Chess by the numbers
What is the Shannon number?
The Shannon number is 10120: Claude Shannon’s 1950 estimate of the number of possible chess games. He used it to show that a computer could never play perfect chess by simply looking at every possible game.
Where it comes from
Shannon, the founder of information theory, wrote one of the first papers on chess-playing computers: “Programming a Computer for Playing Chess”, published in the Philosophical Magazine in 1950. In it he reasoned that a typical position offers about 30 moves, so a pair of moves — one by White and one by Black — has about a thousand possibilities, and that a typical game lasts about 40 such pairs. A thousand to the power of forty is 10120.
How big is that?
The observable universe is often said to contain about 1080 atoms. A computer that had checked a billion games every second since the Big Bang would have looked at fewer than 1027 of them. Shannon’s conclusion still holds: chess will not be solved by trying every game.
Later estimates
Victor Allis estimated in 1994 that the game-tree complexity of chess is at least 10123, assuming 35 choices per move and games of 80 moves. With today’s rules, which end every game after enough moves without progress, François Labelle put the number of all possible games between about 1029,241 and 1034,082. Shannon also estimated the number of positions, at about 1043; the modern estimate is 4.8 × 1044 (see how many chess positions there are).
What has been counted exactly
Exact counts exist for the first fifteen moves, where there are 2,015,099,950,053,364,471,960 games — about 2 × 1021, still unimaginably far below 10120. The first four are drawn on this site’s map; the rest are in how many possible chess games there are.