X² = (x – y) * (x + y)) + y²  (where y is the difference)

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I first realised this mathematical fact

When I planned to purchase some tiles

Of course, I could have got various types

Of colours and sizes and styles.

 

I measured my floor up for area

And settled on 8 inches square

But this would require me to cut some tiles

Not an effective affair.

 

I then contemplated a different approach

After I’d studied a while

Would I need more or would I need less

If I bought a different sized tile?

 

So supposing I used a different size

Let’s say seven inches by nine

That’s longer and shorter on each of its sides

By an inch than the first one of mine.

 

Now the 8 by 8 covered 64 inch

Square, I hope you agree,

While the 7 by 9 could only manage

An area of just 63.

 

But what if I followed on with my plan

And purchased a 6 inch by 10?

This is now longer and shorter by 2

So would it be smaller again?

 

Well, yes, it now comes to 60 sq inch

(I’d need more of these for the floor)

The tile is reduced in area now

By a greater figure of 4.

 

Now already you’ll be ahead of me

So let’s try 11 by 5;

And, sure enough, it’s happened again

We’re now down to just 55.

 

And then I saw what was happening here

Just why I kept making these gaffs

It was clear enough as I measured the tiles

So I’ll try to explain it in maths.

  

When I varied the edges by just one inch

You recollect just how I fared?

The area lessened by just one inch –

The difference of one, and then squared.

 

On the second example the difference was 2

The obvious truth it had glared

At 60 square inches this was 4 less –

The reduction of 2, and then squared.

 

And, of course, when the difference was 3

As at the equation I stared

55 inches was fewer by 9 –

That difference of 3 and then squared.

 

So what had I just discovered here?

That a square made longer and shorter

Reduces in area and, in fact, is not

The same as you’d think it oughta!

 

Such complications and implications!

Would they waste less but cost more?

And all of this thinking had made my head spin

I think I shall carpet the floor.