Mind fun

I've been treating myself lately to a set of lectures on tape.  At first I stuck with pure audio tapes, because they're so much faster to download, and until a few weeks ago, we were stuck with a HughesNet account that was subject to severe daily download limits.  Thirty-minute lectures are about 20MB, but a 30-video is more like 300MB.  Also, I wanted something to listen to while I did handwork, either crochet or painting a series of large signs I've taken on.

(Here's the picture part of the sign I've just done for the State Park, by the way.  That's what we call locally "The Big Tree," and a whooping crane.  You can listen to a lot of lectures while you paint all that detail, but of course long car trips are a good opportunity for listening, too.)

Anyway, how that we finally have a better internet connection, I broke down a ordered a handful of lectures that were available only in the video format.  The really critical images are few and far between, so I can still get my crochet work done, just stopping now and then to stare at the screen.  These are courses from "The Great Courses," and they're uniformly wonderful.  This week, though, I've stumbled on my favorite so far:  a series on how to solve mathematical puzzles.  The lecturer gives me the leap of joy I used to feel only in talking to my father.  He talks about an article  he read in an educational journal, which he admits is the only article in such a publication he ever managed to read from beginning to end (so right away he won my heart).  It described the experience of posing the same problem to a set of gifted kids and a bunch of kids on the vocational track:  how do you weigh a giraffe?

The gifted kids, the article said, were used to looking the answers up and pleasing their teacher.  They couldn't come up with an approach and quickly became anxious and discouraged.  One of the vocational kids suggested, "Let's get a chain saw and cut the giraffe up, then weigh the chunks."  The approach appealed to him, the lecturer said, because it's wrong, it's criminal, it's breaking all the rules.  The good news is, it's a metaphor for math puzzles, where there's nothing really wrong or criminal about breaking the rules.  In fact, "chainsawing the giraffe" is his new expression for the humdrum "thinking outside the box."  He lays great stress on mental tricks to avoid discouragement or anxiety, which will only tend to keep us in a mental rut.  Remember, he advises, that all of us are relatively stupid individually, because we weren't evolved to solve difficult mathematical problems.  Luckily, we're part of a civilization that can amass and transmit an enormous body of knowledge and technical skill, and we should steal ideas whenever possible -- giving credit where due, of course; he's not advocating plagiarism.

Here's one of his first puzzles.  A patient has to take one pill from Bottle A and another pill, identical in appearance, from Bottle B, every day.  Failure to take both pills is fatal, as is doubling up on either pill. The patient pours one pill out of Bottle A, but then carelessly pours two pills out of Bottle B while looking away for a moment.  Now he has three pills in his hand.  He knows that one is an A pill and two are B pills, but he can't tell by looking at them which is which.  How does he take the right dose for that day?  (Update:  And to make the problem harder for Grim:  if you don't take the entire course you'll die, and the pills aren't being made any more, so you can't just throw the three you've got away.)