Upgrade to Chess.com Premium!

Mathematics and chess

  • Last updated on 9/27/13, 5:27 AM.

  • Send to friend
  • | 1109 reads
  • | 434 comments

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


  • 5 years ago · Quote · #181

    Armani323

    if you memorize all of this, then where is the challenge? How can you play against an opponent, who you no has all the positions in his head, memorized? Chess is a game of the mind, every move tells you of you'r opponent's attack and defense strategy, how they think basically. Its much more interesting when you dont no all the position's. Well you gotta at least think about 3 move's ahead, before you make a move, that's my opinion.

  • 5 years ago · Quote · #182

    Sphchow

    I think people have tried to memorize all the chess moves possible but have gone insane...

  • 5 years ago · Quote · #183

    tigerhawk

    Anyone looked into Seirawan chess? The two new pieces make memorizing conventional opening theory a dubious activity, and places more emphasis on tactical play. At least until the game is computerized and analyzed significantly.

    I've played a few games so far, and it's been both interesting and entertaining!

  • 5 years ago · Quote · #184

    liteng890116

    Math should not be used applied only as a counter here.

  • 5 years ago · Quote · #185

    tresumoe

    This is good to know beacause it feels like there is an endless amount of learning which means that there is alwats something to strive for. This leaves no room for boredom as mastery is virtually impoissible.

  • 5 years ago · Quote · #186

    shing87

    i am just wondering, then how many moves you can make first to became or considered to be a grandmaster as possible?...

  • 5 years ago · Quote · #187

    Satch_boogie

    even a super computer cannot do it bcoz after u memorise all the positions u  have to choose the best position after every move .....which is impossible.

  • 5 years ago · Quote · #188

    baldurthegood

    Normally I don't care to comment on things like this but someone above stated that 

    It would not be possible, based on the numbers given. Even if you could "store" one complete chess position in every neuron in your brain, there are only about 100 billion neurons (10xE10) - that is:

    Our current understanding of the brain is that it IS NOT like a computer, that is to say your memory does not have a certain limited amount of space or capacity, our memories are chemical in nature, based on complex chemical relationships which are more like chess positions themselves than a gig of ram... it is thought that the human memory, in theory, could be infinitely complex because it is based also on the complex chemical relationships in the brain, not stored in space. I like to think that an amazing person like kim peek or the pi sevant mentioned above could become god like at chess. Of course the rest of us are left to utilize just a fraction of our total theoretical brain power. I realize there are a lot of contradictions in here, like having a fraction of infinite mental power :S welcome to reality, it doesn't always make sense.

    but thats just my two cents :)

  • 5 years ago · Quote · #189

    vwato4

    actually the game was designed as evertthing in our world to balance out in the long run.. Therefore EVEN STEVEN...DRAW if each side make the correct move just as in Tic-Tac-Toe

  • 5 years ago · Quote · #190

    Deranged

    It gets even harder when you have to take into consideration pawn promotions to certain pieces.

  • 5 years ago · Quote · #191

    wickana

    Teachers always told me chess and mathe are alike.Is this how?

  • 5 years ago · Quote · #192

    yogesh249

    Very nice and interesting discussion !!!

    Cracking Chess !! ehhhhh, NOT POSSIBLE.

    Even if it is cracked, who the hell can remember so much of combinations !!

    But if chess is cracked, then the result will be known in advance, if 2 super computers are playing amongst themselves...Well that would be interesting to know...

  • 5 years ago · Quote · #193

    block1312

    crekers has been the most "inteligent" game to be jet cracken.  in had something like 10~25~ chess has 10~75~ an dits calcuatred it wold take millions of years to caunt by super compuresr.

  • 5 years ago · Quote · #194

    osmosis92

    technically, there's a lot of move possibilities

    but there's far fewer plausible combinations (even a mediocre player won't start off with a move like a4 or h4 and later in the game, no player would make moves that get pieces killed uselessly or compromise position uselessly)

    and there's only a few perfect combinations (meaning every side makes the best move possible)

    so it depends on how you look at it!

    and please don't crack chess: it would ruin the game!

  • 5 years ago · Quote · #195

    GrAy_MaTtEr

    Does this include the fact that you can have up to 10 knights on the board guys? or 9 queens/ or 4 knights ad 5 queens?!/

  • 5 years ago · Quote · #196

    yogesh249

    remember....

    Have u ever given a thought that why would a mature player would not move a4 or h4 in first move?? Because, he has learned from his experience....

    while computers do not have memory...so this scenario is not directly applicable to them.. You need to implement Artificial Intelligence to a highest level....

    Very interesting thread...

  • 5 years ago · Quote · #197

    findployk

    i wonder how much the math is skewed if say the opening is e4 d5 2 exd5 one less piece but how many less possible positions?

  • 5 years ago · Quote · #198

    MickeyWentz

    “All that matters on the Chessboard is good moves”
    (Bobby Fischer)

  • 5 years ago · Quote · #199

    manuelclayford

    cant understand thisUndecided

  • 5 years ago · Quote · #200

    mamad87

    so we can say that probability of being 2chess games (with minimum 20 moves) strictly like each other is null!

Back to Top

Post your reply: