# Mathematics and chess

• Last updated on 9/27/13, 5:27 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.

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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.

Also, see this.

• 2 years ago · Quote · #281

It seems like much of the math revolves around probability theory and statistics. I wonder if anyone has touched on the geometry involved with openings, piece development and formations. Learning about the rate of change with respect to time when a player moves his pieces can also help science learn how the brain processes information.

• 2 years ago · Quote · #282

It seems like much of the math revolves around probability theory and statistics. I wonder if anyone has touched on the geometry involved with openings, piece development and formations. Learning about the rate of change with respect to time when a player moves his pieces can also help science learn how the brain processes information.

• 2 years ago · Quote · #283

interesting!

• 2 years ago · Quote · #284

I love math, sometimes i say with a fiery passion..... and I'm actually doing a school project relating the concepts of chess to math. This reminds me of how the number of possible ways to order a deck of cards is 80658175170943878571660636856403766975289505440883277824000000000000.
Pretty interesting. Found it at http://www.schuhmacher.at/weblog/52cards.html

• 2 years ago · Quote · #285

very interesting! Glad to know that!

• 2 years ago · Quote · #286

interesting!

• 2 years ago · Quote · #287

• 2 years ago · Quote · #288

>>If both sides play at a rating far beyond now with super computers in like 50 years do u guys think every game will be drawn or will white always win b/c gets 1st move?<< I think it is always a draw.

• 2 years ago · Quote · #289

Very mathematical and scientific. I never knew this before.

• 2 years ago · Quote · #290

>>If both sides play at a rating far beyond now with super computers in like 50 years do u guys think every game will be drawn or will white always win b/c gets 1st move?<< I think it is always a draw.

• 2 years ago · Quote · #291

wow...

• 2 years ago · Quote · #292

only about 5 000 000 possible games? You must be kidding, that is complete nonsense, the number is much much bigger.

• 2 years ago · Quote · #293

5000000 possible games is complete nonsense. The number is much much bigger.

• 2 years ago · Quote · #294

Chess is a mental torture but it's a real great game.

• 2 years ago · Quote · #295

No, 5000000 possible logical games. Obviously, not every move is accounted, because if someone just moves their horse back and forth while the opponents queen takes every thing, then that isn't a logical game.

• 2 years ago · Quote · #296

simple, if chess were cracked, then FIDE would make 10x10 boards with 4 rooks and 10 pawns perplayer, calculate those posible moves!

• 2 years ago · Quote · #297

holy intergalacticical vectron based quanticator loopitator chess people.

• 2 years ago · Quote · #298

Chess will be cracked ...One day !!

• 2 years ago · Quote · #299

if the maximum number of logical positions is a mere 141 milion, why isn't someone having a supercomputer crunching those positions and finding the best moves for them????  After that hapens, any move that is shown to be better can simply updated.  Thus creating a database of all logical positions.  I bet this website has already seen all of those positions many times over!

• 2 years ago · Quote · #300

chess is easy game, if you know how to play your opponent !!!!!