there must be a lot of folks with 'not enough work to do' that they have time to peruse this post, ponder it, and speculate (some actually based on sound reasoning), on the total number of possibilities in chess.

    I'm not enough of a mathematician to comprehend entirely, but this does make a fun read.

I am curious though, as Eniamar wrote that the average length of a chess game is 80 moves and average length of a GO game is 150 moves.

What is the source of those two numbers??  

not to change the subject of this forum, but simply to ask:  what is the source of those numbers, and if the chess 'average # of moves' is wrong, then is the average listed for GO also wrong??

It's a lot harder than you might think to count LEGAL positions. With 32 pieces, your calculations can make sure the right set of pieces is on the board i.e. 2 kings, 16 pawns, etc. However, there are a number of additional constraints such as: two kings must not occupy adjacent squares; white pawns cannot occupy the 'a' rank.

good point, Artfizz,  but also, don't forget, while you cannot place two kings on adjacent squares; since you cannot place a king where it can be captured by any other piece, that will significantly reduce the remaining possible positions as well.

  I suspect that those who establish calculations for 'maximum' number of positions ignore those stipulations and just look at physically possible positions, not the number of possible "legal chess" positions.

also, you would have to reduce the positions by those that - for instance - have all of one side's 8 pawns lined up on two adjacent files as that isn't "legal chess" possible if the opponent's other pieces and pawns are on the board as well.  For every pawn that changes to a different file, you would have to remove something from the opponent's side.

Calculating the number of legal positions with JUST 2 KINGS on the board is instructive. One approach to counting TOTAL legal positions is to start by counting physically possible positions (say 10⁵⁰ ) and then subtract positions which are invalid, for the types of reasons you mention.

The 2 kings case illustrates many of the difficulties of the overall problem: duplicated positions; the chessboard lacks rotational symmetry; context information (in this case: just checks - but in the general case: pawn position invariance, en passant, castling, promotion, captures, white moving first).

It's quite possible to limit the number by the possible pawn structures achievable by the max amount of pieces captured by them. However, it is just plain impossible to find a way to give an exact number to how many positions contains positions with both sides in check. Just one side in check is no problem, that side can be assumed to be at the move.

I do however think the number can be estimated by generating 10000 positions or so and calculate the ratio of legal/possible positions. We know the number of possible ones. This will give introduce a certain uncertainty in it, but that cannot be avoided. The more details taken care of before the simulation the more accurate will the total be. As the standard deviation varies as the square root of the number of trials, one really doesn't gain too much by increasing the number of generated positions too high.

I also ignored that pawns could be promoted, but that just changes the ratio. It's not hard to include that, it's just to do the same calculations for all new sets of pieces.

cuendillar,

you cannot have pawns promoted unless there are fewer than 16 remaining opponent piece/pawns total.  because all pawns are opposed by a pawn - none can 'queen' unless something is captured, and that will drastically change the number of possible positions

(or will it?  heck, I don't know, this is straining my neurons.  I need to find a kid who needs an old guy to take him fishing - now there is an enterprise worth my time - matching wits with a fish while getting outdoors with a kid whose parents are both working and the kid needs someone to sit with and observe how to match wits with a fish.  and while waiting for the fish to bite, we might just play a game of chess.)

I am not a statistician.  probably that is why I'm thinking of going fishing.

maybe GM Larry Kaufman will weigh in on this topic. (the statistical part, not the fishing part).   He is a statistician.

Doh, am i feeling stupid now! However, the method I deviced should work nonetheless. Just to make sure I didn't make any more mistakes, the formula I used was "configurations of squares occupied" * "possible orders of the pieces, respect taken to multiplicities" = 64!/(32!*(64-32)!) * 32!/(2!^3*8!)^2 = 63!/(32!*8!*8!)=4 634 726 695 587 809 641 192 045 982 323 285 670 400 000

my head hurts.  I'm going fishing.

...just say infinity, its a lot easier than all these calculations

But that answer is infinitely far away from the truth...

 Okay, I'm sure this has been done before, because it's so simple, but I can't resist:

There are four ways to place the white king in a corner, and for each of those there are 60 ways to place the black king (4 * 60)

There are 24 ways to place the white king on an edge and not on a corner, and for each of those there are 58 ways to place the black king (24 * 58)

There are 36 ways to place the white king not on an edge, and for each of those there are 55 ways to place the black king (36 * 55)

4 * 60 + 24 * 58 + 36 * 55 = 240 + 1392 + 1980 = 3,612

ichabod801,

but what happens when you add more pieces to the board ???

can you calculate those in your head ? and keep track of them all without missing any and without duplicating any?

hahaha.

If I could keep things in my head, I'd be playing chess OTB instead of here.

Of course, you could take the two kings calculation and use it to make a two kings and a pawn calculation. You could keep adding pieces and make a new calculation every day. Assuming you could make those calculations, it would take you over two years to calculate all the possibile positions, without accounting for pawn promotion.

On the pawn promotion issue, the pawn, although has changed into another piece, is still a unique piece from the others.

One must eliminate the possibility of any pawn changing into a King and must account for all combinations of all the other pieces as a promotion and their space occupying moves.

So, once we calculate the permutations without a promoted pawn, we must go back and add another layer of permutations accounting for pawn promotions and its' permutations.

Another discussion - http://www.chess.com/forum/view/general/how-many-different-chess-positions-are-there - reached many of the same conclusions as this one; which led me to pose this question: which will be solved first: chess or chess.com?

There are 2^2600 posible chess games....

Artfizz, if there are as many as 50,000 chess players here, and up to nearly 500,000 members who signed in,

then the variable ranges from 50,000 to 500,000 (as of today.)  we will call that variable     X

Sleeper postulated that there could be   2^2600 possible chess games, that would mean that there could be as many as   500,000 ( 2^2600) possible ideas to solve on chess.com

Since that is too large a number for most minds to remember, relearning occurs and "history repeats itself" and it is possible that chess.com will never be solved.

while chess, with a finite number of moves possible, theoretically could be solved, the sheer enormity of some of the numbers involved, precludes humans from realizing the 'solution' of chess.

My head hurts, I need to go fishing.

Retguvvie, my numbers came from http://en.wikipedia.org/wiki/Game-tree_complexity.

Note that I might have needed to be more specific in saying that the game is 80 half-moves long, on average and this doesn't appear to be too far of a stretch from the actual truth.

Also if one wants to dig into the math a bit deeper on that page, basically draughts, Chess, and Go are ridiculously more complex than the others listed because their game-tree complexity requires a deterministic(exponential) turing machine to solve versus a polynomial one for more trivial games.

well, buddy, 80 half moves in chess is only 40 moves, not 80 moves as you stated initially.

The number of possible moves, recorded in short algebraic, is tiny. 

64 non-capture pawn moves; 64 non-capture each for rook, knight, bishop, queen and rook. Throw in capures, checks, diambiguations, en passant. About a few thousand; tops.