A Spider-Fly Problem with a Surprising Solution

I’ve long been familiar with the spider and fly problem and its solution. Although, after reading this article, I realised that I had been focused on an incorrect solution. This was indeed surprising to realise that I’d been deluded all these years. The problem can be stated as follows: a spider and a fly are on opposite walls of a 30 × 12 × 12 meter room. The spider is 1 meter above the floor, the fly is 1 meter below the ceiling. They are both 6 meters from adjacent walls, as shown in Figure 1. If the fly does not move, what is the shortest distance the spider can crawl to reach it?

Figure 1

A sensible first attempt would be to travel straight up (or down) and across. For example, straight up the spider’s wall (11 meters), along the roof (30 meters) and down to the fly (1 meter). See Figure 2. This gives a total distance of 42 meters.

Figure 2

What I believed to be the shortest path is shown in Figure 3 and it is clearly not the shortest path!

Figure 3

In fact the shortest path requires the spider to cross five of the six internal surfaces and this is shown in Figure 4. The shortest path can be seen to be 40 metres.

Figure 4

The site from which this information is taken has some nice animated gifs of the rectangular prism’s unfolding, so the reader is encouraged to visit. The author of the article is Russell Lim, a high school teacher in Melbourne.