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# Difference of Odd Squares

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*

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Age 14 to 18

Challenge Level

Let's take a look at the difference of squares of odd numbers:

$$11^2 - 5^2=96$$

$$5^2 - 3^2=16$$

$$7^2-3^2=40$$

**Find the difference of some more squares of odd numbers. **

What do you notice about your answers?

Can you prove your conjecture?

You can find some hints on how to construct a proof in the Getting Started section.

**Other questions to think about:**

Once you have had a think about this problem, you might like to think about these questions.

- What happens if we take the difference of squares of even numbers?
- Can a number which is a multiple of $8$ be written as the difference of squares of even numbers?
- What happens if we take the difference of the square of an odd number and the square of an even number?
- Which numbers can we write as a difference of two squares?
- Which numbers can we not write as a difference of two squares?