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### Number and algebra

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### Working mathematically

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### Advanced mathematics

# What's Possible?

### Here are some questions to consider:

**Can you prove any of your findings?**
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Age 14 to 16

Challenge Level

*What's Possible? printable worksheet*

*You may be interested in Hollow Squares which offers an alternative way of thinking about the same underlying mathematics.*

Many numbers can be expressed as the difference of two perfect squares. For example, $$20 = 6^2 - 4^2$$ $$21 = 5^2 - 2^2$$ $$36 = 6^2-0^2$$

How many of the numbers from $1$ to $30$ can you express as the difference of two perfect squares?

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

What about the difference between the squares of 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 express as the difference of two perfect squares?

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

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?