(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.)
14 comments:
You weigh a giraffe the same way you weigh the missus: on the elevator scale.
Oh, and the obvious answer to the pill problem: throw the pills away, and measure a new dose. Pills are cheap.
The giraffe solution is applicable...
NOT for weighing the missus, for preparing and taking your needed medication.
Enjoyed the elevator scale. I'll have to remember that one and never, ever, pull such a stunt. I like sleeping indoors what with the exoskeleton taking a chill so easily.
Tex, your painting looks mahvelous, absolutely mahvelous! And I believe I'll get some use from the Great Classics link. Thanks.
The solution is extremely simple and straightforward, and never would have occurred to me.
It occurred to me probably due to my watching W.B. cut up her allergy medicines, vitamins, etc.
Since retirement, I find I'm a subscriber to ye old big hammer methodology. It's almost as if I've outgrown minutiae! *snerk*
It only didn't occur to you because you were thinking of it as a math problem. If you'd had a child in front of you for whom you were sorting the medication, you'd have thrown it out and started over without having to think about it at all.
You take new pills, and while doing so, send the old pills to the lab for analysis, hoping you get the results back in time to save your life.
(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.)
Oh, fine.
You know you have 1 A pill and 2 B pills in your hand. If you divide all three pills in half, and take 1/2 of each, you therefore know you have correctly taken 1/2 the A dose, and the full B dose. Divide a second A pill in half, eat half of it, and set the other half aside. You are now good for today.
The three "half" pills you had also contain 2 halves of a B dose, and one half of an A dose. Put them with the other half of the A dose, and retain for your next dose.
Exactly. You have 1-1/2 doses in your hand, which is inconvenient. So make it 2 full doses by adding an "A" pill, then cut all four pills in half. Voila: two A halves and two B halves.
I think that's elegant. After you repair the broken symmetry, a solution becomes easier. He uses a lot of doubling and pairing techniques, and attention to odd and even components. Good speaker, too. He used to coach math teams, and I can imagine he was superb at it, because his pleasure is infectious and his exposition is clear. But I have to hit "pause" a lot and scribble on a notepad to catch up, which makes it hard to crochet.
"you repair the broken symmetry, a solution becomes easier.
Yup. That is a technique used by almost all coding widgets. For instance when one is faced with a large application that incessantly eats memory. You play little games like tossing together a script file to search through the sources for the number of memory allocations (malloc function calls) within the code and compare that count to the number of memory release (free function in C language) calls in the code.
As in most things, balance is good, but sometimes, the large hammer of tossing out excessive wreckage and starting afresh is better, if not as satisfying.
With such an exacting regimen there were probably equal numbers of pills in the bottles to begin with. It might be easier to count the numbers of pills left. (I've cut pills before, and some are hard to get split evenly. Even with a pill splitter.) Of course that assumes knowledge not given in the problem...
The problem with counting the number of pills is that it doesn't add to your knowledge. You already know that you got a double dose of B. Counting the pills will only confirm that -- it won't tell you which of the three in your hand is an A.
Yours truly was up too late last night, and I don't think I've fully recovered today...
Believe me, I understand. It's an occupational hazard in the warrior-poet business.
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