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Journal Articles International Computer Games Association Journal Year : 2022

## An upper bound for the number of chess diagrams without promotion

Daniel Gourion

#### Abstract

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.

### Dates and versions

hal-03483904 , version 1 (16-12-2021)
hal-03483904 , version 2 (17-11-2022)

### Identifiers

• HAL Id : hal-03483904 , version 2
• DOI :

### Cite

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⟩

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