# 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.

• 3 years ago · Quote · #201

cant understand this

• 3 years ago · Quote · #202

so we can say that probability of being 2chess games (with minimum 20 moves) strictly like each other is null!

• 3 years ago · Quote · #203

703687441776640000000000000000000000000000000000000000000000

• 3 years ago · Quote · #204

no.... chess...in it's perfection is always a draw...because even if you trade that 1/2 a move for a pawn....with perfect play it is still a draw if you are a pawn down

• 3 years ago · Quote · #205

this thread is interesting. I wish we could ask good will hunting.

• 3 years ago · Quote · #206

Have you ever studied your favorite opening. Now you know all the right moves and some bad moves for your opponet. Now somewhere along the line he plays something not in any book and now you bewildered. I study alot and when I play live online chess I"ve seen the most bizarre moves that I've never seen in tournament play.  I played my Master friend the other day. By move 30 I had a decisive advantage according to my FRITZ 12 software. But on move 31 I made a move that looked great but lost immediatly. So even knowing the first 30 perfect moves is no guarntee on winning, that's what makes chess great is the uncertainty.

• 3 years ago · Quote · #207

Although, the amount of moves that are not out right dumb is far less than that given number. You don't have to memorize every move... just every not stupid move. I think it could be possible.

• 3 years ago · Quote · #208

to a large extent, expertise in chess implies reducing the number of possible moves--pruning the decision tree.  this is why chess masters can play faster than amateurs, and part of the reason why chess masters can memorize a position instantly--better chunking.

by this measure, by the way, chinese chess (xiangqi) is more complex--40 or so orders of magnitude more complex.

• 3 years ago · Quote · #209

i love science...actually math,,bcoz i like math more than the others lesson...

and i'd like play chess too,,,lol

• 3 years ago · Quote · #210

I can remember e4 e5 Nf3 Nc6 Bb5. Seems like I have a long way to go then!

• 3 years ago · Quote · #211

Ok, i'll send you the new guestament soon.

• 3 years ago · Quote · #212

ouch

• 3 years ago · Quote · #213

I think you are a little wrong in this math: in the beginning, pawns can move two spaces or one space. Now obviously, depending on the different moves you do, other moves you cannot do. For example, 16 pawns with the ability to go 1 or 2 spaces give the pawns a total of 32 different moves, adding that with the knight's 4 make 36. Next you said black's moves total up to 400 positions because 20 x 20= 400, however, with 36 moves for white, 36 move are also available, creating this scenario: 36 x 36= 1296. This makes the entire article faulty and mathematically incorrect.

• 3 years ago · Quote · #214

I think you are incorrect. Pawns can go forward 1 or 2 spaces, creating 32 possibilities (16 pawns x 2 choices= 32 possibilities for pawns). Add that with the knight's possible moves and you get a total of 36 moves. The fact that you overlooked this tells me that you probably overlooked the fact that pieces can capture and or some pieces can be stopped from moving to certain squares

.

Obviously, in this position, the pawn can't move, nor can the rook, limiting the possibilities. In no offense to you, but I believe this serves as a counter example.

• 3 years ago · Quote · #215

"The total number of chess positions is about 2x10 to the 46 power. "

Yes, maby true, but the number of logical chess positions are far lower,to a magnitude of ???, as most logical chess games do not go much past 35 move and logical 60 move game are a rarety anding that as a logical game progresses piece are capured reducing the possible positions even more. Taking into account that there are quite a few logical games that end in the range of 4 to 35 moves this reduces the possible positions even further.

My estimate,for possible positions, is far lower than the figure metioned above, because of the limitaions metioned in the perviouse paragraph. My geustamate is that there are no more than a million possible logical postitions in chess.

• 3 years ago · Quote · #216

if you add in the special moves:  en pessant, castleing, and pawn promotion (which was not in the calculation), than there would be about 5x72 to the 84 power possible chess combinations.  That's amazing, not even computers have mastered and perfected there chess to the max!

• 3 years ago · Quote · #217

shut up. math nothing to do with chess.my friend failed math but hes a GM

• 3 years ago · Quote · #218

I think if chess got solved it would be a draw

• 3 years ago · Quote · #219

There is a mathematical algorithm to play a perfect game of chess, its the same algorithm that computer chess programs use today (minimax algorithm).  If we had a computer fast enough, with enough memory, it would play a perfect game of chess, the only problem is we don't, and probably never will.  The mathematical algorithm for optimal play already exists, its just too resource intensive be used to solve chess.  This algorithm can be used to solve other simpler games however, like tic-tac-toe for example.

• 3 years ago · Quote · #220

This is a sort of combination.. so much to unfold about Chess.