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Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?


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

Russian Cubes

Age 14 to 16
Challenge Level

Why do this problem?

This problem invites learners to consider a familiar geometrical setting (the cube) and think more deeply about how you can utilise its properties to analyse a problem, and develop and share their ideas and deductions.

Possible approach

Invite the learners to colour nets of cubes (using just two colours) so that they can be made into different cubes. Ask them how they decide whether two cubes are identical and allow time for the group to investigate. This process could be shortened by using linking squares of two colours.

Allow groups to share their findings and encourage others to challenge them so that further reflection can take place.
  • How do they know they have them all?
The surprising result that there are just two, helps with the final part of the problem and here you might wish to encourage the sharing of different approaches to providing a convincing argument. Learners may chose to do this practically (several cubes each partially painted to show the possibilities at each stage) or more theoretically by defining sides and edges and talking about the freedoms and constraints arising at each stage.

Key questions

How do you know you have all the possibilties?

Possible extension

Consider using three colours instead of two. How many more possibilities does that open up?
The problem Proximity might be tackled next.

Possible support

Making nets and creating different cubes may be essential for learners to be able to viusalise and remove redundant examples. Focus on the discussions around what you could do to one cube to make it look the same as another before actually doing it.