circular chains of chinese dice

• Main Authors: Eduardo Piza Volio, Leo Schubert
• Format: Article
• Language: Spanish
• Published: Universidad de Costa Rica 2010-04-01
• Series: Revista de Matemática: Teoría y Aplicaciones
• Online Access: https://revistas.ucr.ac.cr/index.php/matematica/article/view/312

In this paper we study Chinese dice, mathematical objects similar to ordinary dice but allowing repetition among their face values. We say that a die A is preferred over a die B (written A B) if A wins more frequently than B does. We study first the existence of circular chains of three dice A, B, C (those that A B C > A) using a mixed integer programming algorithm. Then we generalize the problem to n-dimensional dice—that is, dice of n faces, with n ? 4—and we search circular chains of length m (with m ? 3) using a simulated annealing algorithm. We compare some different objective functions and obtain good solutions to the problem with very efficient algorithms. Finally we obtain a theoretical result concerning the existence of circular chains in the general case.