Chess by the numbers
How many chess positions are there?
About 4.8 × 1044 legal positions, according to John Tromp and Peter Österlund (2022). Far fewer appear early on: 20 after the first move, 400 after one move each and 72,078 after two moves each — even though 197,281 different games lead to those 72,078 positions.
Positions after each move
A position here means where every piece stands, whose turn it is, and the castling and en passant rights — the things that decide whether a position has “repeated” under the rules. Counted that way:
| Moves played | Different positions | Games that reach them |
|---|---|---|
| 1 | 20 | 20 |
| 2 | 400 | 400 |
| 3 | 5,362 | 8,902 |
| 4 | 72,078 | 197,281 |
| 5 | 822,518 | 4,865,609 |
| 6 | 9,417,681 | 119,060,324 |
| 7 | 96,400,068 | 3,195,901,860 |
| 8 | 988,187,354 | 84,998,978,956 |
| 9 | 9,183,421,888 | 2,439,530,234,167 |
| 10 | 85,375,278,064 | 69,352,859,712,417 |
| 11 | 726,155,461,002 | 2,097,651,003,696,806 |
Why games outnumber positions
Different move orders often lead to the same position — chess players call this a transposition. 1. e4 e5 2. Nf3 Nc6 and 1. Nf3 Nc6 2. e4 e5 are two games but one position. After four moves the 197,281 games reach only 72,078 positions, so on average each position can be reached in almost three ways; after eleven moves the games outnumber the positions nearly three thousand to one. Every opening page on this site gives the same two numbers for the four moves after it.
All legal positions
Counting every legal position one by one is impossible, so Tromp and Österlund numbered a slightly larger set of positions that is easy to count, took random samples from it and checked how many were legal. Their estimate is (4.822 ± 0.028) × 1044 legal positions, with 95% confidence. Seventy years earlier Claude Shannon had guessed about 1043, from the number of ways to place the pieces on the board.
That is a lot of positions — but tiny next to the number of games, which is at least 10120. See how many possible chess games there are.