A smart way of finding the square of 'something-and-a-half'.

For example, I can tell you right away that 712×712=5614, without having to do

152×152=2254=5614.

How do I do it?

Well, draw a square with side 712and find its area. One way to do it is to cut up the square and reassemble it like this:

 

The area of the square is equal to that of a 7×8rectangle, together with the 12×12square, and so it is 5614altogether.

[Alternatively, if you know the difference of two squares identity:

(712)2−(12)2=(712−12)×(712+12)=7×8

So (712)2must be (12)2more than this: 5614]

If you were persuaded by my explanation for 712×712being 5614, you can now probably state the value of, say, 512×512, and know why it is so.

In effect, you now know that n-and-a-half squared is equal to n times n+1, and a quarter, for any positive integer* n: and why it is so. But I didn’t write out a general, formal proof of this generalization – I only gave one example, with n=7(although I did present it very carefully …).

* Actually this generalization is not only true for integers, but my demonstration with712was easier to write because 7and 8are integers.