# 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.

• 22 months ago · Quote · #321

• 22 months ago · Quote · #322
cooooooooooool
• 22 months ago · Quote · #323

Not as good at chess as I would like.  I make too many blunders.  Lack of concentration.  ADHD as a child (moderate degree)

• 22 months ago · Quote · #324
simply beautiful
• 22 months ago · Quote · #325

I'm curious where those number came from.  http://oeis.org/A083276 agrees for the first few numbers but then gets much larger.  If there are really only 140 million possible positions, they could all fit in the memory of most modern computers and chess would have been solved a long time ago.

• 22 months ago · Quote · #326

It isn't possible positions. It is logical positions. You would need a computer like Deep Blue, that could logically reason in chess, and one that could document every possible position on its own. It'd be tough. To get all possible combos, it'd be nearly impossible and then to have a computer logi-fy? all of the positions would take millenium.

• 22 months ago · Quote · #327

moonlight....zhhhhzzzzzzzzzzzzz

Okay, what am I doing?

What if you had so many people against one and each one of the people were to memorize some chess positions or like write them down?

That would be so funny

• 22 months ago · Quote · #328

lol

• 21 months ago · Quote · #329

"...giving the approximation of 4,670,033 maximum logical possible games, thus making chess very playable"

This "fact" is obviously erroneous - offhand I'd say it's missing at least 3 zeroes and quite likely many many more

My CB BIG DB 2011 is updated to about 5,340,000 games so far regardless of how many are considered "playable" or not.

• 21 months ago · Quote · #330

OMG!!! this is chess???

• 21 months ago · Quote · #331

I have already memorized every possible move and combination there is in chess.  That is why my rating is so high.  :):)

• 21 months ago · Quote · #332

The quickest game is only 2 moves and can get there in 8 combinations. What is highest  number of moves of a legal game of chess, can be a win or draw depending on final postion ? I get confused when the 50 move comes in , so am multipy by 49 or 50 ? total pawn moves= 48,captures =29

• 21 months ago · Quote · #333
okinawaoly wrote:

The quickest game is only 2 moves and can get there in 8 combinations. What is highest  number of moves of a legal game of chess, can be a win or draw depending on final postion ? I get confused when the 50 move comes in , so am multipy by 49 or 50 ? total pawn moves= 48,captures =29

Technically, there is no upper limit to the number of legal moves in a chess game since it is optional whether a player claims a draw after 50 moves without a capture or pawn move.  If neither player decides to claim the draw then they can go on until one of them dies, I guess.  (I don't know who wins if one of them dies -maybe it's a draw.)

That said, if we assume that every time a player can claim such a draw, he does, it's a little bit of a tricky problem.  Whether the first capture/pawn movement occurs on white's 50th move or black's 50th move depends on whether white, after his 50th move, can leave black in a position that she is forced to capture a piece or move a pawn.  If she is not left in such a position she can write down her intention to make a normal move and then claim the draw.  So if white can force the position on black, he doesn't have to capture a piece on his 50th move, but if he can't, then he does.

Generally I don't think it will be easy for one player to force another player to capture a piece or move a pawn.  In a hypothetical world where that can never happen, white would have to capture/move a pawn on his 50th move, then black would have to caputre/move on her 99th move, white on his 149th... there are 49.5 moves in every cycle.

If somehow all the pawns get promoted before being captured (maybe impossible), then I think there are actually 96 pawn moves (48 for each colour).  The 30th capture ends the game in a draw.  If I haven't made any mistakes that's (96+30)*49.5=6237 moves!

• 20 months ago · Quote · #334

Mathematics and Chess!!

• 19 months ago · Quote · #335

This number, 4.6 million, sounds ridiculously low.  I'd love to see the author's definition of 'logical position' as I would believe that definition would change dramatically as the game progressed.

• 19 months ago · Quote · #336

Interesting!

• 18 months ago · Quote · #337

cool

• 18 months ago · Quote · #338

I still think the number would be more than 140,100,033, because think of 32 different pieces, 64 different squares, how each piece moves, and how many places each piece can move to. That's alot. Just a thought.

• 18 months ago · Quote · #339

cool

• 18 months ago · Quote · #340

It would be a draw. No doubt about it. The advantage of the first move is not enough to win.