Sprouts game on compact surfaces

📝 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.

💡 Analysis

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.

📄 Content

arXiv:0812.0081v3 [math.CO] 2 Jul 2013 SPROUTS GAME ON COMPACT SURFACES JULIEN LEMOINE - SIMON VIENNOT 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.

1. Introduction Sprouts is a two-player game, which needs only a sheet of paper and a pen to enjoy. Rules are extremely simple, and can be found for example in Martin Gardner’s article of 1967 [4] (when the game was invented). The player who makes the last move wins, and since the number of moves of a game with p spots is finite (at most 3p −1), the game cannot end in a draw. It implies that one of the players has a winning strategy. Most of the interest of Sprouts is to find the player having a winning strategy, but it is made difficult by the game’s complexity. The first manual analysis only achieved to find a winning strategy up to p = 6 spot, and it necessitated a lot of reasoning to consider all the possible cases. The first program of Sprouts enabled Applegate, Jacobson and Sleator [1] in 1991, to extend this analysis up to p = 11 spots, and to formulate the following conjecture: Conjecture 1. The first player has a winning strategy in games starting with p spots, if and only if p is equal to 3, 4 or 5 modulo 6. Later computation in 2007 [5] proved this conjecture to be true at least up to p = 32 spots. In this paper, we generalize Sprouts to other surfaces than the simple plane of a sheet of paper. First of all, let us remark that the game is equivalent on the sphere and on the plane. Indeed, if we consider a final Sprouts position after the game was played on a plane, we can “wrap” the plane around a sphere (an adequate mapping is the stereographic projection). This means that the exact same game could have 1 2 JULIEN LEMOINE - SIMON VIENNOT been played on the sphere. Conversely, a similar argument shows that any game played on the sphere could have been played on the plane. Figure 1. Stereographic projection Sprouts natural playground is a surface: if we tried to play Sprouts in a volume, it would be always possible to connect two given spots with a line, and any game would end in exactly 3p −1 moves. Interestingly, we can consider other surfaces than the sphere and in this paper, we generalize Sprouts to any compact surface. We describe first how to play Sprouts on compact surfaces, by listing all the possible moves, then we give our first results on the winning strategy, obtained by adapting our program of [5]. We applied the same programming methods as in [5], simply generalizing some functions to the case of compact surfaces.
2. Basic notions on compact surfaces 2.1. Sprouts. We call position a Sprouts game at a given time: this is a graph G embedded in a surface S . We call region the union set of a connected component of S −G and of all the parts of G touching it. Because of the rule forbidding to cross an existent line, a move is always done inside a given region. After a move, the region remains unchanged, or breaks into two new regions, so that the total number of regions can only increase during a game. Inside a region, we call boundaries the connected component of the part of the graph G touching the considered region. For example, the position of figure 2 has been obtained from an initial position with 3 spots, A, B and C. A has been linked to B, creating D, then A to D creating E. There are two regions in this position: the first one has only one boundary, which can be written “ADBDE”, and the second one has two boundaries, “ADE” and “C”. A life is a remaining possibility for a point to be linked to another point: initially, a vertex has 3 lives, and if k lines (k ≤3) are linked to a vertex, it remains 3 −k lives. Figure 2. A simple position SPROUTS GAME ON COMPACT SURFACES 3 2.2. Orientable surfaces. The sphere will be denoted S, the torus T. The con- nected sum of two surfaces S1 and S2 will be denoted S1♯S2. The connected sum of two tori (or torus with two holes) will be denot

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