A generalized die is a list (x1; :::; xn) of integers. For integers n 1, a b and s let D(n; a; b; s) be the set of all dice with a x1 ::: xn b and P xi = s. Two dice X and Y are tied if the number of pairs (i; j) with xi < yj equals the number of pairs (i; j) with xi > yj . We prove the following: with one exception (unique up to isomorphism), if X 6= Y 2 D(n; a; b; s) are tied dice neither of ...

For integers n 1, a b and s let D(n; a; b; s) be the set of all lists of integers (x1; :::; xn) with a x1 ::: xn b and P xi = s. There is a graph G(n; a; b; s) with vertex-set D(n; a; b; s) in which two vertices are adjacent if one is stronger than the other in a natural dice game. In The prevalence of “paradoxical” dice [Bull. Inst. Combin. Appl. 45 (2005), 70-76] the dice which are isolated v...