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Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

Rationals Between...

What fractions can you find between the square roots of 65 and 67?

Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?


Age 14 to 16
Challenge Level


Why do this problem?

This problem offers an excellent opportunity for students to practise visualisation and apply an idea normally only used in 2D geometry to a 3D case. Learners will have to consider carefully how to communicate their methods for testing combinations and that they have considered all possibilities.


Possible approach

This printable worksheet may be useful: Tet Trouble.

In silence, write three lengths on the board (for example 3 units, 6 units, 7units) and accurately draw a triangle with sides of corresponding lengths. You could use a dynamic geometry package to do this.

Do it again with three more lengths.

And again but instead of drawing the triangle put a question mark. After some thinking time, encourage a member of the group to come up and draw the triangle.

Finally, list three lengths that will not work followed by a question mark and after time has been taken to realise the impossibility, discuss why this is the case as a group.

Now pose the problem.

Working in small groups the challenge will be to employ systematic approaches as well as applying the triangle inequality.

Take opportunities to pull together different ideas for recording, including the use of nets and working systematically.

Key questions

  • How do you know you have tried all possibilities?
  • Is it possible to construct more than one tetrahedron?
  • Can you find six lengths which will give more than one tetrahedron?


Possible support

Use construction straws of equivalent lengths to make (or fail to make) triangles and tetrahedra.
Alternatively, draw nets and cut them out to see if they 'work'.

Possible extension

The problem Triangles to Tetrahedra requires students to work systematically to generate all possible tetrahedra from four particular triangles.

Or there is the problem : Sliced . This is a challenging next step in this kind of visualisation.