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≤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.