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Go and mathematics
The game of Go is one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration for mathematical research. Chinese scholars of the 11th century already published work on permutations based on the go board. In more recent years, research of the game by John H. Conway led to the invention of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals〔(Go Infinitesimals )〕 being a specific example of its use in Go).
==Legal positions==
Since each location on the board can be either empty, black, or white, there are a total of 3N possible board positions on a board with N intersections. Tromp and Farnebäck showed that on a 19×19 board, about 1.2% of board positions are legal (no stones without liberties exist on the board), which makes for 3361×0.01196... = 2.08168199382... ×10170 legal positions ''"of which we can expect all digits to be correct"'' (i.e. because the convergence is so fast).〔("Combinatorics of Go" ), J Tromp, G Farnebäck - Computers and Games, 2007〕 It has been estimated that the observable universe contains around 1080 atoms, far less than the amount of possible legal positions. As the board gets larger, the percentage of the positions that are legal decreases. Go (with Japanese ko rules) is a two player un-bounded EXPTIME-complete game.〔Hearn 2006〕 Rule variations that place a polynomial bound on the length of the game produces a PSPACE-complete game.〔Papadimitriou 1994〕 The complexity of Go with superko rules remains an open question.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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