Networks/graph theory

How many different colours would be needed to colour these different patterns on a torus?

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

Without taking your pencil off the paper or going over a line or passing through one of the points twice, can you follow each of the networks?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry