# 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.

### Comments

• 5 years ago · Quote · #81

You lost me. I left with the guy who went to get the beer

• 5 years ago · Quote · #82

INTERESTING!!!

• 5 years ago · Quote · #83

lol nerds

• 5 years ago · Quote · #84

MATHEMATICScannot cracked CHESS. MATH is only pure calculation........no emotion involved. CHESS is a combo of both. But if ever CHESS is cracked, then it wiil no longer be considered a sport but a another variant of MATH as a subject many students hate or love. how about instead of taking Algebra or Geometry or trigonometry , u enrolled on CHESSEBRA, CHESSMETRY or CHESSNOMETRY?

• 5 years ago · Quote · #85

There would be no need to memorize 95% of all those as they are blunders or poor moves easily countered by good tactics. :)

• 5 years ago · Quote · #86

My contribution to this might be minimal for all the- from weird fact, looking to impress- type of people but it will be widely accepted by the rest- down to earth, enjoying chess and real life- group. Well even IF Quantum infinite-loop fractal vectors are enough to 'crack' chess, non human being will ever be able to read all the possible games , just because he will run out of time. Yes the phenomenon-dimension, time; strikes again and saves the world (all humans) from the unbeatable man, dominator of the game, absolute monarch player.

I hope i made you laugh, i hope i made you think.

Hope is the absolute sin.

• 5 years ago · Quote · #87

If I'm not mistaken, haven't the Russians specially trained their chess students to memorize positions and what to do in which position?  Of course, once you find something that wins once, you can win twice... and playing each other would be pointless, seeing as you know what the other person would always do.  There is no way to 100% win against everything with every strategy.  It's possible to lose, no matter what opening you pick.

• 5 years ago · Quote · #88

i think mathematics cannot cracked chess.  math is pure calculation........ no emotion involved. CHESS IS A COMBINATION OF BOTH!

• 5 years ago · Quote · #89

wow...                                                      life is full of surprises!!!

• 5 years ago · Quote · #90

I just dont understand that why do people talk about memorizing every chess position,even after reding this article.

After 5 moves the game has just begun and it has to go a long way.

And anyway how can they memorize the best move.

How do they know its best.

Better study than memorize moves.

• 4 years ago · Quote · #91

interesting numbers.

• 4 years ago · Quote · #92

I remember something about 100+ move minor peice combinations calced by super cpus... puts any chess player to shame.

• 4 years ago · Quote · #93

interesting, but with one move you can erase 20 or 30 moves in one like Bb4+

• 4 years ago · Quote · #94

glad to know also ,thanks

• 4 years ago · Quote · #95

thats nice

• 4 years ago · Quote · #96

it always adds up in chess...

• 4 years ago · Quote · #97

i love math-if anyone has any maths puzzles please could they send some to me

thanks

• 4 years ago · Quote · #98

Checkers has been completely solved so we still have some time left.

Read it here<<<<<<<<

• 4 years ago · Quote · #99

Chess may not be cracked anytime soon, but it is starting already. Just look at Chess.com's Game Explorer. It records move variations and shows the likely moves to give you a better chance at winning. I really wish they would take that out.

• 4 years ago · Quote · #100

quantum infinite-loop fractal vectors.  I can't stop laughing.

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