I normally got poor grades in school, mostly because I couldn't be bothered to do homework. One day, in calculus class, the teacher decided to taunt the class by announcing "I will award a grade no less than one letter grade below what you get on the final exam." At first, he enjoyed the expected groaning and grimaces, until he saw me grinning madly and realized he'd made a tactical error. He and I both knew I would ace the final, and now he'd just promised me a B for doing it, even if I didn't do anything else for the rest of the semester!

The final was tough, and I did ace it. There was one final question that was this big scary horrible-looking integral that refused to yield to any of the usual techniques for simplification. I refuse to be scared off by marks on paper, and I figured I had plenty of time and nothing better to do. So I dove in and integrated the mess. And sure enough, the whole thing fell apart in the process, leaving the final answer to be something trivial like "2". It turns out the star pupils both were too frightened by the monster equation and didn't even attempt it!

My mom was going back to school to prepare to re-enter the workforce, and we ended up deciding to take the same math class. After a few weeks, the teacher called us both in. She explained that she was a little curious how we did things. We always sat together, and had the same surname, and when she'd hand out a test, she'd get back a lot of copies of essentially the approach she'd just taught, and two oddballs, from us. At first she thought we were copying from each other, but then she noticed our solutions were not only unrelated to what she was teaching, but to each other.

For example, she pulled out our most recent test, which contained a problem involving proving that the four points provided were the corners of a square. I had done something involving line lengths and reciprocal slopes. My mom had shown that the diagonals were the same length and perpendicular. Whereupon I pointed out that this didn't constitute a proof of squareness - unless the diagonals were perpendicular bisectors, you could have a figure known as a "kite", not a square. The teacher was boggled she hadn't even seen the flaw in the reasoning, but let my mom keep the credit for her answer.

In calculus class, the teacher outlined a problem with two yards of specific sorts of shapes, the idea being to enclose the maximum area with a given amount of fence. After a while, he asked what areas people had gotten. Mine was more, so I said so. He said I had the wrong answer. I maintained that my answer was correct, and I could prove it. Sure enough, he called me up to the board to show everybody. I started by drawing the diagram – with the two yards sharing a section of fence. As soon as I drew that, he knew what I had done. He pointed out that no-one had attempted that approach in all the years he had taught the class. He also pointed out that the problem was much harder to solve that way. Heh heh heh.

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November 2013

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