How many chess games are possible? What's counted, and Shannon's number

Q: How many chess games are possible?

So many that nobody has counted them. Only the start is counted exactly: 20 possible games after one half-move, 400 after two, and almost 69.4 trillion after ten. The often-quoted 10¹²⁰, a one with 120 zeros, is a rough estimate from 1950, not a count.

What has been counted exactly

A half-move is a move by one player: 1.e4 is one half-move, and 1.e4 e5 is two. White can start in 20 ways and Black can answer in 20 ways, so after two half-moves there are 400 possible games. It grows quickly from there.

Computers have worked out exactly how many games are possible, but only a little way into the game. Here are some of the numbers.

Two games are different if the moves differ. A game that is already over isn't counted in the later rows; the first checkmates come after four half-moves.
Half-moves	Possible games	Different positions
1	20	20
2	400	400
3	8,902	5,362
4	197,281	72,078
5	4,865,609	822,518
6	119,060,324	9,417,681
8	84,998,978,956	988,187,354
10	69,352,859,712,417	85,375,278,064

Why are there fewer positions?

Different move orders can lead to the same position. So from the third half-move on, there are fewer positions than games.

After 1.e4 e5 2.Nf3. After 1.Nf3 e5 2.e4 the pieces stand exactly the same: two different games, one position.

Shannon's number: a rough estimate

In 1950 the mathematician Claude Shannon wrote a paper about how a computer could play chess. He wanted to show that it couldn't calculate everything. A typical position has about 30 possible moves, so a move by White and a move by Black give about 1,000 possibilities. A typical game runs about 40 moves before someone resigns. That makes 1,000 to the power of 40, or 10¹²⁰.

This is often called the Shannon number. It isn't the number of possible games, and Shannon didn't say it was. It's a conservative estimate for a typical game, made to show that chess can't be worked out move by move. Longer games exist, so the true number is larger. Nobody knows exactly how much larger.

How many positions are there?

Shannon also made a rough estimate of the positions: about 10⁴³. There is a better one today. In 2022 John Tromp and Peter Österlund worked out how many conceivable positions there are, drew a random sample of them, and checked which could arise in a legal game. The result is about 4.8 × 10⁴⁴ positions, with a margin of error of 0.03 × 10⁴⁴. So this is an estimate too, but one with its method on the table.

Do they ever run out?

Yes. Under FIDE's rules a game is drawn without either player having to claim it if the same position has appeared five times, or if both players have made 75 moves with no pawn move and no capture. So no game can go on forever, and the number of possible games is a finite one. The rules are under draws.

Try it yourself: play a game, or see how others have continued in the openings.

Sources

OEIS: A048987 (possible games after n half-moves) and A083276 (positions)
Chessprogramming Wiki: Perft Results
Wikipedia: Shannon number
Claude E. Shannon: Programming a Computer for Playing Chess, Philosophical Magazine, 1950; A Chess-Playing Machine, Scientific American, February 1950
John Tromp: ChessPositionRanking (GitHub, with Peter Österlund's check of the positions)
FIDE: Laws of Chess, effective January 1, 2023 (Article 9.6)
The first three rows, and the number of games on the fourth, are also counted in this site's own test

The facts on this page come from the sources above and, where possible, were checked across several sources. Where the sources disagree, we say less or say what each one says. A number that is an estimate says whose it is, and a counted number comes with its date.

Updated November 9, 2026