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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


  • 4 years ago · Quote · #241

    aspen101

    There is a very good possibilety that chess can be cracked. This should not be an negative to anybodies ego but a plus as it would show how ingenious we are. Once cracked the gifted should be able to learn how this was done and apply those teqneus to beat any computer as a computer is only a machine, even though it is a powerful machine.

  • 4 years ago · Quote · #242

    aspen101

    This 100 billion neuron thing i doubt, have you counted them?

    Show me that Physiscists and his calculations.

  • 4 years ago · Quote · #243

    aspen101

    I love math to.

  • 4 years ago · Quote · #244

    aspen101

    An abacus a post-it input from a lot of people a lot of time and we have some answers. Its not the final answer, hopefully soon. This calculation could drive me mad !

  • 4 years ago · Quote · #245

    kamileon

    The gentleman that could rattle off pi for nine hours, even at 1 digit p/sec would only manage to recite 32,400 digits. Even two digits p/sec it would still be 62,800 digits. Nowhere near every possible position written about in this article. However, I believe that if he could use his talent to memorise positions to a certain opening. maybe an opening that could be achieved no matter what white or black played. For example, Mr Pi could decide, as white to always open with Kings Indian attack (1.Nf3,2.g3,3.Bg2,4.0-0,5.d3,6.Nbd2,7.e4). This is an opening that white could achieve regardless of what black replies, unless black goes out of his way to stop it, or, as black, Mr Pi could always open with Kings indian defence (Nf6,g6,Bg7 . Not necessarily in this move order). Then, I believe, Mr Pi would be a worthy adversary. However, regardless if we mere mortals do indeed crack the " Secret of Chess", I don't believe it would be the end of the chess itself, but a new step in her evolution. Chess, over history, has seen alot of changes and developments, and cracking chess may be just another change that will someday eventually happen. Once we figure out the 8 by 8, we might have to go 9 by 9, adding a second Queen and a ninth pawn. Crunch those numbers!!

  • 4 years ago · Quote · #246

    Chess_Abolisher

    you lost me at ply

  • 4 years ago · Quote · #247

    FCannella

    Chess is a microcosm of the Universe. Millions of ways forces act on each other. controlling the action of the forces, that's the key.................

  • 4 years ago · Quote · #248

    MrDurdan

    its the Human element!

  • 4 years ago · Quote · #249

    prasadinamdar

    UnBelievable

  • 4 years ago · Quote · #250

    MBClevenger

    this is great stuff! I wonder...I'd like to see God play chess. :-D
  • 4 years ago · Quote · #251

    kyldyl

    :)

  • 4 years ago · Quote · #252

    david266

    its insane how smart ppl are man

  • 4 years ago · Quote · #253

    MikeDoyle

    Nice new addition to the original article!

  • 3 years ago · Quote · #254

    Irishman964

    good point, digdugdiggySmile

  • 3 years ago · Quote · #255

    Irishman964

    no way, Phobetor!

  • 3 years ago · Quote · #256

    nonowho

    okay how will I use that in my chess play?

  • 3 years ago · Quote · #257

    MikeDoyle

    Can't see that you will.

  • 3 years ago · Quote · #258

    IWantThatPawn

    "Lots of chess games end between moves 3 and the final move and the pieces decrease as they are captured." Really? Lots of games end between the 3rd move and the final move? I'm really sarcastically excited about that!

    And, to Frankdawg: if both sides played perfectly, they would perfectly counter each other's attacks, and thus draw.

  • 3 years ago · Quote · #259

    Aslanjohn

    You can prove anything with Maths.  For example:

    A Proof That 2 = 1:

    1.   X=Y                       let x=y
    2.   X*X = XY                  multiply by x on both sides
    3.   X*X - Y*Y = XY - Y*Y      subtract y*y from both sides
    4.   (X+Y)(X-Y) = Y(X-Y)       factorise
    5.   (X+Y) = Y                 divide by (x-y) on both sides
    6.   Y+Y = Y                   replace x with y
    7.   2Y = Y                    factorise
    8.   2 = 1                     divide by y on both sides 

  • 3 years ago · Quote · #260

    nk2014

    lol, nice

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