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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.

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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


  • 5 years ago · Quote · #141

    paulosni

    its fantastic that a simple game of 64 spaces can generate so much speculation about math posibilities. It makes me wonder about the whole universe and its creator. How wise and powerful He is and how miserable we are in comparision with him.  Jesus Christ is the Savior of the world. Think that.

  • 5 years ago · Quote · #142

    learninchess

    well even if people do crack chess they will have to crack chess 960,find some openings for 960 and wellI love 960 and fischer and maths and chess!!!!!

    SmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmileSmile

    64 smiles

  • 5 years ago · Quote · #143

    rabbit100id

    i'd just like to get to 1600

  • 5 years ago · Quote · #144

    gaby1234

    I Love Chess But I hate Math!!!

  • 5 years ago · Quote · #145

    Caspa

    Chess has been cracked

    Otherwise how do the computer players work?

    They have a certain code programmed in to them so the know what to do in every single one of the several millon billion possible positions in chess.

  • 5 years ago · Quote · #146

    MINTAKASTAR

    chess is a semi-chaotic game! because during the game the posibilities decrease when pieces are captured!

  • 5 years ago · Quote · #147

    METAPHORE

    Fantastic !

  • 5 years ago · Quote · #148

    Tiny2

    ei,pple with figures gift,go on do ur thingCool

  • 5 years ago · Quote · #149

    timlawson

    To decide the perfect game, we need to consider EVERY possible result - it could turn out that by virtue of having the first move, white is, in fact, in zugswang with perfect play and therefore, assuming absolute correct play, black wins every game.

    You could say that Whites best move is, for arguments sake, 1.e4. Black's best response is (again for example) 1....e5.

    If White is in Zugswang to begin with, why not just play 1.e3 (and hope black blunders by playing 1.....e5??. White simply plays 2.e4 and gets him/her self out of zugswang. If black were to play another move, eg 1.... e6 to counter white's attempt to get him/her self out of zugszwang

    It is for this reason (and I am talking about the sheer variation early on in any game) that it is impossible for either White OR Black to win (i.e. for chess to be "cracked").

    Due to the nature and mechanics of the game (control of the centre - and it would be impossible to dispute that this is a flawed theory!) and due to the fact that pawns can move either one or two squares and due to the fact that there are so many combinations of board position, the winner of every game should usually be the stronger player.

    Another consideration - about "cracking" chess... chess has rules and one of those rules are time limits. You would therefore need to have a computer so powerful (immeasureably more powerful than anything we have today) to be able to analyse deeply enough to beat the best players in the world on a regular basis.

    Okay, deeper blue beat GK a few years ago - but anyone who has seen Game Over will know that GK and his team didn't have access to any of the machines previous games or analysis - information that would have been available in any human v human encounter. For me, this was not a decent comparison and it would be interesting to see the same machine take on humans now with analysis available of even those few games. Of course, computing technology has moved on since then but the mechanics of the game have stayed the same.

    It would be a most remarkable achievement if a computer were able to "solve" chess even with an unlimited time to analyse from move 1. But it won't happen because the answer easily stares us in the face - if no one makes a mistake, it will most likely be a draw.

    For humans to have such a grasp (i.e. IM/GM or especially super GM) of such a complex game is in itself a stunning indictment of the power of the human mind. For humans to be able to create a box of electronics which can not only emulate humans but even beat the best and, perhaps, solve the thing entirely is possibly an even more stunning indictment of the human race.

    What a shame we keep investing all this intelligence in warfare and the accumulation of material things....

    No, I'm not a philosopher, but it's an interesting subject to me!

  • 5 years ago · Quote · #150

    salimb

    How many possibilities can a neuron store? I would venture many more than one.

    It also might be that memory is stored as interactions between neurons, in which case the total number that can be stored may be more than 2x 10E45.

    However, retrieving the proper move from storage might take more than would be practical.

    Also, you do not really need to store all the positions. Heuristics such as 'passed pawns must be pushed' will give the best move in the vast majority of positions (which is why an average player will play the best move very often). All you have to do is remember exceptions.

  • 5 years ago · Quote · #151

    michaelwraphael

    10^170...

  • 5 years ago · Quote · #152

    pawngenius

    I'm a math and chess genius.

  • 5 years ago · Quote · #153

    sure_shock

    (x-a)(x-b)(x-c)..........(x-z)= ????????....,i love math,music RELIGION and chess...sometimes...flirting with girls...hehehe..

  • 5 years ago · Quote · #154

    vibhu8dan

    nice article i really like it

  • 5 years ago · Quote · #155

    chirp55

    wow, that is move than I can count to

  • 5 years ago · Quote · #156

    franknstein

    This is why even the best computer programs cant be perfectly sure of any given position.Rybka 3 manages to calculate 200000 positions per second, with the help of 64 processors.Still it cannot grope all of chess positions.Amazing.

  • 5 years ago · Quote · #157

    jeffroj51

    The other factor in this is our own individual personality and psychology. We all know from experience that chess is about two individuals battling it out over a board. Many of our choices of moves have to do with our style and approach. We don't just apply memory or logic (although they are critical factors in our games). We use personal choice, emotions etc. How many times have we made moves based on fear, pressure, sudden confidence, even frustration?

  • 5 years ago · Quote · #158

    icx

    Too bad life's too short to try them all.

  • 4 years ago · Quote · #159

    ness456

    That's why every chess game you play is always different

  • 4 years ago · Quote · #160

    kalos

    Chess will definitely be 'solved' someday, i'd wager within twenty years.  The Chinese game of Go, however.... now that one would be a bugger :)

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