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Difference of Odd Squares

Age 14 to 18
Challenge 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. 

What do you notice about your answers?

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.