An old chestnut
goes like this:
Four bugs are placed at the corners of a square with one-foot-long sides. Each bug is initially facing the next bug clockwise around the square. At a certain moment, all four bugs start walking. Each bug continually walks toward the bug it is facing, adjusting its direction as necessary. All four bugs walk at the same constant speed.
When the bugs reach the center of the square, how far has each one walked?