MMN-659

# Nontransitive dice sets realizing Paley tournaments for solving Schütte’s tournament problem

**S. Bozóki**, Computer and Automation Research Institute, Hungarian Academy of SciencesBudapest, Hungary,`bozoki@sztaki.mta.hu`

## Abstract

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 Sch¨utte 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 a Paley tournament, that is known to
have 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.

Vol. 15 (2014), No. 1, pp. 39-50

DOI: 10.18514/MMN.2014.659