Single-Player and Two-Player Buttons & Scissors Games

TL;DR

This paper analyzes the computational complexity of the Buttons & Scissors game, establishing NP-completeness for certain parameters and polynomial solvability for others, and introduces two-player variants that are PSPACE-complete.

Contribution

It provides a detailed complexity classification of Buttons & Scissors under various constraints and introduces new two-player versions with high computational complexity.

Findings

NP-complete for 2 colors but polynomial for 1 color

NP-complete if each color appears at most 4 times, polynomial if at most 3 times

Two-player versions are PSPACE-complete

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.