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Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

A Long Time at the Till

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem provides an introduction to advanced mathematical behaviour which might not typically be encountered until university. The content level is secondary, but the thinking is sophisticated and will benefit the mathematical development of school-aged mathematicians. It will be of particular interest to students who want to learn to think like mathematicians and can be used at any point in the curriculum. It will need to be used with students who are already used to engaging with sustained mathematical tasks.
Essentially the task involves carefully reading and then reflecting upon the merits of two very different solutions to a 'difficult-to-solve-but-easy-to-understand' problem. This is of value because mathematicians don't simply stop once an answer is found; reflecting on the method of solution is a key part of advanced mathematical activity. It will help train school students in the art of assessing their own solutions, which will inevitably lead to better performance in exams.
Note: The Full Solutions are to be found here - just click on the 'View full solution' link.

Possible approaches

This task ideally requires at least two students to work together so that ideas arising can be discussed. We suggest two different ways of using the problem:
1. Filling time for early-finishers/mathematics club
Print out a few copies of the problem and solutions to have to hand. Give them out to groups of keen  early-finishers to consider in 'spare' lesson time over the course of a week. Give them space to discuss the two solutions, help each other to understand the subtleties and then to discuss the relative merits of the solutions. The problem will automatically generate discussion amongst students, but you might like them to 'report' back to you or others with things that they have discovered or explored.
2. Whole-class activity
Set the background task itself as a homework problem with a fixed time-limit, stressing that only a partial solution is expected. Students should come prepared to report on the ways that they tried to solve the problem and the things that they have discovered about the problem.
Back in the lesson, group students into pairs or fours. Hand out printed copies of the solutions. Give the groups half an hour or so to try to understand the solutions with the explicit task of writing down 5 short bullet points which explain the key aspects of the solution method. Some students will prefer to discuss solutions together as they work through them whereas others will prefer to work alone. Both approaches are fine, so you might wish to group students according to their preferred style.
Next spend 10 minutes sharing the different lists of bullet points to create a 'shared' list for each problem on the board.
Spend the remanining time back in groups considering the suggested variations on the background problem. Note that some of these are significantly easier problems to solve because of their simplified prime factorisation. As a focus for the activity, set the explicit task: "which of the variation problems would you choose to solve, and why?"

Key questions

What are the 'key steps' in the solutions, and what are the 'details'.
Can you follow the overall 'strategy' of the two solutions?
Which of the two solutions seems more 'reusable' for similar variants on the background problem?
Which of the two solutions do you prefer? Why?
Of the suggested variants, which seems likely to be the easiest to analyse? Why would you think that?

Possible extension

A simple-to-set extension is to ask students to solve one or more of the suggested variations on the background problems
Another more sophisticated extension is to ask: what would make a variation of the background problem difficult or easy to solve? Can you create a much simpler problem which has a unique solution?

Possible support

Recall that we only recommend that you use this task with students already used to sustained mathematical engagement with tasks.
To help students to get started with thinking about the background task, suggest that they work in pence and convert the two conditions into equations involving whole numbers. Stress that the sum will be $711$ but the product will be $711,000,000$ due to multiplying by $100$ four times. Suggest also that prime factorisation will be useful and a clear recording system will be necessary to keep track of calculations.
In assessing the solutions encourage students to go through the solutions carefully line by line and to ask for clarification when there is a line that they do not understand.