Tied dice
نویسندگان
چکیده
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 which ties all other elements of D(n; a; b; s) then there is a third die Z 2 D(n; a; b; s) which ties neither X nor Y .
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