LIPIcs.FUN.2024.14.pdf
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PackIt!: Gamified Rectangle Packing
Authors
Thomas Garrison 
,
Marijn J. H. Heule ,
Bernardo Subercaseaux
-
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Volume:
12th International Conference on Fun with Algorithms (FUN 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Fun with Algorithms (FUN) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-05-29
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Acknowledgements
We thank FUN2024 reviewers for their feedback and suggestions. We also thank Richard Green, for his comments and his blog post about our paper. The last author thanks Abigail Kamenenetsky for her help with a web implementation of the game.
Cite AsGet BibTex
Thomas Garrison, Marijn J. H. Heule, and Bernardo Subercaseaux. PackIt!: Gamified Rectangle Packing. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.14
https://doi.org/10.4230/LIPIcs.FUN.2024.14
Abstract
We present and analyze PackIt!, a turn-based game consisting of packing rectangles on an n × n grid. PackIt! can be easily played on paper, either as a competitive two-player game or in solitaire fashion. On the t-th turn, a rectangle of area t or t+1 must be placed in the grid. In the two-player format of PackIt! whichever player places a rectangle last wins, whereas the goal in the solitaire variant is to perfectly pack the n × n grid. We analyze necessary conditions for the existence of a perfect packing over n × n, then present an automated reasoning approach that allows finding perfect games of PackIt! up to n = 50 which includes a novel SAT-encoding technique of independent interest, and conclude by proving an NP-hardness result.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Combinatorics
- Theory of computation → Discrete optimization
Keywords
- PackIt!
- rectangle packing
- SAT
- NP-hardness
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References
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