White takes back his last move, then he mates in three.

[Rafael Moisevic Kofman, 1958, published on "Schachmatnyi Bjulleten"]

  White takes back 0-0-0 and then plays 1.0-0-0, 2.dc or 2.Rxd2 and 3.Rd8# (1...Kf8.2.Rd8#). Black can`t castle because of lack of pieces to allow the following moves: ab,bc,cd,de,ef,ef,fg,g8R.

^ Your solution seems like the only logical one (since there is no mate in 3 if Black CAN castle, clearly) however why must 0-0-0 have been the last move?

  "White takes back his last move, then he mates in three" means that white is free to choose his last move, that is no move must be the last move. So, if white, for example, takes back Kb1-c1 and plays 1.dc threatening 2.Rd8#, then black plays 1...0-0 and no mate in three.

Got it. So you could say we are looking for a "possible" solution rather than one which must be legal. Nice one!

Correct, maksym1. Congratulations :)