user: How many possible board combinations are there on a regular chessboard? Please show the step by step process to end up at the solution.

assistant: This is a difficult problem so lets establish some ground rules, constraints and rules, and to break it down step by step. This solution will use combinatorics.



user:  Ground rules:

- We will not account for whether the board can be reached in a real game of chess (ie. Assume the players have infinite time to reach each board state).
- We will account for pawn promotions.
- We will account for pawns not being able to be in the top or bottom row.



assistant:  Constraints and rules:

1. To start with there are 10 of each type of pawn, 2 rooks, 2 bishops, 2 knights, 1 king, and 1 queen.
2. Each pawn cannot occupy row 1 or 8.
3. Each pawn can be promoted to any other piece of there own color except another pawn or a king.
4. Pieces can be captured.



user:  Steps:

1. **Calculate the total number of possible states ignoring the constraints:** Since each cell can either be empty or occupied by one of twelve possible types of pieces, there are 13^64 possible board states.

2. **Apply rule #1 to determine the number of possible states with the initial distribution of pieces:**
	- The number of ways to place the pawns is 64 choose 8 for each color, or C(64, 8), which is approximately 4.4262 * 10^9 each.
	- There are 100 ways to place each king.
	- There are 100 ways to place each queen.
	- For the other pieces, there are 2 of each, so the number of ways to place them is (64 choose 2) * 6, or approximately 6.7134 * 10^19 for both colors together.
	- Therefore, the total number of possible states with the initial distribution of pieces and ignoring constraints is C(64, 8)^2 * 100^4 * C(64, 2)^6 ~= 1.3152 * 10^47

3. **Apply rule #2 to determine the number of possible states with the top and bottom rows free of pawns:** This step is simply changing the previous calculation of the number of ways to place the pawns so that it is 48 choose 8 for each color, which results in a total probability of 9.5594 * 10^44

4. **Apply rule #3 to determine the number of possible states after promotion:** To calculate the number of possible states after promotion, we need to consider the number of pawns that can be promoted and the number of possible pieces they can be promoted to.

	- For each player, there are at most 8 pawns that can be promoted (excluding those that have been captured).
	- Each of these pawns can be promoted to one of four pieces (queen, rook, bishop, or knight) of the same color.
	- Therefore, the number of ways to promote the pawns for each player is 4^8, or 65,536.

	To calculate the total number of possible states after promotion, we need to rewrite our formula to with the pawn distribution from rule #2 and the whole distribution multiplied by 4^8^2 to account for the pawn promotion opportunities. Therefore, the total number of possible states after promotion is: (C(48, 8) * 8)^2 * 100^4 * (C(64, 2) * 4^8)^6 ~= 4.8472 * 10^75

5. **Apply rule #4 to determine the number of possible states after capturing:** To calculate the number of possible states after capturing, we need to consider the number of captures that can occur and the number of possible pieces that can be captured.

	- Each player starts with 16 pieces and can capture up to 15 opponent's pieces. (Cannot capture the king)
	- For each piece that is captured, there are 63 possible cells on the board where it could have been located. (Because of the king)
	- Therefore, the number of ways to capture the pieces for each player is 63^15 * 16.
	
	To calculate the total number of possible states after capturing, we need to modify our formula from step 4 to include the capturing possibilities. Therefore, the total number of possible states after capturing is: ((C(48, 8) * 8)^2 * 100^4 * (C(64, 2) * 4^8)^6 * (63^15 * 16)^2 ~= 1.9016 * 10^132
	

In total the number of possible board states following the rules is 1.9016 * 10^132. This answer is slightly different then some other answers to the question as it makes different assumptions and accounts for different factors but it is in the range of other accepted answers. If you have any other questions or need me to explain any steps or constraints in more detail feel free to ask.

assistant: I don't understand what that final number means. Can you write it normally or explain it?