在紧凑的表面上发芽游戏
Sprouts game on compact surfaces
摘要：
Sprouts是一款双人拓扑游戏，由Michael Paterson和John Conway于1967年发明。游戏开始时在一张纸上画p个点，最多持续3p-1步：走最后一步的玩家获胜。Sprouts是一个非常复杂的游戏，最著名的手动分析只能找到一个获胜的策略p = 7点。最近的计算机分析达到了p = 32。标准的游戏是在一个平面上玩的，或者相当于在一个球体上玩。在本文中，我们推广和研究任何紧曲面上的对策。首先，我们描述了一个紧凑的表面上的可能的移动，以及如何在程序中实现它们。然后，我们证明了我们只需要考虑有限个曲面就可以分析任意紧致曲面上的p点博弈：如果我们取一个亏格大于某个极限亏格的曲面，那么这个曲面上的博弈等价于某个更小曲面上的博弈。最后，通过计算机计算，我们观察到在可定向曲面上获胜的玩家似乎总是与在平面上相同的玩家，而在非可定向曲面上有显著的差异。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
