# 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 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[1] 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 show 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).[2] Keep in mind that the universe just has 1082 atoms in it at most. 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.[3] Rule variations that place a polynomial bound on the length of the game produces a PSPACE-complete game.[4] The complexity of Go with superko rules remains an open question.

Game size Board size N 3N Percent legal Maximum legal game positions[5]
1×1 1 3 33% 1
2×2 4 81 70% 57
3×3 9 19,683 64% 12,675
4×4 16 43,046,721 56% 24,318,165
5×5 25 8.47×1011 49% 4.1×1011
9×9 81 4.4×1038 23.4% 1.039×1038
13×13 169 4.3×1080 8.66% 3.72497923×1079
19×19 361 1.74×10172 1.196% 2.08168199382×10170
21×21 441 2.57×10210

## Game tree complexity

The computer scientist Victor Allis notes that typical games between experts last about 150 moves, with an average of about 250 choices per move, suggesting a game-tree complexity of 10360.[6] For the number of theoretically possible games, including games impossible to play in practice, Tromp and Farnebäck give lower and upper bounds of 101048 and 1010171 respectively.[2] The most commonly quoted number for the number of possible games, 10700[7] is derived from a simple permutation of 361 moves or 361! = 10768. Another common derivation is to assume N intersections and L longest game for N^L total games. For example, 400 moves, as seen in some professional games, would be one out of 361400 or 1 × 101023 possible games.

The total number of possible games is a function both of the size of the board and the number of moves played. While most games last less than 400 or even 200 moves, many more are possible.

Game size Board size N (intersections) N! Average game length L NL Maximum game length (# of moves) Lower Limit of games Upper Limit of games
2×2 4 24 3 64 386,356,909,593[8]
3×3 9 3.6×105 5 5.9×104
4×4 16 2.1×1013 9 6.9×1010
5×5 25 1.6×1025 15 9.3×1020
9×9 81 5.8×10120 45 7.6×1085
13×13 169 4.3×10304 90 3.2×10200
19×19 361 1.0×10768 200 3×10511 1048 101048 1010171
21×21 441 2.5×10976 250 1.3×10661

The total number of possible games can be estimated from the board size in a number of ways, some more rigorous than others. The simplest, a permutation of the board size, (N)L, fails to include illegal captures and positions. Taking N as the board size (19×19=361) and L as the longest game, NL forms an upper limit. A more accurate limit is presented in the Tromp/Farnebäck paper.

Longest game L (19×19) (N)L Lower Limit of games Upper Limit of games Notes
1 1 361 361 White resigns after first move
50 2.1×10126 7.5×10127
100 1.4×10249 5.6×10255
150 6.4×10367 4.2×10383
200 1.9×10481 3.2×10511
250 8.2×10587 2.4×10639
300 2.8×10684 7.8×10766
350 3.6×10760 1.3×10895
361 1.4×10768 1.8×10923 Longest game using 181 black and 180 white stones
400 n/a 1.0×101023 Longest professional games
500 n/a 5.7×101278
1000 n/a 3.2×102557
47 million n/a 10108 361^3 moves
1048 n/a 101048 1010171 Longest game

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[9] and no computer is projected to compute anything close to a trillion Teraflops (one yottaflop), any number higher than 1039 is beyond possibility of being played.

## Positional complexity

A game of Go

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 are taught to beginners and help to limit the scope of variation in actual play while deepening strategy.

## Notes

1. ^ Go Infinitesimals
2. ^ a b "Combinatorics of Go", J Tromp, G Farnebäck - Computers and Games, 2007
3. ^ Hearn 2006
4. ^ Papadimitriou. Computational Complexity. 1994
5. ^ Tromp 2005
6. ^ Allis 1994
7. ^ AGA – top ten reason to play Go
8. ^ Tromp 1999
9. ^ The Future of the Universe