This problem involves a significant 'final challenge' which is
broken down into a sequence of three groups of mini-challenges. The
mini-challenges are not arranged in any particular order within
each group and the problem is structured such that students are
likely to 'discover' some of the mini-challenges for themselves as
they strive to solve other mini-challenges.

These notes are designed for classes who are able to work in
groups.

At the outset all challenges are hidden to the learner to
maximise the chance of discovery for the learners.

The purpose of this is two-fold: first to scaffold learners to
help them solve a difficult challenge; second to show that
mathematics is a natural subject where certain questions naturally
arise through the consideration of other questions. This will
firstly help students to structure their mathematical thinking and
secondly to help them to realise that mathematics is not externally
or meaninglessly imposed.

There are three groups of 3 mini-challenges and the final
challenge. Leave these all hidden to begin with.

Throughout the challenge the focus will be on constructing
clear, concise proof and on thinking of possible extension
questions.

Very able students might wish to start on the Final Challenge,
but it will be good to give them a single mini-challenge and see
where their thinking and invention takes them. Indeed, the best and
most inventive students might even 'discover' the final challenge
for themselves.

It is suggested that the following approach be taken

1) (10 minutes) Students individually given one of the first
mini-challenges to think about and work on. Spread the different
mini-challenges amongst the group -- don't allow students to see
any of the other challenges or talk about this with their
neighbours. Students are to think about their challenges and
explicitly write down any other thoughts or questions that arise.
Encourage those that think they have an answer to construct the
clearest proof possible or to think about possible
extensions.

2) (5 minutes) A selection of students to describe their
challenge, proofs and other questions arising. It is likely that
some of the questions arising will be the problems other students
were working on directly.

3) (10 minutes) Ask the class to organise themselves in pairs
so as to get insight into solving their mini-challenge or proposed
extensions. It might be that students pair with people working on
the same mini-challenge or pair with someone who was thinking about
a related problem. Give the pairs a new challenge from the second
grouping to work on.

4) Repeat step 2 and group into 4s, whilst giving the fours a
mini-challenge from the third grouping to work on.

5) Throughout encourage the class to propose their own
extension questions.

At some point students might solve their challenges or pose
the final challenge for themselves. When appropriate move the
discussion onto the construction of a clear proof of the final
challenge. This will be ideally a group effort.

As you think about your mini-challenge, what questions and
extensions arise?

Complete the sentence: I am finding this task difficult
because ...

Complete the sentence: I wonder if ....

Complete the sentence: I would be more able to solve my
challenge if I knew ...

Can you explain your proof clearly in words?

Solution of the final challenge on its own is a tough
challenge.

This task is designed for group work -- encourage groups not
to move on until all in the group understand.

Some students might be uncomforable with posing their own
questions or verbalizing their difficulties. Encourage an
atmosphere where all questions and difficulties are valid.