Difference of Odd Squares

Age 14 to 18

Challenge Level Yellow star

Here are some hints to help you with the proof:

A general odd number can be written in the form

$2n+1$ where

$n$ is an integer (whole number).

You will need two different (and completely unrelated) odd numbers.
Note that

$2n+1$ and

$2n-1$ are consecutive odd numbers, so these are not general enough to use.

The difference of two squares formula might (or might not!) be useful:

$A^2-B^2 = (A+B)(A-B).$

You may be able to write the expression as

$4(...)$ and so know that it is a multiple of

$4$ . To find another factor of

$2$ consider the rest of the expression - can you prove that this must be even?

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