# zbMATH — the first resource for mathematics

Nontransitive dice sets realizing the Paley tournaments for solving Schütte’s tournament problem. (English) Zbl 1313.05150
Summary: The problem of a multiple player dice tournament is discussed and solved in the paper. A die has a finite number of faces with real numbers written on each. Finite dice sets are proposed which have the following property, defined by Schütte for tournaments: for an arbitrary subset of $$k$$ dice there is at least one die that beats each of the $$k$$ with a probability greater than $$1/2$$. It is shown that the proposed dice set realizes the Paley tournament, that is known to have the Schütte property (for a given $$k$$) if the number of vertices is large enough. The proof is based on Dirichlet’s theorem, stating that the sum of quadratic nonresidues is strictly larger than the sum of quadratic residues.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 11A15 Power residues, reciprocity