An upper bound for the number of chess diagrams without promotion

Daniel Gourion (1)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon

In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by $2\times 10^{40}$. 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 $4\times 10^{37}$. 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

Domains: Mathematics [math]; Combinatorics [math.CO]; Computer Science [cs]; Computer Science and Game Theory [cs.GT]; Mathematics [math]; Information Theory [math.IT]

Daniel Gourion. An upper bound for the number of chess diagrams without promotion. International Computer Games Association Journal, 2022, 44 (2), pp.44-55. ⟨10.3233/ICG-220210⟩. ⟨hal-03483904v2⟩