Single-Player and Two-Player Buttons & Scissors Games

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 $F\le 3$. 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.

21 pages, 15 figures. Presented at JCDCG2 2015, Kyoto University, Kyoto, Japan, September 14 - 16, 2015