Prove that White is not allowed to castle here A friend recently gave me the following weird chess problem to solve. I'm afraid that I'm struggling to see the key idea that would ultimately count as proof. From the position, intuitively speaking, it looks like White should be allowed to castle, but I'm sure to be missing something here.
Prove why white is not allowed to castle in this position.
[FEN &quot;1n2k3/p1pqppp1/1p1p1n2/3N2p1/1PB3bP/2P1PN2/1P1P1PP1/2BQK2R w K - 0 1&quot;]
This is actually a rather typical retrograde problem, just start with the most basic observations:

We see that black is missing both rooks and the f8 bishop. Given black's pawn structure it's easy to see that neither the f8 bishop nor the a8 rook could have escaped the structure, thus they must have been captured on the 8th row (by a white knight for example). This leaves us with the h8 rook, for which there's only one plausible choice left, namely having been captured on the b4 square by the a3 pawn.

As for white, the only missing piece is the rook, which must evidently have been captured on the g5 square by the h6 pawn.

Now we arrive at the real question: which rook could have logically been captured first, the one on g5 or the one on b4? Surely if the black h8 rook is to be captured on b4, it must first have been freed by the white rook being in turn captured on g5, but that's impossible, as the h8 rook being taken off on b4 is the very thing that must have freed white's a1 rook in the first place! Which leaves us with one remaining logical possibility: White's h1 rook must have been the one coming out and subsequently captured on g5, which then in turn frees black's h8 rook, to come out and be taken off on b4 by white, which finally freed the a1 rook.

Thus the rook you see currently on h1 is actually the rook coming all
the way from a1 (once freed), with the original h1 rook having been
lost on g5. Given this sequence as the only possible logical sequence
leading up to the current position, white cannot castle, as for
castling to be allowed, neither the king nor the rook must have
moved.


These kinds of puzzles usually go under the category of retrograde analysis, where unlike usual chess puzzles where the intent is to find the best moves/tactics or mates, in retrograde problems it's rather often just a logical question that can be answered by considering the basic rules of the game, and trying to find the only logical sequences (not move by move though, just the necessary ideas) that could have led to the position given in the puzzle. So if you're not familiar with these kinds of problems you may indeed find it difficult to attack them at first. Retrograde problems in chess are really fun to solve, specially as soon as you get the handle of them and start solving the real difficult ones. The solving process is really similar to that of a detective solving a crime, since if you think about it, you're given some clues and trying to figure out what could and could not have happened ;)

Addendum after some comments:
(This is regarding which of the black rooks may have been taken off on b4, although this fact leaves the proof unperturbed.) The a8 rook is trapped on the 8th row, as the only possible way for it to escape the 8th row would be via the h-file, which once opened (i.e white losing the h1 rook on g5) would first let the original h8 rook out. Of course you can always assume the h8 rook and f8 bishop were for example first taken off by a knight, then the a8 rook was the one coming out to get captured on b4 (e.g. Nc6, Qe6, Kd7 to open the 8th row), but for the purpose of the puzzle it won't matter, as in either scenario white wouldn't be able to castle as the same arguments hold.