X2 = ((x – y) * (x + y)) + y2
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 shorter and longer 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
A coverage of just 63.
But what if I followed on with my plan
And purchased a 6 inch by 10?
This is now shorter and longer 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
As at the equation I stared
At 60 square inches this was 4 less –
The difference of 2, and then squared.
And, of course, when the difference was 3
The obvious truth at me glared
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.