Tile-based knot assembly with Celtic!

Authors

• Divya Bajaj, The University of Texas Rio Grande Valley
• Ryan Knobel, The University of Texas Rio Grande Valley
• Juan Manuel Perez, The University of Texas Rio Grande Valley
• Rene Reyes, The University of Texas Rio Grande Valley
• Ramiro Santos, The University of Texas Rio Grande Valley
• Tim Wylie, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

7-22-2026

Abstract

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.

Comments

This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. 

Recommended Citation

Bajaj, Divya, Ryan Knobel, Juan Manuel Perez, Rene Reyes, Ramiro Santos, and Tim Wylie. "Tile-based knot assembly with Celtic! D. Bajaj et al." Acta Informatica 63, no. 3 (2026): 29. https://doi.org/10.1007/s00236-026-00533-2

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Acta Informatica

DOI

10.1007/s00236-026-00533-2