# Is the number of possible chess games infinite?

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?)

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)

• If you define a "game" as the history of moves, any game which allows repetition has infinite possible games. Snakes and Ladders has infinite possible "games". If you're interested in the complexity of solving a game, ignore the history of moves and look at the number of possible states the board can be in. – Schwern Jan 11 '15 at 18:18
• Note: computer science people would immediately object to "infinite, for all practical purposes." It is remarkably dangerous to "round up" to infinity. Generally speaking, when they make the mistake of doing so, someone rapidly breaks their algorithm by showing that it wasn't actually an infinity that they were dealing with. In encryption, it is not unheard of to have algorithms that seemed "unbreakable until heat death of the universe" which were broken due to a few tricks which decreased the problem size by 10^80 or more – Cort Ammon Feb 18 '15 at 6:46
• If I'm not in error you're referring to the TV show Person of interest, right? What they mean is by foreseeing the next possible moves you have to create a decision tree to calculate all possibilities. When Harold refers to the 'second move' he means looking two moves ahead (your's and the opponent's; in computer science this is 2th level of depth of the tree). So without doing the calculations I belief it might be correct. At least it must be a huge number though. – CMPSoares Jan 28 '16 at 3:08
• You may find this video interesting. youtu.be/Km024eldY1A – Jivan Scarano May 25 '16 at 19:21

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.

• @Tony Ennis: I edited. But actually no, the 50-move rule does not guarantee the game will end, because the players can also choose not to claim the draw. – Fate Jan 11 '15 at 12:25
• Could you please edit your answer so it reads as an answer, rather than a discussion with yourself? You start by making the incorrect claim that the number of games is finite, then correct yourself. Just make the correct claim first; if you then want to say "But if the players always take a draw as soon as they are entitled to then..." that's fine. – David Richerby Jan 11 '15 at 13:42
• Actually, as of July of last year, there is a 75 move rule that is mandatory. So the 50 move rule does not guarantee an ending to the game, but the 75 move rule does, although the longest game increases to 17,697 plies. Given an average branching factor of 35, one might estimate the possible number of games at 35^17697, or about 10^27000. – Deedlit Jan 12 '15 at 6:24
• Even if we disregard both the 50- and 75-move rule, if the players play on without claiming a draw, at some point a threefold repetition has to occur. I don't know if taking a draw here is required, but I would consider a finite number of distinct games with infinite possibilities of repeating a finite number of possible games, for the purpose of this question. – 11684 Feb 22 '17 at 21:54
• JFYI, and similar to the 50- and 75-move rule issue, the threefold repetition is not mandatory, but there exists a five-fold repetition rule that is mandatory. – Ignacio Calvo Nov 14 '17 at 14:44

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.

• Yeah, impressive numbers. Funny we can't compute above 14 moves... I wonder how many moves can be computed for Go... Three? :) – landroni Jan 10 '15 at 16:54
• It doesn't need to be "considered infinite for all practical purposes" since it actually is infinite. Although the 50-move and threefold repetition rules allow either player to claim a draw, they don't automatically end the game. – David Richerby Jan 11 '15 at 13:41
• @landroni Go is probably easier to compute than chess. There are 361 one move games, 361*360 two move games, and 361*360*359 three move games. The number of four move games depends on if suicide is allowed. If it is, then there will be 358 possible fourth moves, unless black's first two stones take white's first stone in the corner, in which case there are 359. So 361*360*359*358 + 8 four move games. If suicide is not allowed, then there are 361*360*359*358 - 8*358 four move games. You can continue in this fashion, separating into cases - 14 moves is probably doable with computer effort. – Deedlit Jan 12 '15 at 8:25
• Note that this is 13 half-moves, or ply - seven moves by white and six by black. – RemcoGerlich Jan 14 '15 at 8:31
• @DavidRicherby: the 75 move rule (new since july 2014) is mandatory though. – RemcoGerlich Nov 27 '15 at 21:53

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

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

• Could you include an overview of the methodology here? – landroni Feb 5 '15 at 15:03

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.

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.

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

• Well, if two friends who don't care about any official rules like to play a nonsense game and call it chess, they can! But should we call it chess for the purposes of this site? A position with only two kings is an immediate draw. – RemcoGerlich Nov 27 '15 at 21:55

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).

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.