An upper bound for the number of chess diagrams without promotion

Article type: Research Article

Authors: Gourion, Daniel^{; *; **}

Affiliations: Laboratoire de Mathematiques d’Avignon, Avignon Universite, France

Correspondence: [**] Corresponding author. E-mail: [email protected].

Note: [*] Laboratoire de Mathematiques d’Avignon, Avignon Universite, France. E-mail: [email protected].

Abstract: In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by 2×1040. This number was obtained by restricting both bishops and pawns position and by a precise bound when no chessman has been captured. We improve this estimate and show that the number of legal diagrams is less than 4×1037. To achieve this, we define a graph on the set of diagrams and a notion of class of pawn arrangements, leading to a method for bounding pawn positions with any number of men on the board.

Keywords: Shannon’s number, chess, state space

DOI: 10.3233/ICG-220210

Journal: ICGA Journal, vol. 44, no. 2, pp. 44-55, 2022