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arXiv · 2610.15654

Saving the Queen is hard: Hive is EXPTIME-complete

Francesco Carboni·Daniele Muscillo
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Abstract

Hive is a boardless two-player strategy game in which the pieces form the playing surface. We prove that deciding whether White can force a win in single-Queen $n$-Hive, the generalization with exactly one Queen Bee per player, is EXPTIME-complete, even when all pieces are already placed. Our hardness proof reduces from unbounded two-player Constraint Logic: Grasshoppers encode edge orientations, while the One Hive rule enforces the vertex constraints. We obtain the EXPTIME upper bound by considering positions up to translation. The result answers the single-Queen question raised by Andel and Rin. Keeping the same inventory and movement rules, it also gives EXPTIME-completeness for multi-Queen variants. To the best of our knowledge, this is the first use of unbounded two-player Constraint Logic to prove EXPTIME-completeness for a natural problem.

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Francesco Carboni, Daniele Muscillo. 2026-10-11. Saving the Queen is hard: Hive is EXPTIME-complete. https://arxiv.org/abs/2610.15654

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