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## 'What's Possible?' printed from http://nrich.maths.org/

Many numbers can be expressed as the difference of two perfect
squares. For example, $$20 = 6^2 - 4^2$$ $$21 = 5^2 - 2^2$$

How many of the numbers from $1$ to $20$ can you express as the
difference of two perfect squares?

### Here are some questions to consider:

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?

You may want to take a look
at Plus Minus next.