(Editor’s note: today I am officially moving, and will not have internet access until some point tomorrow. I am writing a couple of pieces ahead of time that will publish themselves during New Year’s Day to keep things “fresh.”)
I tend not to discuss too much personal stuff here, but with it being New Year’s Day and most readers probably (and properly) too hung over to bother with blogs, I figured it’d be a good time to talk about some things that are utterly non-time-sensitive and completely non-political and non-controversial, yet one or two people might find interesting — and, perhaps, might be able to tell me if these ideas and discoveries of mine are unique, or if I’ve spent 30 years of spare brain cycles just re-inventing others’ work.
I’ve always had an affinity for the simpler forms of math — arithmetic, algebra, and geometry. Once it gets much beyond beginning to intermediate geometry, I begin to lose it, but my brain somehow can solve such things a LOT faster than most people. (I know this because when I demonstrate it, they look at me oddly and ask “how did you do that so quickly?”) I’m no Rain Man, but I am fast.
I’m going to stick the rest of this “below the fold,” so as to not bore everyone.
It’s largely a matter of practice. I tend to “play” with numbers in the back of my mind nearly all the time, just for kicks. For some reason, my subconscious finds it entertaining, and that keeps it out of trouble.
One of the things I’ve always been fascinated by is square numbers. I’ve gotten to the point where I “know” the square of every number up past 100, and can provide it within a second or so if challenged. (For example, 77^2 is 5,929.) And while playing with those numbers, I’ve discovered a lot of things I find fascinating.
For example, a lot of people know that the set of square numbers is simply the sum of all the odd numbers. 1, 4, 9, 16, 25 are 1, 1+3, 1+3+5, 1+3+5+7, and 1+3+5+7+9. The “trick” to know when to stop is when you reach the last odd number before twice the number you are squaring. In the last case, to get the answer to 5^2, you stop at ((5*2)-1), or 9.
Another odd thing I’ve discovered is that square numbers can always be expressed as a multiple of the number 7 plus a factor of the number 2 (1, 2, 4, 8, 16, 32), and never as a multiple of the number 7 minus a factor of the number 2. I have no idea why this is, but it holds true to every square number I’ve tested. (And yes, I’m including 1 and 4 — you just have to start out with 7*0.)
The number 7 has some other very odd properties. I will touch on those later.
But my proudest achievement was a formula I discovered, a little “trick” that helps me calculate square numbers. It’s this equation:
x^2 = (x+y) * (x-y) + y^2
I showed this to a high school math teacher once; she promply solved it, and showed me that I had proven x = x. I was very fond of her, but she missed the point: this was a method, not a solution. Her reducing it simply proved that I didn’t “cheat” and it was valid. And the point of the method is to let you calculate square numbers without a calculator.
Suppose that you are trying to find out the square of 97. That makes x = 97 in the formula:
97^2 = (97+y) * (97 -y) + y^2
Since my math teacher proved that y is, in the end, irrelevant, you can assign it any value you like, since they’ll all go away by the end. I always choose a value for y that makes the math simple — in this case, 3. So, once y is plugged into the equation, it becomes this:
97^2 = (97+3) * (97-3) + 3^2
97^2 = 100*94 + 3^2
97^2 = 9400 + 9
97^2 = 9409
Fire up your calculators and check for yourselves. It’s correct.
For my next posting, I’ll take square numbers to what was, for me, the next logical step: Pythagoran Triples.