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### Number and algebra

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# Russian Cubes

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Age 14 to 16

Challenge Level

- Problem
- Student Solutions
- Teachers' Resources

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.

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.

How do you know you have all the possibilties?

Consider using three colours instead of two. How many more
possibilities does that open up?

The problem Proximity
might be tackled next.

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.