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Redblue

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?

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Diagonal Trace

You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

Hamilton's Puzzle

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In 1859, the Irish mathematician Sir William Rowan Hamilton devised a puzzle with a regular dodecahedron made of wood. Here is a dodecahedron:

dedecahedron solid and wireframe

He labelled each of the vertices with the name of an important city. The challenge was to find a route along the edges of the dodecahedron which visited every city exactly once and returned to the start.

Here is a graph which represents the dodecahedron. Can you see how each of the 20 vertices, 30 edges and 12 pentagonal faces is represented in the graph?

Dodecahedron Schlegel diagram with 20 cities on it

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?

Can you find any other ways of making this journey?

Here is a simpler network of countries:

network of six countries

How many different ways are there of visiting each of these countries once and only once, beginning and ending at Australia?