# 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.

• 4 years ago · Quote · #121

theoretically, it would be a draw if both sides played perfectly because even though white goes before black, black gets to respond to white where as white moves before black has revealed what he is doing. It would always be a draw

• 4 years ago · Quote · #122

Before the major computer cracks of the Game it was stated by historians  that in addition to practice, memory is a very important factor with the

Grand masters.  But the context of what is memorized for the most part

are the games containing important positions. As reflected in so many of the notes, the games , the named openings and their players are part of the citation.

A profound fact of psychology is reflected in the Jewish taboo on mentioning the name of the Deity. That is, if you name a thing you have power over it,

( You cannot have power over whom you cannot name, which was why Baruch Spinoza's ontological argument was heretical))

• 4 years ago · Quote · #123

One day someone will calculate the true value of perfect chess play. You know, like mate in 216 moves or draw...But that kind of chess will lose any sense

• 4 years ago · Quote · #124

cool

• 4 years ago · Quote · #125

All things being equal,any one player will either win,lose, or draw.

This I am certain.Now THAT is heavy!!!

• 4 years ago · Quote · #126

the computed combinations of moves from opening to a checkmate is 10^127, (10 raised to 127th power)..and that is greater than all the numbers of atoms in known universe which is just 10^75

• 4 years ago · Quote · #127

if on a beginning position, the black king would exchange its square with his queen ..and it comes to a time where supercomputers are advance, there might be a posibility that every game would meet into a draw...disregarding which side moves first

• 4 years ago · Quote · #128

wow u must be smart to know that kind of stuff

• 4 years ago · Quote · #129

whoa! staggering!!!

i love chess!!!!

• 4 years ago · Quote · #130

perhaps at the most advanced states of arcane chess theory and calculation, the game begins in zugzwang, and in fact white is at a disadvantage for being forced to make the first move.

• 4 years ago · Quote · #131

thats the number of position in just a few moves?ok counting and memorizing those positions is like looking to the galaxy and counting all the stars.

• 4 years ago · Quote · #132

Math Rox

• 4 years ago · Quote · #133

I hate math!!!

but if I can memorize it I would be very great

• 4 years ago · Quote · #134

hey people, if you memorized that all, would you be able to defeat a grand master?

• 4 years ago · Quote · #135

wow hats a huge number!

• 4 years ago · Quote · #136

The numbers prove why Chess is so interesting.

• 4 years ago · Quote · #137

I don't understand the breakdown of it all someone has to explain this to me

• 4 years ago · Quote · #138

Look at it this way: what are the probabilities of chess games being indentical after four moves, then after seven moves. Then consider the total number of recorded games that have been played. What are the chances that any two games of 20 or more moves in all history being identical.

Considering that we now count only games played according to book openiings and variations say, 5 percent the chances that the first four moves played in 24 hours might be 3 in 1000 but for 7 moves one in 16000.

• 4 years ago · Quote · #139

With these astounding numbers of possible moves I fully grasp that chess won't ever become "stale" or that the game will ever be cracked.

Sometimes, when I'm is a cosmic mood I like to think that the game was given to us  when humanity was seeded from elsewhere when I'm in that frame of mind to think that it happened that way.

I definitely think that we aren't the only intelligence around whether I'm in this cosmic mood or not.

• 4 years ago · Quote · #140

"Chess is a sea in which the gnat may drink and the elephant may swim" - Arab proverb