Tall, Dark, and Mysterious

1/10/2005

Lights and fuses

File under: Queen of Sciences. Posted by Moebius Stripper at 10:59 pm.

3D Pancakes has a nice logic puzzle I’d never seen before:

Suppose you have two fuses—strings soaked in alcohol and gunpowder—and a lighter. Each fuse is carefully calibrated to take exactly one hour to burn from one end to the other. However, different portions of the fuses burn at unpredictably different rates, sometimes more slowly, sometimes faster. How would you measure exactly 45 minutes?

The answer’s at Ernie’s, along with a nice generalization of the problem: What time intervals can we measure exactly with one-hour fuses?

This reminds me of one of my favourite problems of this type:

In one room, there are three 100 W lightbulbs; they are all turned off. In another, there are three light switches. You don’t know which switches control which bulbs.

You are allowed to move lightswitches three times (”up” counts as moving a lightswitch once), and you are allowed to enter the room with the bulbs once. How can you tell which switches control the bulbs?

I should send these to the 13-year-old kid I mentored last year; they’ll drive her crazy, but she likes being driven crazy by things like this.

Edited to add that the lights all begin off and that they’re not energy efficient; thanks, Marc and Daniel Lemire.

If it was good enough for Newton and Leibniz, it should be good enough for my students

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 3:33 pm.

I absolutely loathe the way that my class’s calculus textbook (Larson, Hostetler, and Edwards, in case you’re wondering) presents antiderivatives 15 pages before introducing the problem of finding areas under curves. This approach leaves the hapless instructor showing students how to crank out antiderivatives - which are denoted by this weird squiggle (an elongated S, but who’s keeping track?) - before moving on to the ostensibly unrelated problem of computing areas - which also happen to be denoted by the same weird squiggle, but this time with sub- and superscripts. Only two chapters later do we see that, hey, there’s actually a reason that we use the same squiggle to denote both quantities. In other words, we’re motivating a subject via notation. Given the option, I try to motivate a new topic by its historical development; when that’s impractical, I try to motivate it via the applications we’re interested in for the course. Bypassing the why of a subject offends me, and I think that much of students’ dislike of mathematics can be chalked up to their never having seen what brought about this topic or that one. It’s all a bunch of formulas and data and notation, which God in His infinite wisdom decreed would be part of the mathematical canon.

In any case, last week, when I got around to showing that there’s a relationship between areas and antiderivatives, my students’ reactions could be generalized as well, yeah, duh, of course there is, otherwise why would you have presented the two topics together? It all seemed so unsatisfying: the Fundamental Theorem of Calculus should astound people, dammit. Presumably it astouded its discoverers, and they were probably more tired and more hungry than my students when they first saw the connection. I felt as though I’d explained the punch line of a joke before telling the rest of it: by the time I was done, everyone got it, but no one was laughing.