Single-Player and Two-Player Buttons & Scissors Games

Kyle Burke; Erik D. Demaine; Harrison Gregg; Robert A. Hearn; Adam Hesterberg; Michael Hoffmann; Hiro Ito; Irina Kostitsyna; Jody Leonard; Maarten Löffler; Aaron Santiago; Christiane Schmidt; Ryuhei Uehara; Yushi Uno; Aaron Williams

arXiv · 1607.01826

Single-Player and Two-Player Buttons & Scissors Games

Kyle Burke·Erik D. Demaine·Harrison Gregg·Robert A. Hearn·Adam Hesterberg·Michael Hoffmann·Hiro Ito·Irina Kostitsyna·Jody Leonard·Maarten Löffler·Aaron Santiago·Christiane Schmidt·Ryuhei Uehara·Yushi Uno·Aaron Williams

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Abstract

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.

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Kyle Burke, Erik D. Demaine, Harrison Gregg, Robert A. Hearn, Adam Hesterberg, Michael Hoffmann, Hiro Ito, Irina Kostitsyna, Jody Leonard, Maarten Löffler, Aaron Santiago, Christiane Schmidt, Ryuhei Uehara, Yushi Uno, Aaron Williams. 2016-07-06. Single-Player and Two-Player Buttons & Scissors Games. https://arxiv.org/abs/1607.01826

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