Why do this problem?
This problem involves a significant 'final challenge' which is
broken down into a sequence of mini-challenges. The mini-challenges
are not arranged in any particular order 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
These notes are designed for classes who are able to work in
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.
You can print out cards of the statements here
There are 6 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
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 are likely to 'discover' the final
challenge for themselves.
It is suggested that the following approach be taken
0) The context introduced so that every one understands the
1) (10 minutes) Students individually given one of the
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.
It is not expected at each stage that the challenges will be solved
-- merely that difficulties and other questions 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 and some of the difficulties 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. 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
4) Repeat step 2 and group into 4s.
5) Throughout encourage the class to propose their own
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
Complete the sentence: I am finding this task difficult
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. The main extensions might be:
What happens if you start the process with a different
Can you relate this process to any other mathematics you know
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.