for two usual dice is
[tex] (x + x^2 +x^3+x^4+x^5+x^6)^2 [/tex]
[tex]= x^2 (1+x)^2 (1+x+x^2)^2(1-x+x^2)^2[/tex]
which can be split into
[tex]x (1+x) (1+x+x^2)= x+2x^2+2x^3+x^4 [/tex]
[tex] x (1+x) (1+x+x^2)(1-x+x^2)^2 = x+x^3+x^4+x^5+x^6+x^8[/tex]
giving dice of face values: 1,2,2,3,3,4 and 1,3,4,5,6,8. It can be checked that all other combination of indices would not result in two products with exactly 6 terms (after counting multiplicities).