### Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

### What's Possible?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

### Marbles in a Box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

# Difference of Odd Squares

##### Age 14 to 18Challenge 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.

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?

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