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Net Perimeter

Problem

The following net can be folded-up to produce a cube and it can verified that the perimeter of this particular net is 14 units.

Altogether 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.

It might seem surprising that all of the nets have exactly the same perimeter: 14 units.

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

Imagine starting with a solid cube. In order to flatten the solid it would be necessary to cut along its edges. For example, take a moment to convince yourself that the cube below would unfold to produce the net found in the top left diagram above: the bottom square would remain fixed on the surface, the left and right faces would drop down, and the remaining squares would uncurl out towards the rear.

It should be clear that in producing a properly joined net you would not be allowed to form a "loop" when cutting along the edges. That is, you could not make a cut return to a previously visited corner, otherwise it would cause a square to become separated.

Similarly it would be 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.

In graph theory we call the type of path our cutting must follow a spanning tree: a fully connected series of edges (cuts) that visits each vertex once and forms no loops. It can be seen then that each possible spanning tree corresponds with one of the eleven distinct nets above.

As there are eight vertices in a cube, such a tree would comprise of seven edges. 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.

Problem ID: 360 (11 Oct 2009)     Difficulty: 2 Star

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