Total possible chess positions?

[AlvaroBegue]

John Tromp claims to have a tighter bound of around 10^45.888 : http://homepages.cwi.nl/~tromp/chess/chess.html

[Gerd Isenberg]

Ahh, thanks Álvaro - will soon update it.

[SMIRF]

There also has been a simplified estimation, based on a Huffman coding approach: http://www.10x8.net/chess/Zahlen.html

[Angrim]

In the solving chess kickstarter topic I posted a rough figure for the total possible (not necessarily legal) chess positions, that of 3.837824955509396e+78 possible positions.

A friend of mine got the figure of 1.9605347643076107e+71 by doing 13^64 (each square of the board can be in one of 13 different states - 6 white pieces, 6 black pieces and being empty).

Since my maths are evidently off, what is the actual figure?

Matthew:out

I don't have a complete answer, but some additional considerations. If you are just looking at the positions of the pieces we can reduce the
number greatly from your friend's estimate, since most of the squares are empty. Start with a bitmap of which squares are empty(64 bits, no more than 2^64 possible values)
then for the non-empty squares you have at most 48 squares that have something on them for 12^48 possible values. so 2^64 * 12^48 possible positions.
You can reduce it a lot further using facts such as there are no more than 16 pieces of each color, and only 1 king on each side.
But then there is the fact that a chess position is more than just the position of the pieces, the history also matters. 50 move rule and
triple repetition both depend on it. When you take those into account, the number of unique positions gets a whole lot larger.