Copyright © University of Cambridge. All rights reserved.

This problem starts by asking students to find which numbers can be expressed as the difference of two square numbers, and then suggests some possible avenues for exploration. This can then be used as a springboard to generalisations and the use of algebra for justifications and proof. Along the way, students have the opportunity to make use of the
important identity $a^2 - b^2 = (a + b)(a - b)$.

An alternative to this problem which some students may find more accessible is Hollow Squares.

Arrange the class in groups of two or three, and hand each group one set from these cards. Ask them to work in their groups for the first few minutes, looking at what they notice about their three cards, whether they could create other cards that would belong in the same set, and what questions are prompted by them.

Next, bring the class together to share what was on their cards, and what questions occurred to them. Do any of their questions resolve themselves once they see someone else's cards? Make sure conjectures and questions are noted on the board.

One way to begin to resolve these conjectures and provoke new ones is to challenge students to find numbers between $1$ and $30$ which can be expressed as the difference between two square numbers. Encourage them to find more than one solution where possible, and draw attention to systematic ways of working and recording. Once everyone has had some time to find most of the answers,
ask students to share what they have found and tabulate the answers on the board. Note down any new conjectures that emerge at this point.

Allow students to choose a line of enquiry, working on their own or in small groups. The emphasis should be on proving any conjectures they make, whether using diagrams or more formal algebraic methods. A diagrammatic method for calculating the difference of two squares is explored in the problem Plus Minus.

The plenary can involve students presenting their findings to the rest of the class. Expect students to be clear and rigorous in their justifications. Encourage students to challenge any proofs that lack clarity and rigour, and suggest ways of improving them.

One way to round off the work on this problem could be:

"In a while, I'm going to give you a number and ask you to quickly find one or more ways to write it as the difference of two squares, or to convince me that it can't be done."

What do you notice about the difference between squares of consecutive numbers?

What about the difference when I square two numbers which differ by $2$? By $3$? By $4$ ...?

When is the difference between two square numbers odd?

And when is it even?

What do you notice about the numbers you CANNOT make?

Can you prove any of your findings?

Every odd prime number can be written as the difference of two squares. Prove that there is only one way to write an odd prime number as the difference of two squares, and that this is the difference of two successive squares.

120 can be written as the difference of the squares of whole numbers in the following ways:

$$31^2-29^2$$

$$17^2-13^2$$

$$13^2-7^2$$

$$11^2-1^2$$

Can you find all the possible ways to write 924 as the difference of the squares of two whole numbers? Can you prove that you have found them all?

*REALLY CHALLENGING*

Can you describe a way for working out how many ways there are to write any number as the difference of two squares?

Encourage students to list their results in a systematic way, gathering together in separate columns numbers made from squaring two numbers which differ by $1$, $2$, $3$... as started below:

1 | 2 | 3 | |

1 | $1^2-0^2$ | ||

2 | |||

3 | $2^2-1^2$ | ||

4 | $2^2-0^2$ | ||

5 | $3^2-2^2$ | ||

6 | |||

7 | $4^2-3^2$ | ||

8 | $3^2-1^2$ | ||

9 | $5^2-4^2$ | $3^2-0^2$ |