Chess vs. The World
True or false:
There are more possible games of chess than there are atoms in the earth.
For reference, there are approximately 1 0 5 0 atoms in the earth
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10 solutions
To all that said true, I question how you resisted the urge to click error, stack overflow; to all that said false, I question how you resisted the urge to click error, stack overflow.
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I resisted doing so because I know it isn't the right answer. Well, I almost did, though.
But the number 1 0 1 2 0 is just an approximated result if it is assumed that the total number of chess moves played in the game is 40. Sir G.H.Hardy had provided another estimate which is 1 0 ( 1 0 5 0 ) which can have all possible games complying with the chess rules like the 50-move rule, 3-fold repetition, etc.
How come I put 'True,' there are more possible games of chess than atoms, and got it wrong?
What was that "error stack overflow" answer and why was it wrong?
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It... Uh... It's not an answer... In layman's terms, it just means something like "this is too much calculation for me / the calculator / the computer"...
There are far more atoms in the earth, this is a silly question. It shows the inherent ego that comes with most chess '' masters''. There are more atoms in the moon than chess moves
I think there are an infinite number of possible games. My reasoning is as follows: If 1 player is left with a rook and a king and the other has just a king then a checkmate is possible but there is no limit on how many moves it could take.
So there are indeed more possible games than atoms in the earth.
It also depends on how you play the 'repetition of positions' rule. Since there are finitely many pieces and finitely much board space, if any given position is only allowed to occur twice at most (3 and the game is called a draw), then any game is gaurenteed to be finitely long and there are finitely many sequences of game-states -- an INCOMPREHENSIBLY HUGE NUMBER of states, but finite. :)
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occurance of a position morethan twice is not a draw. a draw is occurane of a two positions alternatively for atleast three times that back and forth movement 5 times is a draw
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That's not right, actually. If a position occurs 3 times in game, it is draw. It does not matter if there were several positions in between. This can be hard to spot, but those are the rules. There are 6 ways to draw a chess game. 1. Mutual agreement. 2. Stalemate (when one player has no possible legal moves). 3. Perpetual check (this different than the 3 position rule, occurs when your opponent can and does deliver check on every move and you cannot prevent it). 4. 50 move rule (if no pawn has moved for 50 moves and no captures have been made, the game is drawn). 5. Insufficient mating material for both sides. 6. 3 fold position repetition. as described by Zandra.
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@Thomas Thomas – On another extreme, what if the game is timed, effectively capping the maximum number of moves? A 1 turn game of chess has 20 possible states. 2 turns and you're up to 400. And then it gets more complex very quickly. About how many turns do you need to allow to exceed 1 0 5 0 possible game paths?
I originally proposed this question because I think it's fascinating how large the number becomes and how quickly it does so. Most people think that mind-bogglingly physical things are the biggest things in their experience. But, 52! for example, is about 8 × 1 0 6 7 and that's just the number of ways the card deck you're holding could be arranged. :)
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@Zandra Vinegar – There are computers working on solving this, but it is a fairly complex problem. They have found that after 13 plys (half moves) there are exactly 1,981,066,775,000,396,239 possibilities. They estimate that after 40 moves there are 10^43 possibilities, but that is a fairly crude estimate.
P(40) = 64!/(32! * (8!)^2 * (2!)^6)
I am not sure how they came up with that formula for 40 moves. Seems arbitrary to me. I do not see where the number 40 comes into play. Also, this formula could not possibly account for moves that are illegal due to check, or when it is possible to castle and when it is not.
One thing I can say for sure - chess is a draw if both sides play correctly. I think it is more interesting to ask how many playable positions exist. The approximation for the maximum number of logical possible chess games is 4,670,033.
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@Thomas Thomas – 40 moves used to be the first time control. I think it was 40 moves in 2 1/2 hours. Most of the games would be through by then. Forty may be the average number of moves.
@Thomas Thomas – Amazing! I also have no idea why they've chosen to use the number 40 in this context. If I had to guess, maybe they chose that number because it's close to the observed average length of real games, but there's no real agreement on that fact either. And no reason I can think of to spend too much time focusing on 40.
