I'm not sure I've seen this puzzle before, and I really like it:
We have n keys and n boxes. Each key fits only one box. We shuffle the keys and put one in each box. Then we randomly break open 1≤k≤n boxes. What is the probability that we can unlock all the other boxes?
I've been thinking about the fact that an n×n square grid has as many cells as a triangular lattice with n triangles on each side. Is there a nice continuous bijection between them? When n is odd, you can at least keep one line of symmetry throughout.