# 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.

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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.

• 3 years ago · Quote · #221

nice calculations, math is interestingly everywhere...

• 3 years ago · Quote · #222
A computer will never have joy from chess.
• 3 years ago · Quote · #223

jey righte

• 3 years ago · Quote · #224

71,852 is geometrically correct, but in 1946 Thomas R. Dawson showed that White might have the option of an en passant capture in 232 of these positions. Therefore 72,084 different positions are possible (for two moves by each player).

• 3 years ago · Quote · #225

the area bounded by my my learning curve and the difficulty curve of 2000 rank, tells me that I should win a game after the first 400 lost.

• 3 years ago · Quote · #226

digdugdiggy

even if someone was good enough to know every move they wouldnt always win 100%

My guess would be 80% wins and 20% draws against top competition

• 3 years ago · Quote · #227

how do you win in tic-tac-toe i have to know!!!

• 3 years ago · Quote · #228

cool!!

• 3 years ago · Quote · #229

i'm not good in mathematics... I'm superb!

• 3 years ago · Quote · #230

Math + Chess = :)

• 3 years ago · Quote · #231

wooow!!! that was a new knowledge...

• 3 years ago · Quote · #232

I think an infinite number of monkeys playing on an infinite number of boards along with the quantum vector thingy ought to do it!

• 3 years ago · Quote · #233

I think an infinite number of monkeys playing on an infinite number of boards along with the quantum vector thingy ought to do it!

• 3 years ago · Quote · #234

It's impossible to be perfect. If both black and white were perfect at chess, the government would disect them and remove their brains for study. And then, America's government would  their brains to creat the perfect country and a perfect military (seeing that chess is a war game). And who knows? They might actually start GOVERNING while they're at it. Haha. Sorry it's just so much fun bagging on our government. But ya. Impossible to be perfect. Don't judge me!

• 3 years ago · Quote · #235

"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"

First you should establish what exactly the (decision) problem is. I can think of this statement: given a complete description of a position (like the FEN description) and a color (black or white), does that color have a winning strategy?

By the way, even if it is established that the advantage of white having the first move is enough to win, I still think that won't be the end of chess, because we humans won't be able to memorize all of the positions, and we do make mistakes too :-)

• 3 years ago · Quote · #236

love this game

• 3 years ago · Quote · #237
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• 3 years ago · Quote · #238

>> Even if you could "store"  one complete chess position in every neuron in your brain, there are only about 100 billion neurons (10xE10) [...]

Sorry, but a neural network like a brain is much more complicated than a simple "storage medium" as a hard drive. The "data rate" of our long term memorization is only 0,05 bit/s to 0,07 bit/s. That's only 4320 to 6048 bits (or 540 to 756 bytes) per day... Just imagine you read a book or some articles in this forum... that's much more data. The trick is a very very good data compression. It's lossy... but impressive! Data is simplified... we memorize the "non standard" things. We think and memorize everything in "templates"...

Such "template"-memory can be used for chess to... one memorizes a reduced situation of the board. There are a lot of unimportant peaces on a board... That does not mean they are unimportant at all, but unimportant for the current decision to memorize.

What does that mean? It means: There may exist just a memorizeable number of rules to follow to win the game... And somehow it is what a chess player does: learning the patterns that reduce the complexity.

The next thing is: a person would not need to "learn" every position, but a much reduced subset of it. With every own decision a player can reduce the number of following positions. A common strategy in playing chess is to reduce the number of moves the opponent can do, but increase the own possibilities... In a similar way one could reduce the whole number of positions one has to memorize to win every game to a minimum, when having access to the complete knowlege about all possible chess games.

How ever... the game would not "die" if one would be able to memorize everything. It will always stay a game one can solve by one's own... or just to play for fun. And even if one could memorize everything, one does mistakes when one is under stress. Limit a persons time too much and he/she looses.

If one "cracks" the game, just play it like before and you'll have fun doing it...

• 3 years ago · Quote · #239

This is the most complicated calculation i've ever done, with the help/input from those that came before me. Yet we still don't have an excact number, if this is possible ?

• 3 years ago · Quote · #240

Maby or maby not chess can be cracked. What the math of chess gives us is a better understanding of how complex this game is and why it has been around for at leats a thousand years.