Vexing Vexillological Logic

TL;DR

This paper introduces a new combinatorial game called Flag Coloring, analyzes its complexity, and explores its outcomes on real-world flags, revealing its PSPACE-completeness for multiple colors.

Contribution

It defines a novel flood-filling based game, generalizes it to graphs, and proves its PSPACE-completeness for two or more colors, also analyzing real-world flag scenarios.

Findings

The game is PSPACE-complete for two or more colors.

Outcome classes are determined for flag-based positions.

Many positions on two colors have known values.

Abstract

We define a new impartial combinatorial game, Flag Coloring, based on flood filling. We then generalize to a graph game, and find values for many positions on two colors. We demonstrate that the generalized game is PSPACE-complete for two colors or more via a reduction from Avoid True, determine the outcome classes of games based on real-world flags, and discuss remaining open problems.