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2-digit Square

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?

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Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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Why 24?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

What's Possible?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
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, $$20 = 6^2 - 4^2$$ $$21 = 5^2 - 2^2$$

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.