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We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Platonic Planet

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

Three Cubes

Can you work out the dimensions of the three cubes?


Age 14 to 16
Challenge Level

Why do this problem?

The stimulus for the problem is the engaging context of the construction of the solid. Is this really a regular dodecahedron and how can we be sure?

Possible approach

Without discussing how or why the pentagon is regular, encourage learnersto make pentagons and put them together to make a dodecahedron.
Then challenge them to examine whether the pentagons are actually regular or not.

Key Questions

  • How do you know this is a regular pentagon?
  • What would have to be the case for the pentagon to be regular (all sides and all angles equal)?
  • What do we know?
  • What mathematics do you know that might be useful?


Making this and other solids in similar ways, see the article on constructing platonic solids. The focus for the learners is on reading and following written instructions as much as on gaining greater familiarity with 3D shapes.


The A4 paper part of theproblem.