Problem

This net can be folded-up to produce a cube. This particular net has a perimeter of 14 units.

There are eleven distinct nets that can fold to produce a cube. Which of these nets has the maximum perimeter?

Solution

Here are the eleven nets that will fold to produce a cube.

All of the nets have the same perimeter: 14 units.

In fact, this result can be confirmed by approaching the problem in a slightly different way.

In order to flatten a solid cube you need to cut along its edges. Imagine that the cube would unfold to produce the uppermost net: the bottom square would remain fixed on the surface, the left & right faces would drop down, and the remaining squares would uncurl out towards the rear.

When making nets you're not allowed to form a "loop" when cutting along the edges. You could not make a cut return to a previously visited corner, otherwise it would cause a square to become separated.

It's necessary for each cut to be connected to a previous cut, and each vertex (corner) must be visited by at least one cut, otherwise there would be three edges at a vertex that could not be flattened out.

As there are eight vertices in a cube, such a net would comprise of "seven edges cuts". And as each cut exposes two edges on the perimeter of the net we confirm that all possible nets would have a perimeter of 14 units.