How quickly does chess get big?

White has 20 possible first moves, and Black has 20 possible replies. After just five moves by each player, there are about 69 trillion different ways the game could have gone.

    Possible move sequences after each move

    Normal scale: each gridline adds 10 trillion

    Possible chess move sequences after 1 to 10 moves On a normal scale, the first eight moves look like zero and the line shoots up to about 69 trillion at move 10. On a log scale, the same numbers form a nearly straight line, because the count multiplies by about 20 to 30 times with every move.

    Every move multiplies the possibilities. There are 400 possible sequences after two moves and about 69 trillion after ten. On a normal chart, almost everything before the last step is invisible: the 2.4 trillion sequences after nine moves are only about 3.5% of the final height.

    Neither chart is wrong. The normal scale shows how much bigger the last number is; the log scale shows how steadily it grew. When numbers multiply instead of add, the chart you pick decides which story people see.

    What looked “sudden” to you that was really growing all along?

    How the numbers grow, move by move

    Every move multiplies the possibilities. There are 20 legal first moves for White and 20 replies for Black, so 400 sequences after two moves. As pieces come out, each position offers more choices, and the total multiplies by roughly 20 to 30 times with every move. After ten moves there are 69,352,859,712,417 legal sequences, about 69 trillion.

    On a normal chart, almost everything before the last step is invisible. After 9 moves there are about 2.4 trillion sequences, which sounds enormous, but it's only about 3.5% of the final height. Millions and billions round down to nothing. Switch the chart to log scale, where each labeled gridline is 1,000 times the one below it, and the same data becomes a steady, nearly straight climb.

    Explore more numbers too big to picture

    See Numbers Beyond Comprehension for how millions, billions, and trillions compare, Shared Birthdays for another case where combinations grow faster than intuition expects, and Axis Truncation for how a chart's axis can change the story.

    Related exhibits

    Sources: Chess Programming Wiki, "Perft Results" (the number of legal move sequences from the standard starting position for depths 1 through 10). "Moves" here are single turns by one player (half-moves, or plies); sequences that end early in checkmate are not counted. Chess pieces: the standard "cburnett" set by Colin M.L. Burnett (CC BY-SA 3.0), as used on Wikipedia and Lichess.