Chess begins with a precise simplicity. On the very first move, White has exactly 20 legal options: 16 pawn moves (each pawn can advance one or two squares) and 4 knight moves. Black has the same 20 replies.
That's 400 possible board states after one complete turn each.
Then the game opens up, pieces start interacting, and the branching factor — the number of legal moves available at each position — begins climbing. The question is: by the end of three complete turns for each side (six half-moves total), how many distinct sequences of play have occurred?
Building the Estimate
The first turn is known precisely. White: 20 options. Black: 20 responses. So:
20 × 20 = 400 positions after turn 1
After that, the board starts opening. Take a simple example: 1. e4 e5. That single exchange opens diagonal lines for the bishops, gives the queen a potential path, and creates new attack and defence possibilities. Both sides often now have 25 to 35 legal moves available.
A practical estimate for the branching factor as the game opens up:
- Turn 2 average: ~25 moves per side
- Turn 3 average: ~30 moves per side
So the full six-move sequence:
20 × 20 × 25 × 25 × 30 × 30 = 225,000,000
Roughly 225 million possible sequences in the first three turns.
The Real Number
Chess engines compute exact values called perft counts — complete enumerations of every legal position sequence from a given starting point. These are used to verify that a move generator is completely correct. Any perft error, even off by one, indicates a bug.
The perft(3) value — three full turns for each player — is:
119,060,324
Our estimate of 225 million is about 1.9x too high. That puts it less than half an order of magnitude away from the true answer, which on Magnitudle's scoring translates to roughly 93 out of 100.
Fermi estimate
~225 million
True answer
119,060,324
1.9× too high — less than half an order of magnitude off. The estimate correctly identified the scale.
Why the Estimate Runs High
The reasoning was sound, but reality is messier than a uniform branching factor.
Not every chess position has 25 to 30 legal moves. Checks severely restrict options — sometimes to a single legal reply. Captures reduce the number of pieces available for future moves. Some opening variations are sharply constrained by forcing sequences where one or both players have very few good alternatives.
The true average branching factor across all positions in the first three turns is slightly lower than 25 to 30. Weighting the estimate toward 22 to 25 instead — to account for these constrained positions — would produce a result much closer to the actual figure.
The Deeper Number
Three turns produces 119 million possible sequences. That already feels incomprehensibly large for a game that starts with 20 options.
Chess famously gets much larger from there. Claude Shannon estimated in 1950 that the total number of possible chess games is around 10^120 — a 1 followed by 120 zeros. For context, the estimated number of atoms in the observable universe is around 10^80. There are more possible chess games than atoms in the universe by a factor of 10^40.
Three turns gets you to roughly 10^8. The full game gets you to 10^120. The compounding across 40+ turns from a branching factor of 30 to 35 produces growth that exceeds anything in physical reality.
Why This Matters for Chess AI
The sheer scale of the game tree is precisely why the history of chess AI is interesting. Early approaches tried to enumerate as far ahead as computing power allowed — which, even for modern hardware, barely scratches the surface of the true game tree.
AlphaZero took a fundamentally different approach: rather than searching the tree, it learned to play by playing millions of games against itself and developing intuitions about which positions were strong or weak. It never explicitly computed what every legal sequence looked like. It developed something closer to taste.
Estimation and intuition, not exhaustive enumeration — which is also, roughly, how a good chess player approaches a position.
For more on how huge numbers compound in unexpected ways, see How Big Is a Billion, Really? And for the general technique of reasoning from known starting points to large unknowns, see What Is a Fermi Problem?