X 1 2 3 4 5 6 ...

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

etc.

If you look down the diagonal axis from top left to bottom right, then you get a list of the square numbers - 1, 4, 9, 16, 25 ... What I noticed was that, if you go "northeast" and "southwest" from those numbers, you always get a number exactly one less. That is, if you take a number, and multiply the number one more and one less than it, then you get one less than the number squared. Or ...

(n - 1) (n + 1) = n

^{2}- 1

It turns out to be pretty trivial once you expand out the expression, of course ...

(n - 1) (n + 1) = n

^{2}- n + n - 1 = n

^{2}- 1

But nobody ever bothered to point it out, and I felt a gram of so of smug when I proved it for myself.

There's actually a more general thing lurking here ...

(n - k) (n + k) = n

^{2}- k

^{2}

^{}... which means that if you look at the differences as you continue "northeast" and "southwest" from numbers on the diagonal, you are going to get another series of square numbers.

## 1 comment:

I have recently been sharing with people the amazing number 1089. Was it you who told me about this in the first place Paul?

1089 is 9 x 121 which is 3 squared times 11 squared. Interesting but not earth shattering.

But - it gets better.

1 X 1089 = 1089. 9 X 1089 = 9801.

2X 1089 = 2178. 8 X 1089 = 8712

3 X 1089 etc

But that's only the beginning. Wanna know some more?

By the way I see I need to prove I ma not a robot. I am obviously partially a robot because I often get these wrong. I wish they were not QUITE so difficult.

Robert

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