PUZZLE #4: BUG IN A SHOEBOX (SOLUTION)

An empty shoebox measures 3 in. high, 5 in. wide, and 12 in. long. A bug, starting at one corner of the box, crawls along the inside surface of the box to the opposite corner (i.e., to the one farthest away from the bug's corner). Assuming the bug travels along the shortest path possible, while staying on the surface of the box, how far does it travel?

Let the box be oriented with its largest (5-by-12) rectangular surface on the floor, with one corner of that surface being the bug's starting point, labeled P. Then the finishing point, labeled Q, is the corner of the box that sits 3 inches over the corner R opposite P on the base surface. The obvious answer to the problem is that the bug crawls straight along the diagonal of the base to R, a distance of (52 + 122)1/2 = 13 inches, and then crawls the remaining 3 inches along the edge joining R to Q, giving a total travel distance of 16 inches. This analysis is wrong (and why the puzzle is so devilishly clever). The actual path the bug should take may be more easily seen by mentally cutting along the vertical edges of the box, and then folding the sides down onto the same plane as the base. When this is done, the point P is unchanged, but the point Q has "split" into a point Q1 on one of the short (3-by-5) sides and a point Q2 on one of the long (3-by-12) sides. In this unfolded configuration, the line segments joining P to Q1 and to Q2 both lie on the surface of the original box, and both give rise to a broken-line path from P to Q after the box is folded back up. The distance along the path to Q1 is (52 + (12+3)2)1/2 = 2501/2 ≈ 15.8 inches, definitely an improvement over the obvious solution. However, the distance along the path to Q2 is ((5+3)2 + 122)1/2 = 2081/2 ≈ 14.4 inches. So, to minimize distance traveled, the bug should crawl along the base to a point on the long side closest to Q, rather than to a point on the short side closest to Q.

Here is a variation on this puzzle that has an even more obvious wrong answer; I found it in a 1962 edition of "Mathematics for Pleasure," by Oswald Jacoby and William H. Benson: Instead of a shoebox, we have a room that is 12 ft. wide, 12 ft. high, and 30 ft. long. The bug, starting on one square wall 1 ft. from the ceiling and 6 ft. from either adjoining wall, wants to crawl to the opposite wall, 1 ft. from the floor and 6 ft. from either adjoining wall. How long is the shortest possible path?