Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."
I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."
Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out
x^2 - 1 = (x - 1) (x +1)
So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!
You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we concluded we could take arbitrary steps perpendicularly off-diagonal
(x - n) (x +n) = x^2 - n^2
It's moments like these that make it hard to remember that we need to praise hard work over Smartness. But it's true. Owen thinks about numbers deeply and frequently. Practice leads to creativity, creativity leads to intelligence as his genes encode experience. The boy's brain is growing without bound!