(18.03.2014) This Fairy/Retro original was appeared in the comments of problem n.324 and both authors decided to publish it as a join composition. Welcome to Adrian Storisteanu from Canada! Well - his photo is a little bit unusual, but he told me that he is not so photogenic ;)

 a) The try: 1.Kb3? 0-0-0?? 2.Ka2 Rd6 3.Ka1 Rxa6#, an otherwise exact solution, fails because white's last move must have been with either K or R, invalidating the castling.
   Hence the answer to how many solutions is "0", not "1".

   b) There is a promising set-play, 1...0-0-0 2.Kb3 Rd3+ 3.Ka2 Ra3#.  But random attempts to keep it in the actual solution fail, e.g., 1.Ra6-~? 0-0-0?? etc. – it is black to play, and white's last move must have been with K or R, disallowing castling*.  In the set-play (with white to move) on the other hand, black's last move legitimizes the castling (regardless of prior wR play) via a suicide (a regular capture being impossible, the black pieces standing on light-coloured squares, the wR on dark) such as bBf6?wRa1(+wRa1), giving birth to a brand-new, ready to castle, wRa1.  The same strategy must be employed in the solution:  1.Rxa1(+wRa1)! 0-0-0 2.Kb3 Rd3+ 3.Ka2 Ra3#.
   Therefore the answer here is "1", not "many".

   *There is no black piece on its own rebirth square in the diagram.  If the bR stood on a8 (rather than a6), white would have a non-K/R last move available:  again a suicide, but white now, e.g., wBh1xbRa8(+bRa8), which would then allow all those dualistic 0-0-0 solutions...

   Retro-motivated dual-avoidance.  Two ideal mates, two flavours of 0-0-0 illegality, and in both phases the largely superfluous bR manages to disappear in one fashion or another, and quite economically, from the final mate picture.

   Circe Assassin:  circe variant in which the captured unit is returned to its game-array square, even when this rebirth square is occupied – in which case the occupying unit disappears for good (is "assassinated").  Hence a unit located on its home square cannot be removed:  its rebirth eliminates the captor (who, in effect, commits "suicide").