Chess by the numbers
How many possible chess games are there?
After one move each there are 400 possible games; after two moves each, 197,281; after five moves each, 69,352,859,712,417. Nobody knows the exact number of complete games. Claude Shannon’s famous estimate is about 10120, and under today’s rules, which end every game sooner or later, the total lies somewhere between about 1029,241 and 1034,082.
Counting move by move
Here a “move” is one turn by one player, so after 2 moves both sides have moved once. A “game” is any legal sequence of moves; two games that reach the same position by different routes still count twice (for positions, see how many chess positions there are).
| Moves played | Possible games |
|---|---|
| 1 | 20 |
| 2 — one move each | 400 |
| 3 | 8,902 |
| 4 — the map shows all of these | 197,281 |
| 5 | 4,865,609 |
| 6 | 119,060,324 |
| 7 | 3,195,901,860 |
| 8 | 84,998,978,956 |
| 9 | 2,439,530,234,167 |
| 10 — five moves each | 69,352,859,712,417 |
| 11 | 2,097,651,003,696,806 |
| 12 | 62,854,969,236,701,747 |
| 13 | 1,981,066,775,000,396,239 |
| 14 | 61,885,021,521,585,529,237 |
| 15 | 2,015,099,950,053,364,471,960 |
Each extra move multiplies the count by about 20 at first and by more than 30 later, as the pieces come out and have more squares to go to. The first four moves are small enough to draw: the map shows every one of the 197,281 games around the starting position, and around any position you play.
Whole games: the Shannon number
In 1950 the mathematician Claude Shannon estimated that a typical game offers about a thousand possibilities for each pair of moves and lasts about forty such pairs, which gives roughly 10120 games — the Shannon number. Later estimates are in the same spirit: Victor Allis put the game-tree complexity of chess at no less than 10123.
Is the number finite?
Under today’s FIDE rules it is: since 2014 a game ends automatically after 75 moves by each side without a capture or a pawn move, or when the same position appears five times. François Labelle worked out that the longest possible game is then 8,848½ moves long, and that the number of possible games lies between about 1029,241 and 1034,082.
How long would counting take?
Exact counts exist up to 15 moves, where there are 2,015,099,950,053,364,471,960 games. Even at a hundred million games a second — a fast computer — counting them one by one would take about 640,000 years; on this site the ladder in the side panel times the same count on your own device.