# of possible positions

This one is for the math and computer geeks.

The total number of possible positions that arise after n moves, where n is a single moves played by either side with n = 1, 3, 5, 2n+1 is the white moves and n=2, 4, 6, 2n is blacks move

n     # of positions

1       20

2       400

3       8902

4       197742

5       4897256

6       120921506

7       3284294545

...

...

40      10^43 (estimate)

For n=40, the number arrived at by estimating the number of pawn positions (in the no-captures situation, this is 15^8, multiplying by the possible positions for all pieces, then dividing by two for each of the (rook, knight) pairs that are interchangeable, and dividing by two for each pair of bishops (since half the positions will have the bishops on the same color squares). (However, note that there are more positions with one or two captures, since the pawns can then switch columns) (Source: Mathworld.com).

Interesting, somewhere seemed to say 10^14 in my memory, but this makes more sense, i think.

"  The number of legal positions in chess is estimated to be between 10⁴³ and 10⁵⁰, with a game-tree complexity of approximately 10¹²³. The game-tree complexity of chess was first calculated by Claude Shannon as 10¹²⁰, a number known as the Shannon number. "

source: www.wikipedia.com

How many possible checkmate positions exist within that 10^43 to 10^50?

Does the estimate of number of positions assume a particular board orientation. That is, are mirror positions counted separately?

The following diagram is one of several mate in ten with the given material. Rotating the board produces eight variations of the same position (sixteen if Black is on move, rather than White). Is this one, two, eight, or sixteen positions in the 10^43?

Mirror positions have very little impact on numbers when dealing with numbers of form 10^X - even if each position had 8 mirror images that would not reduce 10^43 to a number below 10^42

I believe that, for instance, your diagram position is one single position in the 10^43 - 10 ^ 50 range. Any change in position of a piece, or addition of additional pieces, or deletion of the queen, would result in unique positions.

I would also note the huge difference between 10 ^ 43 and 10 ^ 50 which seems to be close to the best estimate possible at the current state, and only highlights the (seemingly) infinite universe of possibilities in chess.

Also, consider the objective rules of chess that a game must end in one of the following manners, where human decisions are not taken into account (i.e. not resigning, agreeing to draw, or ending game due to insufficient material)

1. Checkmate

2. Stalemate

3. 50 move rule (insufficient material will end this way)

4. Three time repetition

There is some number that exactly describes the number of possible games that can be played from the starting point. We will not know this number in my lifetime or in the near future (although with computing power increasing as it is, we will eventually get there), but I would sure like to know the exact number.

We can even do away with #4 in my post above, and can change the number in #3 so long as we have some agreement as to when a game ends assuming #1 and #2 are not possible. Otherwise, we would have infinite possible games as an infinite number of games would have infinite number of moves.

So, do we know how many checkmate positions exist? Even a crude estimate?

Let me check, Ziryab.

I will see if I can "stand on the shoulders of giants" and get the answer.

Thanks. That's a good point.

With smaller numbers, of course, it makes a huge difference.

In tablebases, for example, the three piece tablebases need only one solution for the eight unique positions in the diagram, and for the additional eight with the colors reversed. I believe the same eight with Black to move (or white if Black has the queen) is but one more solution (possibly a more complex solution).

There are almost infinitely more possible positions in the Japanese game of Go.

Sorry to be technical but there is no such thing as "almost infinitely more" as there is no "more" when dealing with infinity.

"seemingly infinite" would be accurate as it would take intp account the limits of human conception of large numbers and limited computing power.

I think number of possible positions is greatly enhanced by pawn underpromotions. Each side in theory could via pawn underpromotions have say up to 6 bishops, knights or rooks which vastly increases the number of possibilities.

 Of course this would never happen in real life which shows the theoretical maximum is not necessarily of much practical significance.

I assume tablebases of endgames ignore any positions with more than two rooks or minor pieces for one side..   

A mere 10^171 positions in Go. Substantially higher than the game tree of chess, but still finite. A finite number cannot be infinitely higher than another finite number, as noted by stats_man.

True, but it was a hard concept to express. But your "seemingly infinite" is better :)

Tablebases do not ignore these; they are simple to calculate if they are four or five to one. But with seven piece endgames, even many five to two imbalances remain outside current computing achievements.

 I get chirped at a lot for being technical in such things, but I had a whole math course on infinity and the prof was very picky about how we described it in our work...

I commend your course of action, since any mention of infinity in a math-related discussion is likely to lead to confusion if it's used informally.

This other site says it's 287 billion by the 4th move which is quite a different claim than the 197 thousand in the OP, indicating that unaccounted-for piece captures are much more signficant than speculated. 

http://wiki.answers.com/Q/How_many_possible_moves_are_there_in_the_4th_move_in_chess

https://scholaron.com/