# Mathematics and chess

• Last updated on 3/11/12 5:03 PM.

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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.

• 6 years ago · Quote · #1
I love math!
• 6 years ago · Quote · #2

Very good to know.

I was wondering if someone was gifted enough to be able to memorize almost every chess position, and know what the best moves were in what position. Theoretically, they could win 100%. Its good to know that they would have to be REALLY REALLY gifted to accomplish this, heh.

• 6 years ago · Quote · #3
So then you'd not only have to know all positions, but also know what to play in all those positions... That's about as much as knowing where every atom in the universe is located...
• 6 years ago · Quote · #4

checkers was just cracked.  It took, if I remeber the article correctly, about 200 super-computers linked together to calculate an unbeatable strategy, which could be stored in about 256mb.  I remember when as a kid I "cracked" tic-tac-toe.  Much easier.  Chess I think would be immpossible.  Except maybe in the future using quantum infinite-loop fractal vectors.

• 6 years ago · Quote · #5
maybe its possible for someone to memorize all of those moves and possitions. MAYBE?
• 6 years ago · Quote · #6
Quantum infinite-loop fractal vector. Quantum infinite-loop fractal shmector. I just need an abacus and a post it.
• 6 years ago · Quote · #7

That would be impossible. But someone could memorize a good percentage of them. Ever hear about that guy who memorized enough of the 'number pie' that he could recite it for nearly nine hours! it was on 60 minutes or something, sounded legit.

• 6 years ago · Quote · #8

Chess works because players are in a "phase space" of useful possibles.  This space might be described as a cloud of the past and future possible moves perceived of as legitimately tactical possibles, from the point of view of the attack and defensive positions.

The limited scope of the board, a finite plane with quantized positions, still gives large numbers of possibilities, though nowhere near something like Graham's Number (Weisstein, Eric W. "Graham's Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GrahamsNumber.html).  In fact many of the positions can be discounted, which is what makes chess playable, (there are arguments about whether chess is NP-Hard, it seems to me that is the strongest computers can beat the strongest humans then it is not, but this is my opinion, I am unable to determine if it has been mathematically shown to be NP-Hard), for our purposes as humans the finite subsets of positions properly considered is manageable precisely because only useful subsets are considered.

I am  sure that everyone on the site will be aware that every piece moved affects every other and piece and the influence on it intimately for every move.  This is part of the finite universe scenario, and does not apply in situations which we encounter in everyday life, and so we consider subsets of the possible.  Chess is not infinite in extent, clearly, and the consideration of the movement phase space limits further the considerations we have to ponder while playing.  Without doing this, the possibilities are too large for most minds to consider, the phase space IS very large after all.

I know all of this, but it still doesn't make me any less of an awful player.

• 6 years ago · Quote · #9

Im going for a beer    phew!

• 6 years ago · Quote · #10
imagine if there is a formula in math where you can solve the good move....how i wish there is....
• 6 years ago · Quote · #11

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:

100,000,000,000

and there are 20xE45 possible moves - that is:

20,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

BTW, Physiscists estimate 10xE72 to 10xE87 particles in the entire universe.

• 6 years ago · Quote · #12

Math is like love - a simple idea, but it can get complicated.

Oh , and never forget -

2 + 2 = 5 for extremely large values of 2.

• 6 years ago · Quote · #13
In theory, chess can be cracked.  Why would we want that?  The game would die that day.
• 6 years ago · Quote · #14

here here ill drink to that dont crack chess it will ruin the game

• 6 years ago · Quote · #15
I don't even like the fact that chess openings have been 'cracked' - it makes the game too mechanical. I loooove math (see my username and avatar)! Quantum infinite-loop fractal vectors are probably not enough to 'crack' chess. Even if you sprinkle on some photon-powered computers. Integrating the quantum levels could define more chess position and then there is hope of chess ever being 'cracked.'
• 6 years ago · Quote · #16

If both sides play at a rating far beyond now with super computers in like 50 years do u guys think every game will be drawn or will white always win b/c gets 1st move?

• 6 years ago · Quote · #17
black is always 1/2 a step behind white i think at extemely high lvl play that 1/2 a step will make it so black can never win, and will only be able too lose or draw, if some how we got alien advanced technology and went even beyond that and both sides played absolutely perfectly white would always win.
• 6 years ago · Quote · #18
you gotta take the 50 move rule out at that lvl b/c somtimes it takes more if u make all the right moves.
• 6 years ago · Quote · #19
I have experience that mathematics are good chess players. They are faster in combinations.

• 6 years ago · Quote · #20

Very interesting! Glad to know!