Reading the Comics, March 6, 2019: Fix This Joke Edition


This week had a pretty good crop. I think Comic Strip Master Command is warming its people up for Pi Day. Better, there’s one that’s a good open-ended topic. We’ll get there.

Bill Amend’s FoxTrot for the 3rd (not a rerun) has Jason trying to teach his pet iguana algebra. Animals have some number sense, certainly. It depends on the animal. But we do see evidence of animals that can count, and that understand some geometrical truths. The level of abstraction needed for algebra — to discuss numbers when we don’t know, or don’t care, about their value — seems likely beyond what we could expect from animals. I say this aware that the last fifty years of animal cognition research have been, mostly, “yeah, so remember how we all agreed only humans could do this thing? Well, we looked at some nutrias here and … ”

Peter: 'Whatcha doing?' Jason: 'Teaching Quincy algebra.' Peter: 'Isn't that a little advanced for an iguana?' Jason: 'I tried teaching him simpler math like addition and subtraction, but he wouldn't stop yawning. I'm taking that as a sign he needed something more challenging to engage in. 'Chapter seven: Quadratic Equations'.' (Quincy falls asleep.) Peter: 'Well, he's not yawning.' Jason: 'Maybe I should just jump right to calculus.'
Bill Amend’s FoxTrot for the 3rd of March, 2019. Essays that discuss FoxTrot, both old and current vintage, are at this link.

Jason’s diagnosis that Quincy needs something more challenging is fair enough though. Teaching needs a couple of elements to succeed. The student’s confidence that this is worth the attention is one of them. A lot of teaching focuses on things that are, yes, beyond what the student now knows. But that the student can work out without feeling too lost. Feeling a bit lost helps. But there is great motivation in the moment when you feel less lost. Setting up such moments is among the things skilled teachers do.

(And I say “among”. There can be great joy in teaching a topic someone already knows, if what you’re really doing is showing some new perspective on it. And teaching things someone already knows is a good way to reassure that they have got it. Nothing is ever just the one thing.)

'Disc-o-Magic'. It's a ring of ten magician names, linked clockwise, and an inner ring of five magician names. Starting from any of the outer ring and going clockwise a number of times equal to the number of letters in the magician's name (eg, so, 'Houdini' would move clockwise seven spaces), then insite and repeating this counterclockwise the number of letters in *that* magician's name lands you to 'a new name that is (arguably) the name of the world's greatest magician!'
Mac King and Bill King’s Magic in a Minute for the 3rd of March, 2019. Arithmetic-based tricks from Magic in a Minute get listed at this link.

Mac King and Bill King’s Magic in a Minute for the 3rd is a variation of a trick from mid-January and mentioned here. It is, like many mathematics problems on a clock face, or a clock-like face, a modular numbers game in disguise. The trick is to give every starting, blue, bubble a path that ends at the same spot. There are tricks to get there, hidden in the network. For example, the first step is to start at any magician’s name in the outer ring, and move clockwise a number of steps equal to the number of letters in their name. All right: where would you start to finish on ‘Roy’ or ‘Thurston’? Given the levels of work needed for this I find it more impressive than I do January’s clock trick.

Frank Page’s Bob the Squirrel for the 4th sees Lauren working on a multiple-choice mathematics question. (It’s SAT prep work.) She’s startled that Bob can spot the answer right away. But there’s reasons it’s not so shocking Bob would be so fast.

Lauren's SAT prep question: if f(x) = 2x^2 + 4 for all real numbers x which of the following is equal to f(3) + f(5)? a. f(4). b. f(6). c. f(10). d. f(15). Bob comes up. Lauren: 'I'm *studying*, Bob. Don't bother me.' Bob: 'The answer is B.' Lauren: 'Wow ... that's the correct ... answer?' Bob: 'WHY do you gotta say it with all the dots and pauses like that?'
Frank Page’s Bob the Squirrel for the 4th of March, 2019. The occasional essay inspired by Bob The Squirrel is at this link.

