There Are More Possible Chess Games Than Atoms in the Universe

Quick Answer

The number of possible chess games is estimated at roughly 10^120, known as the Shannon number (after mathematician Claude Shannon). The number of atoms in the observable universe is approximately 10^80. That means there are about 10^40 times more possible chess games than atoms -- a number so large that even writing it out in full would be pointless. Chess, played on a simple 8x8 board with 32 pieces, generates more complexity than the physical universe contains matter.

A Number Too Big for the Universe

Here is a comparison that should break your brain a little. Take every atom in the observable universe -- every atom in every star, every planet, every gas cloud, every galaxy, every speck of dust floating in intergalactic space. That is approximately 10^80 atoms.

Now consider every possible game of chess that could ever be played, from the first move to the last. Not just games that make strategic sense -- every legal sequence of moves, including terrible ones. That number is approximately 10^120.

The ratio between these two numbers is 10^40. That is 1 followed by 40 zeros. To put this in (barely) human terms: if every atom in the universe were itself an entire universe, and you counted all the atoms in all of those universes, you would still not have enough atoms to assign one to each possible chess game.

Chess is a finite game on a finite board, and it contains more possibility than the physical universe contains substance.

Where 10^80 Comes From

The estimate for the number of atoms in the observable universe is derived from cosmological observations.

The observable universe has a radius of about 46.5 billion light-years (accounting for the expansion of space since the Big Bang). Its total mass is estimated at roughly 10^53 kilograms. Most of this mass is in the form of hydrogen and helium atoms, with an average mass of about 10^-27 kilograms per atom.

Dividing total mass by average atomic mass gives approximately 10^80 atoms. This is a rough estimate -- it could be off by an order of magnitude in either direction -- but 10^80 is the standard figure used in physics.

This is already an incomprehensibly large number. If you tried to count to 10^80 at a rate of one number per second, it would take longer than the remaining lifetime of every star in the universe. There are more atoms in the observable universe than grains of sand on Earth (about 7.5 x 10^18) by a factor of 10^61.

And yet, chess has it beat.

Where 10^120 Comes From

Claude Shannon, the father of information theory, first estimated the number of possible chess games in a 1950 paper. He approached it through game tree analysis.

A chess game begins with 20 possible first moves for White (16 pawn moves plus 4 knight moves). Black then has 20 possible responses. After just one move each, there are 400 possible board positions.

After two moves each, the number jumps to roughly 71,852. After three moves each, it is approximately 9 million. The branching factor -- the average number of legal moves available per turn -- is about 30 to 35 throughout a typical chess game.

Shannon estimated an average game length of about 80 half-moves (40 moves by each player). With an average branching factor of 30, the number of possible games is approximately 30^80, which works out to roughly 10^120.

This is the Shannon number, and while subsequent researchers have refined the estimate (some calculations put it higher, around 10^123, when accounting for longer games), the order of magnitude has held up remarkably well for a back-of-the-envelope calculation made in 1950.

Tip

The Shannon number counts possible games, not possible board positions. The number of legal board positions in chess -- the number of distinct arrangements of pieces that could occur in a game -- is estimated at roughly 10^44 to 10^47 (known as the Shannon number for positions). This is a much smaller number, but still enormous. The game tree is vastly larger because the same position can be reached by many different sequences of moves, and each sequence constitutes a different game.

Why Chess Is So Complex

The explosive growth of possibilities in chess comes from the combinatorial nature of sequential decision-making. Each move multiplies (rather than adds to) the total number of possibilities. This is the same mathematical principle that makes password security work -- each additional character multiplies the number of possible passwords by the size of the character set.

In chess, the "character set" is about 30 possible moves, and the "password length" is about 80 moves. Thirty raised to the eightieth power is a number that makes the atom count of the universe look quaint.

Several features of chess contribute to its absurd complexity:

Piece diversity. Six different piece types with different movement rules create a rich interaction space. A game with only one piece type (like checkers) has fewer possibilities.

Board openness. Pieces can potentially reach any part of the board, creating long-range interactions that affect the entire game state. This means distant positions are not independent -- moving a rook on one side of the board can affect the viability of a pawn advance on the other side.

Game length. Decisive games average 40-60 moves, but draws and endgames can extend much longer. The theoretical maximum game length under chess rules (with the 75-move rule and fivefold repetition rule) is estimated at approximately 5,870 moves -- and at that length, the game tree becomes practically infinite.

Has Chess Been Solved?

No, and it almost certainly never will be through brute force.

A game is "solved" when the optimal move for every possible position is known. Checkers was solved in 2007 by Jonathan Schaeffer's team at the University of Alberta after 18 years of computation. With perfect play from both sides, checkers is a draw.

Chess is astronomically more complex. The 10^44 possible positions alone make exhaustive computation impossible with any foreseeable technology. Even if every atom in the universe were a computer calculating one billion positions per second, and they all ran for the entire 13.8-billion-year age of the universe, they would evaluate approximately 10^106 positions -- still falling short of the full game tree by a factor of 10^14 or more.

Modern chess engines like Stockfish and AlphaZero play at superhuman levels, but they do so through heuristic search and pattern evaluation, not by exploring the full game tree. They prune the vast majority of possible moves and evaluate only the most promising lines. They play excellent chess, but they have not "solved" the game.

Chess may never be formally solved. It may be one of those problems where the answer exists in principle but is physically impossible to compute -- not just difficult, but beyond the capacity of any computer that could be built from the matter and energy available in the observable universe.

The 64 squares and 32 pieces generate a mathematical universe larger than the physical one. Every time you sit down to play a game of chess, there is a near certainty that the specific sequence of moves you play has never occurred before in the history of the game, and never will again.

Your chess game is unique in a space larger than the cosmos. Not bad for a board game.