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Age 14 to 16 Challenge Level:
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?
Here are some questions to consider:
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.