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How many possible chess games are there?

How many possible chess games are there?
about 10^120 games; about 4.8 × 10^44 legal positions
There are about 10^80 atoms in the observable universe. There are roughly 10^40 times more possible chess games than that — meaning you could give every atom in the universe its own universe of atoms and still not have enough.

Two different numbers get quoted and they answer different questions. The game-tree complexity — how many distinct games could be played — is around 10^120, an estimate Claude Shannon made in 1950 and which is now called the Shannon number. The number of legal board positions is far smaller and has been computed rather than estimated: about 4.8 × 10^44, calculated by John Tromp in 2021 to within 0.02%. A chess engine searches positions, not games, which is why the smaller number is the one that matters in practice.

How the figure is arrived at

Shannon's estimate: roughly 30 legal moves are available in a typical position, a game lasts about 40 moves for each side, so the tree has about 30^80 ≈ 10^120 leaves. Tromp's position count is a genuine computation using a sampling technique over an encoding of all board configurations, testing legality on a random sample and scaling — which is why it comes with a stated error bar and Shannon's does not.

Other numbers and possibilities counts

Common questions

How many possible chess games are there?
about 10^120 games; about 4.8 × 10^44 legal positions
How is that estimated?
Shannon's estimate: roughly 30 legal moves are available in a typical position, a game lasts about 40 moves for each side, so the tree has about 30^80 ≈ 10^120 leaves. Tromp's position count is a genuine computation using a sampling technique over an encoding of all board configurations, testing legality on a random sample and scaling — which is why it comes with a stated error bar and Shannon's does not.

Related pages

Sources

  1. Calculated on this page — At One Place

How these figures are compiled and checked