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PackIt!: Gamified Rectangle Packing

Authors Thomas Garrison , Marijn J. H. Heule , Bernardo Subercaseaux

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Discrete optimization
Keywords
  • PackIt!
  • rectangle packing
  • SAT
  • NP-hardness

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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.

Cite As Get 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

Author Details

Thomas Garrison
  • Carnegie Mellon University, Pittsburgh, PA, USA
Marijn J. H. Heule
  • Carnegie Mellon University, Pittsburgh, PA, USA
Bernardo Subercaseaux
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

The authors are partially supported by the National Science Foundation (NSF) grant CCF-2108521.

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.

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References

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