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.

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?"

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?

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?

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.