What interests me most is the shape of the combinatorial tree -- the relative degrees of different nodes, extreme cases (board states where there's either only 1 move possible or many more moves possible than usual) and trends across stages of the game.
Regarding your own interests -- is a "playable position" a position that can be reached through some sequence of allowed moves?
Why do you say that chess is a draw if both sides play correctly?
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@Zandra Vinegar – A playable position is one where neither side has incurred any major disadvantage. There are a lot of terrible moves a person can play in the opening leading to a lost position. Most of these are fairly well known, and that is why openings have developed over the centuries with known "best moves". Grandmasters will often play a game for 20 moves before they even start thinking.
It is interesting to note that many openings transpose into positions from other openings. For example, some variations of the Nimzo-Indian Defense will transpose into the Caro-Kann Defense. Although the move orders used to reach such a position are completely different the end result is the same. This is fairly common! So when you ask about the number of possible chess moves it may be more appropriate to ask about the number of different possible chess positions.
Chess has been called a draw by most of the world's leading Grandmasters. A small advantage is not enough to win due to the various ways to obtain a draw. In a recent World Championship game between Carlsen and Anand, Anand was ahead by a whole knight, but due to dynamic possibilities, Carlsen was able to draw the game. Of course, sometimes a pawn is enough of an advantage win a game, but most experts will tell you that you need to create at least 2 advantages to be able to force a win. That just isn't going to happen if both sides play "best moves". Although chess has not been solved by computers (like checkers), with best play from both sides, any game of chess should be drawn. That's why, at the top levels, most games result in a draw.
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@Thomas Thomas – I agree. I have gone through games with a group of good players and a few masters. When people are calm and can help each other is is very hard to win a game of chess. The reason people lose is partially psychological and that is true for most games and sports. . If they get frustrated and the game looks unwinnable people throw in the towel. People don't know they do it but sometimes they will make bad moves just to get the game over with.
@Thomas Thomas – If any of those happen a draw can be claimed- it is not a draw automatically.
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@Rob Williams – True, a draw is "claimed" for 3 fold repetition, perpetual check and the 50 move rule. But if a draw is claimed for these reasons, the game is drawn and your opponent cannot contest this. However, in the case of stalemate and insufficient mating material the game is drawn automatically. It is interesting to note that the 50 move rule was modified when computers started to think about endgame positions. They found that certain endgames positions are winnable but require more than 50 moves to force checkmate (without moving a pawn or capturing a piece). The rule was therefore changed to allow certain exceptions in which 100 moves were allowed with particular material combinations. However, more and more such winnable positions were later discovered, and in 1992 FIDE abolished all such exceptions and reinstated the strict 50-move rule.
Actually it is only a draw if a player offers a draw at that point (same with the 50 moves without a take or pawn advance draw). For that reason, if neither player offers a draw, there is clearly an infinite number of POSSIBLE games
I could be incredibly wrong I'm not a chess expert but I believe that in a real game of chess when only left with a king of you make 50 moves avoiding checkmate it is called as a draw
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It can be claimed a draw by either of the two players, but is not automatically a draw
No. The game ends in a draw if no piece is taken in a 50 moves interval.
If such condition arises, then the game must be won by either player in limited number of moves or the game is declared draw.
There are definitely not an infinite number of games. There are a finite number of positions and repeating the same position enough leads to a stalemate.
Nonetheless this is not a math problem. We aren't supposed to calculate the number of chess games, we are supposed to just look it up.
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Like many questions in math and elsewhere, you could look it up, but you don't need to. Even without knowing much about chess, it's possible to find justification for the lower bound of 1 0 5 0 with your own reasoning. The question "is the number of possible games infinite" is a matter of knowing some trivia/interpreting the rules, but the basic exploration of "how quickly does the game tree grow as players make choices," is very cool.
Well, I guess if one knows how to, they could calculate it! I don't think most people here know how to, though.
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But you don't need to approximate the Shannon number in order to solve this, just prove a much lower bound. :)
I think most of the people solving this are using only their intuition that the number of game paths grows exponentially. And it's fast enough that, since even natural chess games are frequently 40 moves or longer, exceeding a bound of 1 0 5 0 is almost guaranteed. But if " almost " isn't good enough for you, I think most people here know enough to prove it with a bit of elbow grease.
