Can a unique position be deduced if pieces are replaced by checkers (can see piece color but not type)

Question

Inspired by a video where top players guess which famous game a position is from, except pieces in the position are replaced by checker pieces (so they can see the piece color but not type). Of course they remember the games, but if it's a position you don't remember, in which cases can you deduce a position only from the piece colors and locations but not piece types? Assuming the position is reached legally, but you don't know who made the last move or whether the game ended or how it ended.

If you don't need to deduce (don't know or care) who's to move or who has castling rights etc, and only want to know which pieces are represented by which checker pieces on the board. A minimum number of pieces solution might be with only two kings on the board. An obvious case might be the starting position if all pieces are replaced by checkers, but it's not possible to tell if a rook and knight have switched places. I think if the knights are placed in the middle in front of pawns and other pieces are in their starting positions, then it's unique, but I might be wrong. Are there other cases? What's the minimum and maximum number of pieces that can give a unique solution, except for the case with only two kings?

Note: in the video they were often given hints, like which side of the board they are looking at (white/black), sometimes who's to move or what's the last piece that moved, and the type of one or two pieces, or even that white/black resigned afterwards. But considering that would be too many different questions to consider the different cases, I decided to only ask about the basic case - no hints and ignoring who's to move and other information like castling/en passant rights.

Answer 1

Maximum:

[FEN "PppppppP/pppppppp/8/8/8/8/PPPPPPPP/pPPPPPPp w KQkq - 0 1"]

The board must be right side up because the armies cannot switch sides without some capturing.

Given this, all the pawns must be on their initial squares. Since nothing has been captured, the only pieces that could have moved are the Rooks (each shuffling between what used to be called its R1 and N1) and the Knights. So each of the pieces in the corner is a Knight, and each piece on a Knight's square is a Rook.

Moreover we can tell that it is White's move (by the usual parity trick) and that there are no castling rights.

[FEN "NrbqkbrN/pppppppp/8/8/8/8/PPPPPPPP/nRBQKBRn w KQkq - 0 1"]

Answer 2

I claim that Noam’s maximum is also a minimum. I.e. there is no unambiguous position with 3-31 checkers. If a unit is missing then it might be a pawn captured by a pawn and we can have up to 3 promotions resulting. Even if the other 6 columns are pristine, I see no way to constrain these promotions.

Nor is there a trick available with just 2+1 checkers on the board.

So 32 it is, although I think there is more variety possible in the positions than in Noam’s example. Rook pawns can be inched forward a square and knights can be scattered over the board.