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).

中文翻译：

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Brandubh 状态空间复杂度的上界

在国际象棋传入北欧之前，有 Tafl，这是一个与维京人密切相关的非对称战略棋盘游戏家族。本文的目的是研究名为 Brandubh 的爱尔兰 Tafl 变体的组合状态空间复杂性。Brandubh 之所以被选中，是因为它对两名棋手的目标不对称，而且它的整体复杂性远低于国际象棋，这应该使它易于进行强力求解。Brandubh 的规则和特征用于了解游戏的整体状态空间复杂性。状态空间将考虑有效的棋子位置、通用规则集和可接受的最终状态条件。从这些状态推导出强求解 Brandubh 的复杂性的上限。已投入大量精力彻底考虑所有潜在状态并排除游戏中无效的状态。总体而言，解决游戏的上限复杂度约为 10^14 个状态，介于连接四和草稿（跳棋）之间。