If I'm not mistaken, after a certain number of moves in that situation, if the rook can't checkmate the king it's a stalemate
no, there is a 50 move limit
Joe Mansley You are right. The draw is not automatic. One of the players, on their move turn, must claim the draw with the arbiter. If neither player claims a draw the game can go on for ever.
It's probably simpler just to say that there is no limit on the number of moves in a game.
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There is a limit. 50 moves without a pawn move, or a piece captured is a draw. So it's a finite number
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We are talking here the number of moves or possible moves. Let us not consider the limit of the number of moves in order to draw or win. Let us just consider the theoretical value of combination (combinatorics). Let us just focus the mathematical aspect not the technical one. Let us not argue on the latter.
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@Brendix Emata – The game ends in a draw when there is a three fold repetition of any position or there are 50 moves without a capture or a paw move. The mathematical limit on the number of moves possible must take into account the fact that the game ends under these conditions, just like it must take into account the fact that the game ends when a king is checkmated. It isn't clear that you must know the number of possible moves in order to know the number of possible games, however. They are not the same thing.
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@Brian Rasmussen – There is only one automatic draw. If during a time out neither side has sufficient strength to checkmate the other. If both sides only have a king for example. In every other case one of the players must claim a draw.
@Brian Rasmussen – One of the players, on their move turn, must claim the draw with the arbiter. If neither claims a draw there isn't one.
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@James Briggs – The same is true of checkmate. One of the players must claim the checkmate. In fact, a player could make an illegal move and get away with it unless their opponent complained to the arbiter. If one doesn't assume that rules are followed then of course the number of moves possible is trivially infinite. But that doesn't really get you anywhere. For the analysis to be meaningful you have to assume that the rules are followed.
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@Brian Rasmussen – One has to assume the rules are followed but there is no rule against playing on after a a three move repetition. I do it at times in competitive chess. It can be used as a psychological weapon. An illegal move is not a chess move. Once checkmate is obtained any move after that is illegal and therefore not a chess move. I believe that the question should be asked in a better way. Asking how many meaningful moves are possible in a chess game solves the problem.
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@James Briggs – I think it's a difficult question to generalize. I'm curious now about what fractions of the 'possible game' space are carved out by existing or even house rules.
The maximum number of moves in a chess game is 5949, factoring in 50-move and 3-move-repetition rules. Shannon's # was calculated from 40x2-ply move games, so the actual # of positions is exponentially more than 10^120
It is in fact more than all Hydrogen atoms (all atoms) in the observable universe... Hardy's estimate is 10^(10^50)...
Isn't the number of atoms in the universe just 10^80? So yes, the possible number of games of chess, which is 10^120, is bigger than that. 10^(10^50) is an extremely big number, unimaginably bigger than either of the other two numbers. (Think googol vs. googolplex)
I think it's gonna be a loooong time till I see a question like this who's answer is no.
The theory that there are only 10^50 atoms on earth itself is just an assumption.
At the beginning of a chess game there are 32 pieces on the board which contains 64 squares. Already the number of ways these 32 pieces can be distributed on the board is equal to : 64!/32! which is equal to ~4.82E53 which is already higher than 10^50. So no more computation is even necessary.
number of game depends on players so be cool and think deeply on this
In chess the threefold repetition rule (also known as repetition of position) states that a player can claim a draw if the same position occurs three times, or will occur after their next move, with the same player to move. The repeated positions do not need to occur in succession. The idea behind the rule is that if the position occurs three times, no progress is being made. The game is not automatically drawn if a position occurs for the third time – one of the players, on their move turn, must claim the draw with the arbiter. Players will deliberately repeat the position three times to see how much the other player wants a draw. So the number of moves in infinite. Moreover it seems Shannon number is based on meaningful moves not all possible moves.
10^120 games possibilities thanks to CLAUDE SHANNON
The number of possible games of chess is called the Shannon number .
The number of possibilties, 1 0 1 2 0 , is an estimated lower bound on the game-tree complexity of chess. Therefore, it is much much more than the atoms on Earth.