What is the number of legal positions in a chess game?

It is really difficult to find out legal positions but total positions possible is 13^64 , assuming each piece can be positioned in every available square

Old question, but for people who just want a quick estimate they can do with basic math this is an easy way to go about it

If we consider all 32 pieces, we can rearrange them P(64,32) or 64 permute 32 ways. If we then consider 31 pieces, we do the same as above with P(64,31) but this time we must factor in the choice of 31 out of the 32 pieces so we multiply P(64,31) by C(32,31) or 32 choose 31. If we continue along this manner we have the sum from n =2 to 32 of P(64,n)C(32,n) which gives 1.24e54 which is only a few orders of magnitude larger than the number given by Vicor Allis.

Note that in the assumptions, we allowed illegal positions, and double or triple counted certain positions by treating each piece as distinct. But the number is reasonable compared to that above and more importantly is easily attainable with pretty basic math.

Upper bounds on the number of allowable positions are discussed on the Shannon number Wikipedia page.

There is an accepted upper bound of 1.8x10^46 by Shirish Chinchalkar, and a somewhat accurate estimate of 4x10^44 by John Tromp.

By comparison, the number of atoms on earth is about 10^50.