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<p>In this paper we focus on the intersection of tile assembling systems, edge-matching puzzles, combinatorial games, and knot construction and identity. As a basis, we utilize the game Celtic!, which is a 2-player board game where the goal of the game is to construct knots where one knot uses more of a player’s pieces than the other player over all knots. All pieces must build off an existing knot and a valid knot must be closed. We consider three variations: a 0-player self-assembly variation that deterministically places pieces to form a closed knot of some length, a 1-player puzzle variation where the goal is to form a closed knot of some length, and the original 2-player game with restricted pieces. We show these are P-complete, NP-complete (depending on the pieces), and PSPACE-complete (for a first-player win), respectively. We nearly fully characterize the hardness of the 1-player puzzle based on the pieces. We prove these results through standard hardness reductions and with constraint logic. Finally, we note some combinatorial game theory strategies to show certain configurations are a draw through strategy stealing.</p>

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Tile-based knot assembly with Celtic!

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月22日

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  • Divya Bajaj,
  • Ryan Knobel,
  • Juan Manuel Perez,
  • Rene Reyes,
  • Ramiro Santos,
  • Tim Wylie

摘要

In this paper we focus on the intersection of tile assembling systems, edge-matching puzzles, combinatorial games, and knot construction and identity. As a basis, we utilize the game Celtic!, which is a 2-player board game where the goal of the game is to construct knots where one knot uses more of a player’s pieces than the other player over all knots. All pieces must build off an existing knot and a valid knot must be closed. We consider three variations: a 0-player self-assembly variation that deterministically places pieces to form a closed knot of some length, a 1-player puzzle variation where the goal is to form a closed knot of some length, and the original 2-player game with restricted pieces. We show these are P-complete, NP-complete (depending on the pieces), and PSPACE-complete (for a first-player win), respectively. We nearly fully characterize the hardness of the 1-player puzzle based on the pieces. We prove these results through standard hardness reductions and with constraint logic. Finally, we note some combinatorial game theory strategies to show certain configurations are a draw through strategy stealing.