Mathematical analysis of the Royal Game of Ur

Abstract

Abstract Despite many discoveries and proposals for rules for the ancient board game known as the Royal Game of Ur (RGU), no mathematical analysis has yet been performed investigating those rules. In an attempt to fill that gap, this paper presents an initial mathematical analysis of the RGU from an introductory point of view. The paper deduces the overall complexity of the RGU using a state-space and game-tree complexity analysis, allowing the RGU to be compared to the popular games Checkers, Backgammon, Ludo, Chess, and Go. The paper builds upon the fundamental laws of combinatorics and probability to improve the understanding of the game: what patterns should you expect, what moves increase your chance to win, and what moves should you avoid. The paper also presents theorems to predict the probability of future dice rolls and piece movements within the game, allowing basic inferences to be made about strategy in the RGU. The game is further examined by analysing three different influences when determining the best move: advancement and attack (beneficial to the player), and captures (detrimental to the player). These influences are used to deduce explicit equations for the advantage gained by playing each possible move from a position, which allows the formalization of a strategic algorithm to play the RGU.