This beautiful problem reminds me of a book by Raymond Smullyan, "The chess mysteries of Sherlock Holmes". I haven't seen it mentioned before on RHP. In it the great detective solves several mysteries using what he calls "retrograde analysis". Here's a fairly easy example:
"As you see, Watson, neither side is mated-not even in check. And we are given that your side is White. The question now is this: Given that Black moved last, what was his last move, and White's last move?"
Even Watson gets it fairly quickly. Other problems are trickier.