Go and mathematics

Go and mathematics

The game of Go is one of the most popular games in the world and is on par with games such as chess, in any of its Western or Asian variants, in terms of game theory and as an intellectual activity. It has also been argued to be the most complex of all games, with most advocates referring to the difficulty in programming the game to be played by computers and the large number of variations of play.AGA - top ten reason to play Go] While the strongest computer chess software has defeated top players (Deep Blue beat the world champion in 1997), the best Go programs routinely lose to talented children and consistently reach only the 1-10 "kyu" range of ranking. Many in the field of artificial intelligence consider Go to be a better measure of a computer's capacity for thought than chess. [Johnson 1997]

As a result of its elegant and simple rules, the game of Go 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 [ [http://senseis.xmp.net/?GoInfinitesimals 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. Tromp and Farnebäck show that on a 19×19 board, about 1.196% 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). [Tromp and Farnebäck 2007] As the board gets larger, the percentage of the positions that is legal decreases. Go (with Japanese ko rules) is a two player un-bounded EXPTIME-complete game. [Hearn 2006]

From this table, we can see that 10700 is an overestimate of the number of possible games that can be played in 200 moves and an underestimate of the number of games that can be played in 361 moves. It can also be noted that since there are about 31 million seconds in a year, it would take about 2¼ years, playing 16 hours a day at one move per second, to play 47 million moves. As to 1048, since the future age of the universe is projected to be less than 1000 trillion years [ [http://www.windows.ucar.edu/tour/link=/the_universe/Eternal.html&edu=high The Future of the Universe] ] and no computer is projected to compute anything close to a trillion Teraflops, any number higher than 1039 is beyond possibility of being played.

Positional complexity

Many of the commonly seen opening strategies, joseki and tactical shapes which aid skillful play have been developed over thousands of years of play and taught to successive generations rather than discovered through individual play. There are many positional situations in Go which are recognizable by an experienced player that are hard to recognize otherwise. Once players gain knowledge of these patterns in play, they then must ponder how to apply them in accordance with the position of the board as it stands and the recognizable patterns already in place. Thus, the traditions of Go strategical theory utilized by most stronger players and taught to beginners help to limit the scope of variation in actual play while deepening strategy.

ee also

* Game complexity
* Shannon number (Chess)

Notes

References

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* cite web | url=http://query.nytimes.com/gst/fullpage.html?res=9C04EFD6123AF93AA15754C0A961958260 | title=To Test a Powerful Computer, Play an Ancient Game | last=Johnson | first=George | work=New York Times| date=1997-07-29
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External links

* [http://view.samurajdata.se/psview.php?id=87eb7119&page=1&size=full Combinatorics of Go] online viewer
* [http://www.msoworld.com/mindzine/news/orient/go/special/gomath.html Go and Mathematics]


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