Sprouts game on compact surfaces

TL;DR

This paper extends the analysis of the Sprouts game from the plane to all compact surfaces, showing how the game's complexity varies with surface topology and providing computational insights into winning strategies.

Contribution

It generalizes Sprouts to compact surfaces, describes move implementation, and demonstrates that surface genus limits reduce analysis complexity, revealing differences between orientable and non-orientable surfaces.

Findings

Winning player on orientable surfaces matches the plane.

Differences observed on non-orientable surfaces.

Finite surface genus suffices for game analysis.

Abstract

Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots drawn on a sheet of paper, and lasts at most 3p-1 moves: the player who makes the last move wins. Sprouts is a very intricate game and the best known manual analysis only achieved to find a winning strategy up to p=7 spots. Recent computer analysis reached up to p=32. The standard game is played on a plane, or equivalently on a sphere. In this article, we generalize and study the game on any compact surface. First, we describe the possible moves on a compact surface, and the way to implement them in a program. Then, we show that we only need to consider a finite number of surfaces to analyze the game with p spots on any compact surface: if we take a surface with a genus greater than some limit genus, then the game on this surface is equivalent to the game on some smaller surface. Finally, with computer calculation, we observe that the winning player on orientable surfaces seems to be always the same one as on a plane, whereas there are significant differences on non-orientable surfaces.