# Mathematics and chess

• Last updated on 6/5/13 1:51 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.

• 5 years ago · Quote · #61

Scoville: "Ever hear about that guy who memorized enough of the 'number pie'..."

I loves me some of that number pie!

• 5 years ago · Quote · #62
I know only computer can make that calculations with a great programs made by a great programmer that obsessed in chess. That is a great trivia, thanks.

• 5 years ago · Quote · #63

1st: If chess can be cracked, then it isn't worth as a game, so it should be made known. I suppose it can't.

2nd: pi is properly pronounced "pee" (Greek).

3rd: You people are too arrogant to live. Go work in the shipyards! Go die! Tins!

LOL

• 5 years ago · Quote · #64

Who would care if chess was to be cracked? I bet that if I played a player who knew the winning line for white against bkack's best defence, and that at the 50th move I played only the 2nd best move, I could throw off any player.

Same as for opennings, knowledge never replaces skill. Keep that in mind, lazy players who dream of beating kasparov memorizing chess; because if you don't want to learn chess tactics, then you could end up memorizing as many as 2x10E46 positions.

• 5 years ago · Quote · #65

LOL

You have no idea how hard I laughed reading down this...

Anyways, I think that those numbers are fascinating!

However, I don't really support the 'cracking' of chess. I think humans occasionally go a bit to far in their curiosity (for anyone who has read Harry Potter  - Dumbledore saying: "curiosity is not a sin, however we must exercise caution")

I would find it interesting, yet the idea of having a computer that is unbeatable, is annoying, and frankly scary!

If computers can 'remember' every move and every possibility for any game of chess, I would be terrified! The most complicated game that is hugely popular cracked by a series of '0's & '1's.

Plus, who would want to play against it? It totally destroys the brilliance of chess.

Anywhom, that is my 2 cents.

• 5 years ago · Quote · #66
well if the crack chess then ever will just have to start playing casablanca chess hahaha it will take them even longer to break that
• 5 years ago · Quote · #67
I kinda like math.
• 5 years ago · Quote · #68
The values only shows that it is impossible to learn the opening and impractical.
• 5 years ago · Quote · #69
i really really love math its my highest grade!!! lol its the best subject for me
• 5 years ago · Quote · #70
What's unfair is that in every game, despite the best memory a man (or woman) may have, he/she will use not more than some dizaine of those positions.
• 5 years ago · Quote · #71

Did you know that if you put 1 grain of rice on the first square of a chess board then 2 on the next, then 4 on the next, and 8 on the next and so on... till the end... they wouldn't all fit.

• 5 years ago · Quote · #72

One day I'll come back to read this and not get a headache...

• 5 years ago · Quote · #73

So much knowledge, so little grammar.

• 5 years ago · Quote · #74

Paul211,

you need to get your powers straight ;) Saying that 10^525 is almost the same as 10^565 is like saying that the diameter of a hydrogen atom is almost the same as the diameter of the observable universe...

Sorry, couldn't help myself

• 5 years ago · Quote · #75

One day...-_-'

• 5 years ago · Quote · #76

i have been wondering wether chess has finite ot infinite moves. i really dont know. who can help?

• 5 years ago · Quote · #77

Chess with 6 piece endgames has been solved so far, but as far as I know that takes up 1 TB of computer space.  As the retroanalysis continues, the space needed to store this information will only increase exponentially.

On another note, it would be interesting to see exactly what "perfect play" would look like.

• 5 years ago · Quote · #78

i still have trouble with the big hand and the little hand.

• 5 years ago · Quote · #79

I'd guess there are infinite amount of moves seeing as the pieces can go back to the same spot over and over...

Unless you go back to the top where they're talking about distinct chess positions :p

• 5 years ago · Quote · #80

was the number really true??

maybe someone tried to count all the possible moves in chess, how great he is!!

or maybe it is only an estimation..oh my goodness!!

on the 46th power, i can't imagine it...hahaha=D