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arXiv · 2403.12195

PackIt! Gamified Rectangle Packing

Thomas Garrison·Marijn J. H. Heule·Bernardo Subercaseaux
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Abstract

We present and analyze PackIt!, a turn-based game consisting of packing rectangles on an $n \times n$ grid. PackIt! can be easily played on paper, either as a competitive two-player game or in \emph{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 $n \times n$ grid. We analyze conditions for the existence of a perfect packing over $n \times n$, 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.

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BibTeXRIS

Thomas Garrison, Marijn J. H. Heule, Bernardo Subercaseaux. 2024-03-18. PackIt! Gamified Rectangle Packing. https://arxiv.org/abs/2403.12195

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