1. Number Theory is taught by a short man named Csaba. He's rude, makes off-color jokes, loud, throws chalk at people and at the board, and waves his hands a lot. Essentially, he's Mr. Kinberg but for math. Anyway, we were talking about the idea of primes and thinking about primes existing if we restrict our world to just the even numbers. I don't want to get technical, but the short of it is that we concluded, after a very detailed explanation, that 6 is prime in the evens. Then, he turns around and says, "So where's my mistake?" We stare blankly. "Any proof I give may very well be wrong on purpose," he goes on to explain.
Now, for years I've been tutoring math. I do this all the time - I make a big fuss about it. Never has it happened to me. It was awesome and humbling - it didn't help that none of us could find his mistake.
2. Classical Algebra is a 3-week refresher on things like complex numbers, polynomials, etc - stuff that Hungarian students learn when their in womb. A strong tradition in math competitions and education, along with a rigorous curriculum and some number theory means that Hungarian high schoolers can whoop American math majors on the Putnam. This is something that every one of my teachers likes to point out. Back to the story. We're talking about sums of binomial coefficients:
The first expression sums to 2.
Then 5.
11
22
"Can anybody guess the next one?"
"45?"
"Almost, 43. How about the next one? I'll give you three guesses."
Before you peek, take a guess. Actually, take three.
Have you guessed? It's 85.
Now, I don't know about you, but when the professor said three guesses, about 30 students all guessed 85, 86, and 87 in unison. Do you see it now?
If the 5th term is x, then the 6th term will be 2x-1, 2x, or 2x+1. This is true for any kth and (k+1)th terms.
Now, this in itself isn't that interesting. What is (I think) amazing is that we (EDIT: by "we" I mean humans, not that particular group of math students) are so good at finding patterns that, seeing 2, 5 (2x2 + 1), 11 (2x5 +1), 22 (2x11), and 43 (2x22 - 1), we were all able to guess the pattern. It's hardly a pattern at all!
Okay, maybe it's not that exciting. Maybe the pattern is pretty straight forward. Granted, we're math kids - our brain our trained to look for fun patterns. Even so, this got me thinking about how awesome our (EDIT: again, the collective "we" of the species) minds are.
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"Classical Algebra is a 3-week refresher on things like complex numbers, polynomials, etc - stuff that Hungarian students learn when their in womb."
ReplyDelete-So, the class they learn that stuff in.. I guess is homewomb??