An Upper Bound on the State-Space Complexity of Brandubh

TL;DR

This paper estimates an upper bound of approximately 10^14 states for the complexity of Brandubh, an Irish Tafl game, making it feasible for strong computational solving due to its lower complexity compared to chess.

Contribution

It provides the first detailed upper bound estimate of Brandubh's state-space complexity, accounting for valid states and game rules, aiding future computational analysis.

Findings

Upper bound complexity is around 10^14 states.

Brandubh's complexity is between Connect Four and Draughts.

Thorough accounting of valid states was performed.

Abstract

Before chess came to Northern Europe there was Tafl, a family of asymmetric strategy board games associated strongly with the Vikings. The purpose of this paper is to study the combinatorial state-space complexity of an Irish variation of Tafl called Brandubh. Brandubh was chosen because of its asymmetric goals for the two players, but also its overall complexity well below that of chess, which should make it tractable for strong solving. Brandubh's rules and characteristics are used to gain an understanding of the overall state-space complexity of the game. State-spaces will consider valid piece positions, a generalized rule set, and accepted final state conditions. From these states the upper bound for the complexity of strongly solving Brandubh is derived. Great effort has been placed on thoroughly accounting for all potential states and excluding invalid ones for the game. Overall, the upper bound complexity for solving the game is around 10^14 states, between that of connect four and draughts (checkers).