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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Triangular Triples

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.


Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Robert's Spreadsheet

Age 14 to 16
Challenge Level

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.