Why do this problem?
This activity was designed for the 2015 Young Mathematicians' Award
so it might be a particularly useful activity for a small group of your highest-attaining pupils to work on. It is a useful vehicle for developing systematic approaches. It can be used as an activity to encourage children to explain in written or spoken words what it is they have done.
This activity may be used to encourage pupils to ask "I wonder what would happen when . . ." and use their curiosity to take things further with cuboid shapes. If you are interested in pursuing Curiosity further then see the Further Note at the bottom of this page.
Since this activity is aimed at the most confident mathematicians, there will not be much that you have to do to introduce it apart, perhaps, from making the first open box and letting the pupils declare their different ways of answering the first challenge.
Working as a team of four pupils with a maximum of 20 cubes for them to use for the remaining challenges encourages a very thoughtful and systematic approach.
Tell me about how you are getting a solution for the challenge you are working on.
(The main thing when encouraging the pupils to use their skills is to avoid saying things about what you notice and directing them in your way of attempting a solution.)
Learners might like to try the other two challenges that were part of the Young Mathematicians' Award 2015: Dice Stairs and Centred Squares.
You may be interested in the following talks given by Professor Susan Engels, which focus on encouraging curiosity and are available on YouTube:
The Rise and Fall of Curiosity
- the extract from 23.50 to 37.15 on adult encouragement and teacher behaviour is particularly worth viewing
The Hungry Mind: The Origins of Curiosity
- the extract from 8.22 to 12.29 on children asking questions is especially useful.