Problem
Research questions and friction points this paper is trying to address.
Study 2-player Domino game outcomes
Determine undecidability of player victory
Analyze turn order impact on decidability
Two-player Domino games
🤖 AI Summary
This paper investigates the winner-determination problem for two-player domino tiling games on the infinite grid: (1) the standard alternating game, where Player 1 aims to construct a target pattern and Player 2 seeks to block it; and (2) a generalized non-alternating variant, where move order is governed by a balanced word. The authors prove that winner determination is Turing-undecidable under the alternating rule—the first such undecidability result for combinatorial tiling games. They then fully characterize the decidability boundary for the non-alternating setting, establishing necessary and sufficient conditions—expressed in terms of combinatorial properties of balanced words—for the problem to be decidable. Their approach integrates combinatorial game theory, Turing-machine reductions, infinite graph coloring analysis, and combinatorics on words. This work establishes the fundamental computational complexity of such tiling games and, for the first time, systematically identifies how the structural constraints on move sequences govern decidability.
📝 Abstract
We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it. We study the problem of deciding which player wins, and prove that it is undecidable. We also consider a variant where the turn order is not alternating but given by a balanced word, and we characterise the decidable and undecidable cases.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Infinite grid gameplay
Pattern creation prevention
Balanced word turn order
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