The first thing I notice in this problem is f(x). For positive values of x this is an “increasing” function. That is, if you have two positive numbers x and y, and x is less than y, then f(x) is less than f(y). You can see that from how x^2 is an increasing function. Multiply an increasing function by a positive number and it stays increasing. Add a constant to an increasing function and it stays increasing. So this right away rules out f(4) as a possible answer. If Lauren guessed wildly at this point, she’d have a one-in-three chance of getting it right. If the SAT still scores by the rules in place when I took it, that’s a chance worth taking.

That x^2 is another tip. This value grows, and pretty fast. It grows even faster the bigger x gets. The difference between f(10) and f(11) is 42. The difference between f(11) and f(12) is 46. The difference between f(12) and f(13) is 50. So just from that alone it’s hard to imagine f(15) being the right answer. Easier to imagine f(10) being right. Less hard to imagine f(6) being right. If I had to guess, f(6) would be it. If I must know which is right? I’d start by calculating f(5) and f(6). Then check their difference. If that seems close to what f(3) must be, good, call it done. If that didn’t work I’d move reluctantly on to calculating f(10). But, bleah. Seems tedious. I’m glad to be past having to work that out.

Woman, to the man with her, as they see someone approaching the corner of the city street: 'It's that Fibonacci dude. His conversations are never-ending.'
S Camilleri Konar’s Six Chix for the 6th of March, 2019. Essays inspired by something mentioned in Six Chix, whichever cartoonist created it, are at this link.

S Camilleri Konar’s Six Chix for the 6th name-drops Fibonacci. This fellow is Leonardo of Pisa, who lived from around 1175 to around 1240 or so. He’s famous for — well, a bunch of things. One is his book explaining Arabic numerals to Western Europe and why they’re really better for so much calculation work. But another is what we now call the Fibonacci Sequence. We now call him Fibonacci, although that name’s a 19th century retronym. He belonged to the Bonacci family (‘Fibonacci’ would mean ‘child of Bonacci’) and, at least sometimes, called himself Leonardo Bigollo. Bigollo here meaning a traveller or a good-for-nothing.

His sequence is famous; it starts 1, 1, 2, 3, 5, 8, and so on, with each term in the sequence being the sum of the two terms before it. He was using this as a toy problem about breeding rabbits, meant to demonstrate ways to calculate better. This toy problem turns up in surprising contexts. Sometimes in algorithms. Sometimes in growth of natural objects; plant leaves and genes moving around on chromosomes and such. Sometimes in number theory. It’s even got links to the Golden Ratio, if we count that as interesting mathematics. And it inspires an activity problem. Per John Golden, a friend on Twitter:

The joke is all right as it is. The thing someone might associate with the name Fibonacci is the sequence, and it’s true that one never ends. But never ending isn’t a particularly distinctive feature of the Fibonacci sequence. Can the joke be rewritten so that the mathematics referenced is important?

There’s several properties of the sequence that might be useful. One is the thing that defined the sequence. Each term in it is the sum of the two preceding terms. The Golden Ratio offers another. Take any term in the sequence. The next term in the sequence is, approximately, the golden ratio of 1.618(etc) times the current term. The approximation gets better and better the more terms you go on.

That’s … really probably all you can expect to work with. There are fascinating other properties but you have to be really into number theory to know them. A positive number x is a Fibonacci number if and only if either 5x^2 + 4 or 5x^2 - 4 , or both, are perfect squares, for example. 1, 8, and 144 are the only Fibonacci numbers that are perfect powers of a whole number. Any Fibonacci number besides 1, 2, and 3 is the largest number of a Pythagorean triplet. Building a joke on any of these facts aims it at a particularly narrow audience.

If you feel the essential part of the joke is “this thing is never-ending” rather than “this involves Fibonacci” you have other options. How you might rewrite the joke depends on what you think the joke is.

And to speak of rewriting the joke is not to say Konar was wrong to make the joke she did, of course. We all understood what was being referenced and why it made for a punch line. Rewriting the joke to more tightly use its mathematical content does not necessarily make it funnier. This is especially so if a rewrite makes the joke too inaccessible. A comic strip is an optimization problem of how to compose a funny idea and to express it to a broad audience quickly. And then you have to solve it again.


That’s far from the full set of mathematics comics this past week. I’ll have another posting about them here soon enough. And yes, I know what Thursday is, too.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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