"Supplément Problèmes", L'échiquier de France, 1957
14+7. Mate in 2
Analysis: The white P structure requires that all nine missing black men have been captured by Ps, towards the center of the board. One of the two missing white men has been taken on c5 or c6 so that at most one of the two black Ps a and h could leave its original column. Therefore one of the two black Ps a and h had to be promoted (to a1 or to h1) before it could be captured by a white P. Thus one white R had to leave its home square, a1 or h1.
Now that we see that one white R has moved, there is no difficulty showing that the other one could very well have been left undisturbed. So that, the right of castling K-side and of castling Q-side cannot be ruled out, though their conjunction can be.
Therefore you can pick any of the valid solutions: 1. O-O (2. Rf6#) and 1. O-O-O (2. Rd6# or 1 ... Qd3 2. Nxc5#), though you can't say that the problem has two solutions.