Is the number of possible chess games infinite?

Question

This question is somewhat related to Can the total number of possible wins/draws/losses be calculated?, but slightly different.

There is a recent TV show episode that claims that there are "more possible games of chess than atoms in the universe". They go on that "each possible move represents a different game, a different universe [..]"; "by the second move there are 72084 possible games, by the third -- 9 million, by the fourth --- 318 million".

So is the total number of chess games infinite, for all practical purposes given human and technological limitations? And do the above numbers actually hold up to scrutiny? (i.e. What are the estimated possible games by, say, the 10th move?)

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Curiously, Wikipedia seems to be implying that the number of games can be estimated:

the number of possible games [in Go] is vast (10⁷⁶¹ compared, for example, to the 10¹²⁰ possible in chess)

Answer 1

The maximum number of moves in a chess game is not infinite, it's 11797 plies = 5898 moves and a half. This is due to the fifty-move rule.

So no, the number of possible chess games is not infinite.

The maximum number of legal moves in a position is 218. So a crude upper bound for the number of possible chess games is 218^11797 = 10^27586

Wait, actually after fifty moves without any capture or pawn movement the players can also continue playing without claiming the draw...

Article 9.3 of FIDE Laws of Chess states that:

9.3

The game is drawn, upon a correct claim by a player having the move, if:

• he writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move which will result in the last 50 moves by each player having been made without the movement of any pawn and without any capture, or
• the last 50 moves by each playerhave been completed without the movement of any pawn and without any capture.

So I guess the number of possible chess games could be considered as infinite then...

But if you're not interested with the previous theoretical numbers:
The average number of legal moves in a position is around 35, and the average length of a chess game is around 40 moves = 80 plies, so an estimate of the number of "rational" chess games is 35^80 = 10^123
As for the total number of legal positions, it's somewhere between 10^40 and 10^50.

Answer 2

Q1: Yes. The total number of chess games can be considered infinite for all practical purposes. We don't have the technology to brute force over the first 13 moves from the initial position.

Q2: The actual numbers all the way up to depth 13 is known. The exact number of possible positions for the 10th moves is 69,352,859,712,417. Read this Wikipedia article for more details.

There is an attempt for depth 14 but so far the calculation after months and months is still running.

Answer 3

One simple argument that the number of chess games is finite could be as follows.

Due to the 50-move rule, any 50-move subsequence of a given chess game will contain at least one capture or a pawn move. Since there are finitely many pieces on the board, and since pawns can move only finitely many times during a game, the number of moves in a chess game has a finite bound. Since in each move, there are only finitely many possibilities, the numbers of all games is finite.

Note that this argument is almost useless if one wants to get an estimate on the number of possible games. If for nothing else, the only thing I use above is the 50-move rule and how the pieces move, so the repetitions are allowed (max. 50-fold repetitions, of course). Hence, the argument is just theoretical, not practical.

Answer 4

At some point you'll run out of combinations. So the answer is basically no.

Answer 5

According to my calculations is about 10 ^ 134 different variants of the game http://jknow.republika.pl/chessexplorer/szachy.html

Answer 6

50-move rule includes 'upon a correct claim': No claim, no implementation of the rule. Same applies to repetition. Ergo, infinite.

Without a mandatory maximum number of moves, of course.

Answer 7

On understanding FIDE laws-First they are for use with tournament play- so given that information do you understand how FIDE laws doesn't relate to two friends who decide to play? For two friends, who whittle down to two kings only, they can chase each other around the board an infinite amount if they wished. (Plausible-not really, possible-yes)

On FIDE law 9.2 - 50 consecutive moves must be made where there is no pawn moved and no capture made. This obviously wouldn't be a "50-move game" (e.g. 1.e4 would mean another 50 consecutive moves without a pawn moved or capture made)

On FIDE law 9.6 - 75 consecutive moves... Same reasoning that this is not a 75 move game.

One of the first evidence of a recorded game went 14 consecutive moves (1. e4 b6 2. d4 Bb7 3. Bd3 f5 4. ef5 Bg2 5. Qh5 g6 6. fg6 Nf6 7. gh7 Nh5) Even though the 15th was checkmate- if the winner decided not to checkmate he would have still needed 75 more moves to declare the draw in FIDE law 9.6 (with 12 pawns left on the board -I doubt it would have happened in 75 moves)

Respectfully, CFC

Answer 8

Since other answers here point to repetition or similar I wish to modify your question to, "Are the number of possible chess POSITIONS infinite. The answer is "No." The total is very large though and estimated to be about 10 to the 120th power. The total number of atoms in the universe is thought to be only 10 to the 80th power. Wow!

The number 10 to the 134th power given by a previous responder may be correct.

The Chinese game "Go" is even more varied than chess(but boring by comparison since chess has pieces with different abilities, while in Go all the pieces are the same).

Answer 9

I may be looking at this too simplistically but it seems to me the number has to be finite. If we look at the board and the pieces rather than the game of chess and calculate the number of possible variations we can obtain an answer that is finite. Mind bogglingly huge but finite. Given not all combinations are possible in a game of chess the number of combinations in a game of chess has to be less than this finite number and therefore a finite number itself.