Mathematics and chess. [Formules et Theo]

The number of distinct chess positions after White’s first move is 20 (16 pawn moves and 4 knight moves). There are 400 distinct chess positions after two moves (first move for White, followed by first move for Black). There are 5,362 distinct chess positions or 8,902 total positions after three moves (White’s second move). There are 71,852 distinct chess positions or 197,742 total positions after four moves (two moves for White and two moves for Black). There are 809,896 distinct positions or 4, 897,256 total positions after 5 moves. There are 9,132,484 distinct positions or 120,921,506 total positions after 6 moves (three moves for White and three moves for Black). The total number of chess positions after 7 moves is 3,284,294,545. The total number of chess positions is about 2x10 to the 46 power.

"The total number of chess positions is about 2x10 to the 46 power.” Yes true, allthough there is a built in limit in the logical possible positions as the average chess game is about 30 moves, 60 moves and above chess games are a rarity. There are a lot of chess lines that end between move's 3 and move's 30 and the pieces decrease as they are captured, reducing the number of logical possible positions by a magnitude of 2x10 to the 40'th power. In end game situations the material combinations their frequency and the number of logical moves needed to mate or draw are know and it's in the region of a few thousand, limiting the logical possible positions in a end game situation to a few million.

Phase Logical position # of positions

· Initial position * 1
· Opening xxo*oxx +- 36 x 10(6)
· Middle game xxxooo*oooxxx +- 54 x 10(6)
· Middle game xxxoooo**ooooxxx +- 54 x 10(6)
· Middle game xxxooo**oooxxx +- 31 x 10(6)
· End game xxxoo*ooxxx +- 3,2 x 10(6)
· End game xo*ox +- 1,8 x 10(6)
· End positions * +- 32
+- 8.81 x 10(6) + 33

Playable traditional chess games +- 900,033

# Of draw positions +- 125,066 12.5%

· * 1
xxo*oxx
xxxooo*oooxxx
xxxoooo**ooooxxx
xxxxooo**oooxxxx
xxxo8888oxxx +- (32 x 8((2)(4)) Possible logical positions.
xo8888ox +- (32 x 8((2)(3)) The 4x8( 16 8’s) square .
o*ox 32 The Kings draw.
*=draw, o=winning/lose, x=other

The possible logical games are +- 442,533.

My guesstemate is that the logical possible positions are somewhere in the region of +- 8.810,033 , including trans-positional positions, as appose to "2x10 to the 46'th power" distinct chess positions, giving us in the region of +- 450,033 possible logical games, thus making chess much more playable.

Source: http://www.chess.com/chessopedia/view/mathematics-and-chess

http://www.chess.com/chessopedia/view/mathematics-and-chess
La prochaine fois, prière d'indiquer la ou les sources si l'article n'est pas de vous, car sinon ça s'appelle du plagiat.

oussama1305 a écrit:

http://www.chess.com/chessopedia/view/mathematics-and-chess
La prochaine fois, prière d'indiquer la ou les sources si l'article n'est pas de vous, car sinon ça s'appelle du plagiat.

Bonjour Oussama,

C'est un article ou j'ai trouvé dans une disscusion d'echecs, sur Chess.com.
Je ne savais pas qu'il est tiré d'une E.L. ou bien d'une autre place, c'est pourquoi j'ai laissé la source à part pour ne pas attirer de critiques par l'auteur du sujet.. Sinon moi aussi j'ai participé à éditer le sujet là-bas sur Chess.com (...)

C'est édité avec plaisir