The ant and the laddu

On a flat sheet, the shortest path between two points is the straight line joining them. But what if you must stay on the surface of a box and cannot cut through it?

The book's story: a hungry ant lives on the surface of a cuboid, and there is a laddu somewhere else on the surface. Which way should the ant walk? There are endless possible routes, so how can we be sure one is the shortest?

Here is the idea that solves it. Unfold the box into a net. A path on the box becomes a path on the net of exactly the same length, because unfolding does not stretch anything. And any path on the net folds back into a path of the same length on the box.

So we can solve the problem on the flat net, where we already know the answer: the shortest path is a straight line.

That gives a test. Draw the net. Mark the ant and the laddu. Join them with a straight line, and fold back. If a route becomes that straight line, it is the shortest. If it bends on the net, it is not the shortest.

One catch. Sometimes the straight line on one net runs outside the net, across empty paper where there is no face. That is not a real path on the box, so that net is no use.

Unfold the box a different way and the straight line may lie fully inside. So how you unfold matters, and different nets give different lengths.

Working out the length is easy once the net is drawn, because you get a right triangle. In one of the book's cases the two short sides are 24 cm and 32 cm, so d² = 24² + 32² = 576 + 1024 = 1600, and d = 40 cm. To be sure you have the true shortest path, list all the sensible ways of unfolding and take the smallest.