Number of absolute positions in a chess board.

Google says : It is estimated there are between 10^111 and 10^123 positions (including illegal moves) in Chess.

From my programming skills, a 64 bit value can have 2^64 numbers where each bit has 2 values 1 or 0.

Let's put this in the chess board, 64 squares can hold 7 different types of pieces (nopiece, king, queen, rook, bishop, knight, pawn), and 2 colors which means 13 types of pieces (6 * 2 + nopiece).

This will leave us with 13^64 possible absolute positions in a chess board, which is much less than what google says.

Any comments would be appreciated !

There are 16 pawns, 4 rooks, 4 knights, 4 bishops, 2 kings and 2 queens.  Each of those can be in 64 different squares.

This is the representation of absolute possible positions in any piece layout (e.g. you can have 10 kings on the chess board), and this is the answer. It's clear, right ?

I will add that there are actually 13 types of pieces, 6 in  each color + nopiece. So the answer is 13^64.

There are 10^44 legal positions in chess.
https://github.com/tromp/ChessPositionRanking

The vast majority has 3 or more knights, bishops, and/or rooks on one side.

A better estimate is 10^37 positions without promotions to pieces not previously captured.
https://arxiv.org/pdf/2112.09386.pdf 

 This is accurate.

The numbers in the OP are estimates of the number of sequences of moves. Many different sequences lead to the same positions.

The number of legal positions will be much less than 13^64. There must be only one king of each color and they can't be on adjacent squares. At least half the squares must be empty. Pawns can't be on the first or last rank. And many of the remaining positions will be impossible for some reason (both kings in check, unachievable pawn configurations, etc.).

And then we have a sea of impossible positions involving trapped bishop, trapped king, no retractable moves, parity, etc.

It can't be exactly calculated in a reasonable amount of time.

@5

"The numbers in the OP are estimates of the number of sequences of moves."
++The number of possible chess games lies between 10^29241 and 10^34082.
https://wismuth.com/chess/longest-game.html 

@7
"It can't be exactly calculated in a reasonable amount of time."
++ It has been exactly calculated:
8726713169886222032347729969256422370854716254 positions, of which 4.6*10^44 legal.
https://github.com/tromp/ChessPositionRanking 

That’s a new estimate. The OP found an old number, still lots of places on the internet, and took it to be something different than what it claimed to be.

The calculation I made was about an estimated number of possible positions (including illegal ones) in a chess board, where you can place a pawn on the first rank, or a king near the other or maybe 5 kings.

The thing is, it is much less than what google says, because using my calculations you can't fit more than 13^64 possible piece setups in a chess board and google says between 10^111 and 10^123 which is completely false using my approach.

@11
Google is wrong. Your 13^64 is right if you accept more than 16 men per side, more than 8 pawns, 10 knights, 10 bishops, 10 rooks, 9 queens, 1 king per side, pawns on the bottom and top ranks king not having the move in check...

since this comes up on google searches it should be known that tygxc is misrepresenting the contents of the paper he is citing, the paper describes without ANY promotion, and the math inside the paper reflects that.

Shannon's number

Shannon’s number is “game-tree complexity”. Not the number of positions, but the number of sequences (moves resulting in games). The number of positions is much lower.

See https://wismuth.com/chess/statistics-positions.html for an explanation of the difference. Unfortunately, Labelle does not give us an answer to the OP’s question, but does clarify his confusion.