# 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.

***

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.

• 5 years ago · Quote · #241

This 100 billion neuron thing i doubt, have you counted them?

Show me that Physiscists and his calculations.

• 5 years ago · Quote · #242

I love math to.

• 5 years ago · Quote · #243

An abacus a post-it input from a lot of people a lot of time and we have some answers. Its not the final answer, hopefully soon. This calculation could drive me mad !

• 5 years ago · Quote · #244

The gentleman that could rattle off pi for nine hours, even at 1 digit p/sec would only manage to recite 32,400 digits. Even two digits p/sec it would still be 62,800 digits. Nowhere near every possible position written about in this article. However, I believe that if he could use his talent to memorise positions to a certain opening. maybe an opening that could be achieved no matter what white or black played. For example, Mr Pi could decide, as white to always open with Kings Indian attack (1.Nf3,2.g3,3.Bg2,4.0-0,5.d3,6.Nbd2,7.e4). This is an opening that white could achieve regardless of what black replies, unless black goes out of his way to stop it, or, as black, Mr Pi could always open with Kings indian defence (Nf6,g6,Bg7 . Not necessarily in this move order). Then, I believe, Mr Pi would be a worthy adversary. However, regardless if we mere mortals do indeed crack the " Secret of Chess", I don't believe it would be the end of the chess itself, but a new step in her evolution. Chess, over history, has seen alot of changes and developments, and cracking chess may be just another change that will someday eventually happen. Once we figure out the 8 by 8, we might have to go 9 by 9, adding a second Queen and a ninth pawn. Crunch those numbers!!

• 5 years ago · Quote · #245

you lost me at ply

• 5 years ago · Quote · #246

Chess is a microcosm of the Universe. Millions of ways forces act on each other. controlling the action of the forces, that's the key.................

• 5 years ago · Quote · #247

its the Human element!

• 5 years ago · Quote · #248

UnBelievable

• 5 years ago · Quote · #249
this is great stuff! I wonder...I'd like to see God play chess. :-D
• 5 years ago · Quote · #250

:)

• 5 years ago · Quote · #251

its insane how smart ppl are man

• 5 years ago · Quote · #252

Nice new addition to the original article!

• 5 years ago · Quote · #253

good point, digdugdiggy

• 5 years ago · Quote · #254

no way, Phobetor!

• 5 years ago · Quote · #255

okay how will I use that in my chess play?

• 5 years ago · Quote · #256

Can't see that you will.

• 5 years ago · Quote · #257

"Lots of chess games end between moves 3 and the final move and the pieces decrease as they are captured." Really? Lots of games end between the 3rd move and the final move? I'm really sarcastically excited about that!

And, to Frankdawg: if both sides played perfectly, they would perfectly counter each other's attacks, and thus draw.

• 5 years ago · Quote · #258

You can prove anything with Maths.  For example:

A Proof That 2 = 1:

1.   X=Y                       let x=y
2.   X*X = XY                  multiply by x on both sides
3.   X*X - Y*Y = XY - Y*Y      subtract y*y from both sides
4.   (X+Y)(X-Y) = Y(X-Y)       factorise
5.   (X+Y) = Y                 divide by (x-y) on both sides
6.   Y+Y = Y                   replace x with y
7.   2Y = Y                    factorise
8.   2 = 1                     divide by y on both sides

• 5 years ago · Quote · #259

lol, nice

• 5 years ago · Quote · #260

I don't believe that guy memorize pi but rather could calculate the digits as he recited them.