[FEN "7r/pppppKpP/6P1/8/8/8/n7/8 w - - 0 1"]
Assume that the following position was reached in a legal game of chess. In the last 5 moves, no pawn has moved and no pieces were captured. The Black king is invisible – on which square does it stand?
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The black king must be on c8.
The white king reached g8 through g6 and h7 before the white pawns arrived on g6-h7. A white knight took the Bc8 from b6; the black queen was also taken by a knight. The black king remained en e8, and the black rook on a8, until a few moves ago.
The last moves were for instance:
[Title "Invisible Black King solution"] [FEN "r3k1K1/ppppp1pP/6P1/8/8/8/8/2n5 w q - 0 1"] 1. Kh8 Na2 2. Kg8 Nc1 3. Kh8 Na2 4. Kg8 O-O-O+ 5. Kf7 Rh8
Evarglo's answer is correct, but doesn't give any reasoning for why lines that involve queen-side castling are the only viable ones.
The key to the puzzle is noticing that with the pawns in their current position, the White King can't approach it's current location from the White side of the board. Thus it must have just moved from the 8th row. Further the Black King is trapped on the 8th row too and, without castling, the Black Rook is trapped on that side of the 8th row.
This obviously causes problems: if those pawns have been in place for 5 turns, and no pieces have been taken then that means that white has only been moving his King ... and where has it been moving, if the black rook was covering that entire side of the 8th row?
One might think that there had been a black piece on f8/g8, allowing the White King to move around on the 9th row, but the blocking piece would have had to have been taken recently ( the Knight on a2 is too far away to have been 'helping').
The only conclusion left is that the Black Rook wasn't on this side of the Black King, until the last few turns. ... leading to the given solution.