### Why do this problem?

This problem offers the opportunity to explore sequences generated
by square numbers, which in turn can offer practice in algebraic
manipulation or geometric representations to explain results.
Central to the task is the idea of making and proving conjectures,
and the open nature of the task offers lots of potential lines of
enquiry.

### Possible approach

Show the picture of Robert's spreadsheet, or if computers are
available ask learners to create it for themselves. Give them time
to discuss in pairs any patterns they notice or any interesting
questions that occur to them. Then bring the class together to
share things they've noticed, and collect together all questions
and conjectures on the board.

Possible lines of enquiry might be:

- Will square numbers only appear in columns A, D and H?
- Does the pattern of the odd squares in column A continue?
- Will the sequence continue to alternate from column A to D
to A to H?

and so on.

Suggest that each pair chooses a conjecture or question that
interests them, and then allow them time to explore. Stop the class
while they are working, when appropriate, for mini-plenaries to
share what people are working on, and to suggest ways of using
algebra or geometric representations to start to justify
conjectures. Learners could be given some numerical challenges:
"Can you explain where $29^2$ will appear? Or $34^2$? Or
$52^2$?"

Towards the end of the session, bring the class back together
to share what they have been working on and to demonstrate their
convincing arguments or explanations for what they have found
out.

### Key questions

What patterns do you notice?

Will the patterns continue?

Can you prove it?

### Possible extension

Explore what happens when square numbers are highlighted on
different grids.

Some of the patterns can be explained using modular
arithmetic, so reading

this article may
be helpful.

### Possible support

Start with a spreadsheet with four columns to investigate
patterns in odd and even square numbers and their relationship to
multiples of four.

The problem

Remainders looks
at properties of numbers based on remainders, and might be a useful
introduction.