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Mathematics and chess

  • Last updated on 9/27/13, 5:27 AM.

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The number of possible chess positions after White’s first ply move is 20 (16 pawn moves and 4 knight moves).  There are 400 possible chess positions after two ply moves (first ply move for White followed by first ply move for Black). 

There are 5,362 possible positions (White’s second ply move) or 8,902 total positions after two ply moves each. There are 71,852 possible positions or 197,742 total positions after four moves. There are 809,896 possible positions or 4,897,256 total positions after 5 moves.There are 9,132,484 total positions after 6 moves. From move 7 the possible positions stabilize as chess lines end, even from move 2 some chess lines end. There are +-10,921,506 total possible positions after 7 moves. 

The special draw, the King's draw, should occur a minimum of 32 times. The longest recorded game ended in a draw after 269 moves.

***

There is a built in limit in the logical positions as the average chess game is about 30 moves, 60 moves and above chess games are a rarity. Lots of chess games end between moves 3 and the final move and the pieces decrease as they are captured. In end game situations the material combinations their frequency and the number of moves needed to mate or draw are known and it is in the region of tens of thousand, limiting the logical possible positions in an end game situation to hundreds of thousand.  

 Phase                         Classification                ~ # of positions          Moves  

·         Initial position                     *                                     1                            0

·         Opening                        xxo*oxx                +-    5     x 10^6           1   -  5

·         Opening                  xxxooo*oooxxx           +-  40     x 10^6          6   - 10 

·         Middle game       xxxoooo*ooooxxx          +-  45     x 10^6         11  - 15

·         Middle game         xxxooo*oooxxx            +-  40     x 10^6         16  - 20

·         End game                xxxo8*8oxxx              +-    5     x 10^6         21  - 25

·         End game                      xo*8x                   +-    5     x 10^6         26  - 30

·         End game                        o*8                    +-    0.1  x 10^5         31   - Final move

Logical possible positions                                 +- 140.1   x 10^6  + 1 

Possible/playable chess games (Avg game 30 moves)  +- 4,670,033

~# Of total draw positions @ 7% of playable games    +-    326,933

        *=draw, o=winning/lose, x=other, 8=known end game combinations

 A guesstimate is that the maximum logical possible positions are somewhere in the region of +-140,100,033, including trans-positional positions, giving the approximation of 4,670,033 maximum logical possible games, thus making chess very playable.

When compared to the numbers available from online databases the actual number of games played so far , for reasonable players, seem to be somewhere in the region of +-2,910,286 which should be taken as a minimum number for the possible logical games.

See Shannon Number for the Upper bound for Random Chess.

Also, see this.

Comments


  • 6 years ago · Quote · #101

    turbine506

    great facts

  • 6 years ago · Quote · #102

    Akssay

    my gosh

  • 6 years ago · Quote · #103

    jxchessmaster

    10*E72 = 10 times 10 w/72 zeros after it. 10*E87 = 10 times 10w/87 zeros after it. A huge number

  • 6 years ago · Quote · #104

    bjazz

    I`m reminded of the futurama episode where two robots are playing chess: "mate in 147 moves" states the other before the first move, afterwhich black resigns :)

  • 6 years ago · Quote · #105

    herbmonkey

    Computer Vs computer, "Who would win?" Awesome.

    The answer is BLACK. Reason being the first move by white would give away strategy, and narrow future move possibilities by white. Therefore giving Black the immediate upper hand. White now being the 1/2 step behind.

  • 6 years ago · Quote · #106

    sebas4life

    If both sides played perfectly the game would most likely be drawn.

  • 6 years ago · Quote · #107

    Springsman

    hmmmm ok...I'll throw all that in my trivia pile.....

  • 6 years ago · Quote · #108

    Alea-Iacta-Est

    What I'd like to know is what the 20E45 moves statistic takes into account. For instance, is it simply every possible configuration of every piece on the board, regardless of how the pieces got there? Obviously you couldn't have all the pawns on the board switch places with those of the other color, but would that specific configuration be included in the number? In short...how many playable positions are there? It's surely still a massive number, but is it the number given above, or a slightly smaller one?

  • 6 years ago · Quote · #109

    swiniaWkosmosie

    good to know... I've heard that they've done that with checkers, and I hope they'll never be able to crack chess :D

  • 6 years ago · Quote · #110

    Angel8808

    I LOVE Math!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  • 6 years ago · Quote · #111

    jonatz_elise

    ouch nosebleed!! But very very interesting!!Laughing

  • 6 years ago · Quote · #112

    madpawn

    Right, I have just done something fantastic! I have sucessfully memorised all the possible first moves for white!! 

    (celebrate small successes is my motto)

  • 6 years ago · Quote · #113

    Xcelarate

    I think It's brilliant that we have a game that young, middle aged and old alike have something that they can use intelect and have fun at the same time,CHESS!!!!!!!!!!!!!!!!!!!!!!!

  • 6 years ago · Quote · #114

    alahoy

    Oooohh...I'm having a headache, I hate math but I love playing chess. Can we just ask my teacher to play chess instead of math?

  • 6 years ago · Quote · #115

    jrcolonial98

    amazing. It just makes you think how complicated the game of chess is.

  • 6 years ago · Quote · #116

    doctordiamond

    very very scary:|

  • 6 years ago · Quote · #117

    natanku2

    You have a lot of time on your hands to figure that out!

  • 6 years ago · Quote · #118

    jamesamble

    I love math but this is to hard

  • 6 years ago · Quote · #119

    jssedloc

    this is why I give up learning openings!

  • 6 years ago · Quote · #120

    GeniusKJ

    theoretically, it would be a draw if both sides played perfectly because even though white goes before black, black gets to respond to white where as white moves before black has revealed what he is doing. It would always be a draw

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