Retrograde chess problems are logic puzzles which involve trying to figure out what happened previously in a chess game, based only on the actual rules of chess. See for example The Retrograde Analysis Corner, a chess.com discussion, a mathpuzzle.com page, and the amazing book The Chess Mysteries of Sherlock Holmes by Raymond Smullyan.
My wonderful nephew Aaron Trainor has written two very clever retrograde chess problems (the second is actually three problems in one):
Give them a try!
-- Jeffrey Rosenthal