Ok, so I've wondered for a while how many chess board states are valid. At first you might think: ok, plop down every piece in every spot and there you go. But, there are a few things to verify. I'm going to distinguish these into 3 categories: unplayable, playable but unreachable, and reachable.
First, the full set we're subdividing: each square on the board is either empty or has exactly one piece of the standard chess pieces on it. Technically, and this will be relevant later, we'll also include castling rights, and en-passantable pawns as part of a board state (and they will not allowed to be set to obviously false values). A board is "playable" if it satisfies 3 conditions: (1) there are no pawns in the first or eighth ranks, (2) there is exactly one king of each color, and (3) a king is not in check on the other color's turn. These are the boards you could pick up and reasonably play without inventing new rules for how to handle pawns or kings in those situations.
So, we now have the division between playable and unplayable boards. But how many boards that are playable, can be reached in theory from a valid sequence of moves in a chess game? Off the top of my head, there are a few reasons a board might be unreachable: (1) there are too many pieces on the board (either side has more than 16, or there are more than 9 queens or same-colored bishops, or more than 10 pieces of the same type, or more than 8 pawns), (2) too many pieces given the number of pawn captures needed (e.g. if white has doubled pawns and black still has all their pieces), (3) blocked-in pieces (e.g. pawns on b4 and d4 and white's dark-square bishop on a square other than c1 while also either still having all of their pawns so it can't have been promoted).
There are certainly more, that seem a bit harder to codify into straightforward rules. I'll leave it as an exercise for the reader, but here's an example if you want a spoiler.
rnbqkbQr/ppppp1pp/8/8/8/8/PPPPPP1P/RNBQKBNR b KQkq - 0 1
Anyway, I'm hoping to spend the next part of this sequence of blog posts discussing this in greater detail, and trying to define comprehensive combinatorial rules.