Another thing that was never mentioned (as far as I can remember) was the interesting phenomenon in the multiplication square - you know, this thing ...
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) = n2 - 1
It turns out to be pretty trivial once you expand out the expression, of course ...
(n - 1) (n + 1) = n2 - n + n - 1 = n2 - 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) = n2 - k2
... 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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