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# How many different chess positions are there?

• #1

###### This is a hard one, but I thought about it and it could be a new way to look at chess and/or chess computers. Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player? To start, there are 7 different possiblilites that can be on a square at one time. A King, Queen, Knight, Bishop, Rook, Pawn, or an empty space. So that means the max possible number of positions would be 7^64 (number of squares) which is somewhere around 1.2 e 54, or 2.4 e54 if you count whose move it is. But then when you start taking out impossible combinations, such as more than 2 king on the board at once, pawns on the back row, more than 10 of a knight, rook, bishop, 9 of a queen, or 8 of a pawn on either side, it gets smaller. And of course there are other board impossibilites, like 8 pawns on the same side lined up vertically, a double knight check, etc. So has anyone ever heard of someone who has calculated this number? I tried searching for it but it doesn't seem many people have thought as chess this way.Edit: There would be 13^64 maximum possible positions, because each piece could either be black or white.
• #2

somewhere around 12, i think

• #3

Count sand on the beach.  The number will be 5 units greater than that.

• #4

Fourpointo wrote:

This is a hard one, but I thought about it and it could be a new way to look at chess and/or chess computers. Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player?

To start, there are 7 different possiblilites that can be on a square at one time. A King, Queen, Knight, Bishop, Rook, Pawn, or an empty space. So that means the max possible number of positions would be 7^64 (number of squares) which is somewhere around 1.2 e 54, or 2.4 e54 if you count whose move it is. But then when you start taking out impossible combinations, such as more than 2 king on the board at once, pawns on the back row, more than 10 of a knight, rook, bishop, 9 of a queen, or 8 of a pawn on either side, it gets smaller. And of course there are other board impossibilites, like 8 pawns on the same side lined up vertically, a double knight check, etc.

So has anyone ever heard of someone who has calculated this number? I tried searching for it but it doesn't seem many people have thought as chess this way.

What is the point of the fine print?  If you want people to take you seriously, don't force them to get out a microscope to read your question.

• #5

If you really want to blow your mind, calculate how many possible iterations of a chess game there are.  So, white can open up with 20 different first moves, and black can respond with 20 different first moves.  That's 40 combinations after the first 2 moves, it starts adding up quick.  Let us know what you come up with :)

• #6

of the general order of 64! / 32!(8!)2(2!)6   [Shannon]

• #7

According to Shannons number it is 10120 unique games of chess. Thanx Wiki http://en.wikipedia.org/wiki/Shannon_number what did i do before it

• #8

Games and positions are not the same thing.

So, white can open up with 20 different first moves, and black can respond with 20 different first moves.  That's 40 combinations after the first 2 moves, it starts adding up quick.  Let us know what you come up with :)20 * 20 = 400.

Same moves in different order may lead to the same position. To start, there are 7 different possiblilites that can be on a square at one time. A King, Queen, Knight, Bishop, Rook, Pawn, or an empty space. So that means the max possible number of positions would be 7^64 (number of squares) which is somewhere around 1.2 e 54, or 2.4 e54 if you count whose move it is. But then when you start taking out impossible combinations, such as more than 2 king on the board at once, pawns on the back row, more than 10 of a knight, rook, bishop, 9 of a queen, or 8 of a pawn on either side, it gets smaller. And of course there are other board impossibilites, like 8 pawns on the same side lined up vertically, a double knight check, etc.

It must be pretty hard to give the exact number of legal positions, but there is a way of approximating it - generating random positions and counting how many were legal and how many not.

After you've generated and tested, say, a few milions, the percentage of illegal ones must be pretty close to the "absolute truth". Then you simply multiply it by the number of all possible positions. It's still an estimation, but reasonably accurate.

• #9

after 4 moves there are over 70,000 posible positions

• #10

Fourpointo wrote:

This is a hard one, but I thought about it and it could be a new way to look at chess and/or chess computers. Assuming there is a perfect single move for every position, how many would have to be known to create a theoritical perfect chess computer/player?

...

This is exactly what is meant when people refer to "solving chess".

• #11

If we are trying to find the number of ways of filling r positions with with n distinct objects, the equation we would use is n!/(n-r)!. In the case of find the number of possible chess positions r is clearly 64, but n is a bit more difficult to find. At the start of a game there are 32 pieces so n>32 or n=32, but some pieces may get captured so we also have to include empty spaces, let's assume that a blank chess board is a posible postion then we would need at least 64 blank "pieces". so we get n=96.

96!/(96-64)!=96!/32!=96x95x94x...x33

• #12

The number i gave doen't account for pawn promotions, and some postions may not be legal, but it's a decent estimate

• #13

there is only two.... the right one and the wrong one.

• #14

pvmike wrote:

If we are trying to find the number of ways of filling r positions with with n distinct objects, the equation we would use is n!/(n-r)!. In the case of find the number of possible chess positions r is clearly 64, but n is a bit more difficult to find. At the start of a game there are 32 pieces so n>32 or n=32

Pawns are not distinct, for example if you get a position with doubled pawns it can be, say, either the e-pawn above the f-pawn or vice versa (depending on how the position was reached, which we may have no way of knowing), but it's the same position...

One may say that "doubled pawns are rare", but they are rare in REASONABLE positions! (Positions that emerge in real games, where both players tend to avoid doubled pawns), the number of reasonable positions is obviously a very tiny percentage of all legal positions.

Naturally, pawns can also swap files due to captures (say, b-pawn lands on the c-file and its comrade c-pawn on the b-file)

Knights and Rooks are not distinct pieces, either.

If you have position with all 4 Knights and 4 Rooks on the Board, you've got 16 permutations of "which K/R was which [initially]", or in other words, 16 ways to swap them, but it's only one and the same position anyway (hope you see my point, sorry English is not my 1st language ;) )

• #15

Vibovit,

yeah your right i didn't consider that, I'm drawing a blank on how to fix that problem, I don't have much experince with statistics, there may be some way fix that.

• #16

This would be a great challenge to come up with the exact correct answer.  Ask God,  He could tell you.  Even a greater question who cares?

• #17

There is not necessarily just one best move for any given position.  Some positions are symetrical, so they would have at least two possible "best moves".  Actually, any move that maintains material and temporal equality could be thought of as a  "best move".  (Think of the inital position.  1  p e4,  1 p d4, 1 p c4,  and 1 N f4 are all strong and maintain equality while a host of other moves may not be quite as strong but yeild at least a draw with perfect play.)

In any event, I don't think there are enough molecules of water in all the oceans on Earth (that's with the polar caps melted) to equal the number of possible positions in chess, so how could we ever hope to build a computer to store them all?

Next time please use a standard size font.

• #18

I think this would be interesting to try to calculate. People talk about the number of possible chess games, or the number of legal positions after a given number of moves, but you are asking about the number of legal positions, period.

There are actually 13 possible conditions for a given square at a given time: the six types of pieces in each color plus no piece. There are 13^64 positions, including legal, illegal, and impossible. That number includes such positions as a totally empty board and a board filled with 64 black kings. The number of legal positions would be much smaller, and of course harder to calculate.

• #19