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Math Help - Longest possible chess game

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    MHF Contributor alexmahone's Avatar
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    Longest possible chess game

    Has anyone ever calculated the length (number of moves) of the longest possible chess game? I'm talking about a hypothetical game in which both players play as aimlessly as possible, without violating the three-move draw or fifty-move draw rules.
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    In order for such a game actually to be played, both players would have to work at it, so I'm not so sure about the "aimlessly as possible" part. Neither the 50-move rule nor the 3-move draw force a draw. Both players could refuse to ask for the draw.

    However, if you're really asking that the opportunity doesn't come up, then you've definitely got some limitations. The number 49 will likely be a factor of a significant portion of the desired bound. Now, the maximum number of pawn moves possible before a capture is necessary is equal to 32 (say, one side's pawns moving as slowly as possible: 4 positions before hitting the opponent's pawns times 8 pawns). We'll assume that both players can easily avoid the invocation of the 3-move rule up to this point. So now you're already up to 32 x 49 moves possible. You would then have to have 8 captures to move all of one side's pawns to the last rank for promotion, which would, incidentally, allow the other side's pawns to promote. However, you'd have to have at least one pawn of one side being captured. So you now have the full 6 ranks x 15 pawns = 90 pawn moves, along with 8 captures. You would then have to continue capturing pieces, since there would be no pawns left. There are 30 - 8 pieces to capture. So, in total, so far, you have (90+30)49 = 5880 moves. Let's assume that the last of the 30 pieces captured was a queen, to allow for the possibility of checkmate at all times so far. Now you're down to king versus king. Checkmate and stalemate are both impossible, so I think the rules of chess would say that a draw was forced on you. If not, you could, if you wish, go into the number of possible positions occupied by both kings, which is equal to

    4 x 60 (white king occupies a corner)
    + 24 x 58 (white king occupies a non-corner edge square)
    + 36 x 55 (white king occupies an interior square)
    = 240 + 1392 + 1980
    = 3612 possible positions for the kings. Therefore, to the 6174 moves already mentioned, you could add another 3612 + 3612 + 1 = 7225 moves for a grand total of 13105 moves. But that last one is debatable.

    Keep in mind that I am also assuming that, for instance, in this incredibly boring endgame with two kings, it is possible to get all the positions listed. It may not be possible because of the rules of the game. So I merely am attempting an upper bound on the number of moves.

    These are my thoughts. They could be way off, but take them for what they're worth.
    Last edited by Ackbeet; October 8th 2010 at 07:11 PM. Reason: Incorrect assumption about moving pawns past enemy pieces.
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by alexmahone View Post
    Has anyone ever calculated the length (number of moves) of the longest possible chess game? I'm talking about a hypothetical game in which both players play as aimlessly as possible, without violating the three-move draw or fifty-move draw rules.
    Haven't looked deeply into it, but a quick google search reveals many conflicting answers/estimates. What surprises me is that I haven't found a single proposed list of moves to achieve such a game; just broad arguments, basically upper bounds. Thanks for the very interesting question. If I have time I will try to contribute eventually. It could make for an interesting and somewhat time-consuming exercise involving programming.
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    MHF Contributor alexmahone's Avatar
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    A google search reveals that the longest game of chess that is theoretically possible is 5949 moves.
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    Quote Originally Posted by alexmahone View Post
    A google search reveals that the longest game of chess that is theoretically possible is 5949 moves.
    I have seen this figure, among others.

    Disclaimer: I am not really knowledgeable about chess, besides basics.

    Preamble: We assume that a player will claim a draw whenever it is within rules to do so. Otherwise a game could continue indefinitely, and the question would not be interesting.

    It is also worth mentioning the rule "A draw also occurs when neither player has sufficient material to checkmate the opponent or when no sequence of legal moves can lead to checkmate." (Wikipedia)

    And here is a list of chess rules, which are official according to the title: http://www.ericschiller.com/pdf/Offi...hessSample.pdf (pdf)

    (See article 9 starting on page 26.)

    Note: While writing this I wasn't careful about the difference between "move" and "ply" because I wasn't too familiar with the terms. Quote: "a 'ply' is a black move or a white move. Cheesplayers call ten black 'moves' and ten white 'moves' ten moves/twenty plies." (source) But maybe the distinction isn't important here? (Edit: It's important; see below.)

    I think K+R v K (king and rook versus king) offers the most end-game flexibility since the rook can be captured by a king, but a checkmate is also possible. So we can assume the game eventually ends with K+R v K.

    Each pawn can advance at most 6 times, but of course pawns are blocked by opposing pawns. I think we maximize the number of moves by letting each pawn advance 6 times. I count that the event of a pawn capturing another piece must take place 8 times in order to allow this. After accounting for 16 pawns advancing 6 times and getting captured, there are 5 other pieces that can be captured in order to leave K+R v K. Each advance/capture is preceded by 49 moves that supposedly can be done without causing threefold repetition draw. This makes 50 * (16 * 7 + 5) = 5850 moves, with the 5850th move being a capture of some piece. Then K+R v K can dance for 49 moves; and I think at this point, the draw can be called before the 50th move, although I'm not very clear on that. This gives 5899 or 5900 depending on how the draw claim works at the end. I cannot see how to get beyond this number of moves. Note to emphasize: this was assuming that avoidance of threefold repetition is possible (which seems likely).

    Any feedback appreciated.

    Edit: I see my answer matches with Guy Macon's here

    http://www.tomshardware.com/forum/48...est-chess-game

    111799 plies translates to 5899.5 moves. Also there is a strategy given. So apparently the last capture is allowed to take place, and we must consider plies because when the ply that prevents a draw alternates between black and white, we lose half a move.

    Edit 2: Proginoskes mentions here

    http://www.mathkb.com/Uwe/Forum.aspx...le-chess-moves

    that the pattern K+N+N vs. K+P is an exception to the 50-move rule and is officially allotted 100 moves, which could explain where the extra 50 came from. (Although this might be an old rule that has been overturned?) But the number 6148 given in the same post is a complete mystery to me. Reference given is "The Longest Game" (Journal of Recreational Mathematics, Vol 14(4), 1981-82, pp. 241-245), by Sidney J Rubin. (On google scholar the article is listed as being cited by 3 other articles, but that's all I could find.)
    Last edited by undefined; October 8th 2010 at 10:34 PM.
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    I've played computer chess where after 106 moves, the screen froze which I took it to mean that the computer called a draw.
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  7. #7
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    Much easier to calculate the shortest possible game...

    1 g4 e5 (or e6)
    2 f4 (or f3) Qh4
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    An old joke . . .


    Two guys were playing chess in the park.
    I looked at the board and saw that they each had a lone King left.
    They were slowly and deliberately walking their Kings about the board.

    I leaned over to the nearer player and said,
    . . "You know, you can't possibly win this game."

    "Shhh!" he whispered. ."He might make a mistake!"

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