Drawing round the edges of a cube

Below you can see a simple cube, which has eight vertices (or corners) labelled from 1 to 8, and twelve straight edges. The idea of this investigation is to draw a continuous line along the edges (as you might with a pencil) in such a way as to cover all of them with as short a route as possible.

Experiment by clicking on one of the vertices (it responds by growing a little bigger), and then successively clicking on other vertices, with each one being just one edge away from the previous one (so, for example, you could try tracing the path 1 2 3 7 6 2 3 4 1 2 3). Notice that

• The list of successive vertices you go through (the path you take) appears above the cube.
• The first time you go across an edge, it turns red; the second time, it turns green; the third time, it turns blue; and so on.

Now click the "Start Again" button, and try to draw a line with the shortest possible route which covers all of the edges of the cube. When you've covered all of the edges, the program will tell you whether or not you could have done better. If you don't succeed the first time, keep trying!

Think about why it's impossible to cover all of the edges without going over some of them twice. (Hint: look at any vertex - there are three edges meeting there. Suppose you go through that vertex, in along one of the edges and out along another. What does that leave?)

Problem 1

• Write down a path (as shown above the cube) which gives a shortest possible route.
• Explain why it's impossible to cover all of the edges without going over some of them twice.
• How many edges did you have to go over twice in your shortest route? Can you explain why it isn't possible to go over fewer edges twice? (Hint: Write down the vertices at the ends of the green edges when you've finished doing a shortest route. Write down the vertices where you started and ended. What do you notice? If you find this problem too hard, then carry on with the rest of the page - you may find it easier at the end).

Drawing round the edges of an octahedron

Now let's try to do the same thing with an octahedron, which is a double pyramid on a square base.

Problem 2

Experiment with the Octahedron until you've managed to draw a line with the shortest possible route which covers all of the edges (it's much easier than it was with the cube).

Problem 3

• Write down a path which gives a shortest possible route.
• What's different about the Octahedron which makes it possible to cover all of the edges without going over any of them twice?

Drawing around the edges of an "envelope"

Experiment with drawing around the edges of the edges of the shape above, and then try this final problem, which will test whether you've understood the lessons of the cube and octahedron.

Problem 4

• Write down a path which gives a shortest possible route.
• Now suppose that you have to start at vertex number 4. Why do you have to go over some edges twice? What's the smallest number of edges you have to go over twice? Write down a path (starting with 4) which gives a shortest possible route. (Of course, the program will tell you that you could do better - it doesn't know that you're being forced to start at vertex 4!)