# Mathematics and chess

• Last updated on 6/5/13 1:51 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.

• 4 years ago · Quote · #161

hmmm

• 4 years ago · Quote · #162

The total number of chess positions is = ?

• 4 years ago · Quote · #163

If there was an answer to any of this, and we'd worked it out, we'd have cracked the perfect opening move orders for white given any response. Therefore we'd have been clever enough to devise something even more complex. As a kid, a friend and I devised 'double chess'; where the board was 16x16 and we'd start with 2 armies each, but with 3 Queens & 1 King - It was stupidity of the highest order!

• 4 years ago · Quote · #164

Math teachers or even guys who have PH'ds in maths dont make brilliant chess players, chess is for the brave not guys who hide in dark corners playing with calculators....lol seriously any1 can play high level chess....ANY1....PRACTICE PRACTICE PRACTICE...u catch my drift.

• 4 years ago · Quote · #165

2+2=4

• 4 years ago · Quote · #166

2+2=4

• 4 years ago · Quote · #167

what means "distinct" ?

• 4 years ago · Quote · #168

I dont think that if both sides played absolutely perfectly white would always win. playing first is not such an advantage. Think about a position where white has a pawn on e2 his king on e1 and black a pawn on e7 and his king on d8.Thats a clear draw whoevers turn is .Sometimes even a pawn difference is not enough to win the game.I think that if black has a pawn and a king and white has a king in font of the pawn then it is draw. Also think about evans gambit or any other gambit. sometimes giving a pawn doesnt mean either winning or losing. Also there are positions that although one side has achieved a material advantage it might be a lost position.Playing second is not that bad to deprive you of at least drawing the game .Thats what i personally believe.However i think that your potential of winning is somewhat higher.But it all depends on what your opponent will play.Thats what I think but it might possibly be an always winning possition for white if played absolutely correctly by both sides.Think about having a bishop and king versus king.again it is draw but the point is that if you gain a bishop advantage early on then you most probably have a winning position.

• 4 years ago · Quote · #169

Why ruin a lovely game with mathematics,when the game can be relaxing?

• 4 years ago · Quote · #170

For those interested (think you all might be) IBM created a program calld BIG BLUE. However for those who remeber even that super computer program making millions of computation per second for every move lost (I thinl) one game and had a draw against the current world champion of the time.

• 4 years ago · Quote · #171

So, what you are really trying to say is, after the first 3 or 4 moves chess is basically unpredictable and no single game is likely to repeat itself. Was all the complicated mathematical jargon really necessary?

• 4 years ago · Quote · #172

well what about if white is in zugzwang on the first move

• 3 years ago · Quote · #173

how many positions after 7 moves allow mate

• 3 years ago · Quote · #174

CHESS IS CANDY FOR THE MIND

• 3 years ago · Quote · #175

I'm actually quite surprised about this article, since I love mathematics and I'm considering the possibility of myself in the far future of taking both Pure Mathematics and Mathematical Physics as majors. Cracking the code for any possible moves? That sounds Number Theory to me with Game Theory. Specializing in Number Theory, Topology, Complex Analysis and Chaos Theory, Set Theory. I would love to try solving Rieman Hypothesis and Goldbach Conjecture.

• 3 years ago · Quote · #176

Math is fun!,

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• 3 years ago · Quote · #177

That's about 576000000000000000000000000000000000000000000000, i'm not kidding.

• 3 years ago · Quote · #178

chess is geting to much like math and less like a game.I know that it would be hard  to crack chess bat what about chekers...

• 3 years ago · Quote · #179