The Amazing Nature of Go

Go is a Chinese game for 2 players played on a 19x19 board. A move consists of placing a stone anywhere on the grid of the board. Thus there are 361 ways to make the first move, 360 ways to make the second, etc. Altogether, there are 361! possible games to be played. The complexity of Go is far superior to that of chess, for the number of possible moves in chess at any given moment is well below 100.

Let us consider what it means to attempt to exhaustively play through all the games of Go.

361! = 1,437,923,258,884,890,654,832,362,511,499,863,354,754,907,538,644,755,876, 127,282,765,299,227,795,534,389,618,856,841,908,003,141,196,071,413,794,434,890, 585,968,383,968,233,304,321,607,713,808,837,056,557,879,669,192,486,182,709,780, 035,899,021,100,579,450,107,333,050,792,627,771,722,750,412,268,086,775,281,368, 850,575,265,418,120,435,021,506,234,663,026,434,426,736,326,270,927,646,433,025, 577,722,695,595,343,233,942,204,301,825,548,143,785,112,222,186,834,487,969,871, 267,194,205,609,533,306,413,935,710,635,197,200,721,473,378,733,826,980,308,535, 104,317,420,365,367,377,988,721,756,551,345,004,129,106,165,050,615,449,626,558, 110,282,424,142,840,662,705,458,556,231,015,637,528,928,999,248,573,883,166,476, 871,652,120,015,362,189,137,337,137,682,618,614,562,954,409,007,743,375,894,907, 714,439,917,299,937,133,680,728,459,000,034,496,420,337,066,440,853,337,001,284, 286,412,654,394,495,050,773,954,560,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

This number exceeds 10⁷⁶⁸! Now from what we know about the universe there are in the order of 10 billion galaxies each containing about 10 billion stars. This adds up to 10²² total stars in the universe. The sun weighs in the order of 10⁴⁰ kg (mass of all 9 planets in the solar system is negligible). If we take the Sun to be the average star by weight, and the total mass of the universe is estimated to be 10 times the mass of the stars, we obtain the total mass of matter in the universe to be in the order of 10⁵³ kg, or 10⁵⁶ g. Suppose all the matter in the universe consisted of the lightest periodic element (H) in its elementary (uncombined) form. Then the total number of moles present in the universe would be in the order of 10⁵⁶; each mole contains in the order of 10²³ molecules, making the total number of atoms in the universe 10⁷⁹. Suppose we could attach a 1GHz processor to every atom in the universe, solely dedicated to the task of playing Go, and suppose it took one clock tick of the processor to play out one game of Go in its entirety.

Each molecule would then play 10⁹ games of go per second or in the order of 10¹⁶ games per year, since there are 31,557,600 seconds in a year. All of the molecules in the universe would then be able to play 10⁷⁹*10¹⁶=10⁹⁵ games per year, or 10¹⁰⁷ games in a trillion years.

So what part of our task would be accomplished after a trillion years in such a setup? We can calculate that the fraction of games played by the every atom in the universe with a 1GHz processor dedicated exclusively to playing Go for a trillion years would be approximately 10¹⁰⁷/10⁷⁶⁸=10^-661, which is indistinguishable from zero, even in the sciences concerned with high levels of precision. Thus even the resources of the entire universe did not bring us remotely close to completion of this gigantic task.

As you can see, playing all Go games is impossible by any means imaginable, and the number of possible games in Go is, for all practical purposes, infinite.