During a recent stay at a cosy little hotel in Germany, we had a shower cabin with a square footprint (ABCD) and a sliding door (ab) that operated such that its endpoints a and b slid along the sides of the square, so point a moved along AB, while point b moved along BC.
We wondered if a well-rounded hotel guest – let’s call him Pythagoras (hint, hint) - could fit into the shower cabin and close the door behind him. Specifically, if the side length of the square is 1 meter, what is the maximal radius of a rigid rotund guest that could fit in and close the door?
Answers on a postcard. (This really happened, and the young mathematician in the family worked it out. However, I suspect that somebody, eg Martin Gardner, must have come up with this riddle before.)