Reading the Comics, May 12, 2020: Little Oop Counts For More Edition


The past week had a fair number of comic strips mentioning some aspect of mathematics. One of them is, really, fairly slight. But it extends a thread in the comic strip that I like and so that I will feature here.

Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 10th continues the thread of young Alley Oop’s time discovering numbers. (This in a storyline that’s seen him brought to the modern day.) The Moo researchers of the time have found numbers larger than three. As I’d mentioned when this joke was first done, that Oop might not have had a word for “seven” until recently doesn’t mean he wouldn’t have understood that seven of a thing was more than five of a thing, or less than twelve of a thing. At least if he could compare them.

Penelope, leading to the library: 'If you're going to keep coming to school with me, Alley, we've got to catch you up. You must learn to read.' Alley Oop: 'Hey! I can read.' Penelope: 'Really? How is that possible?' Alley: 'Well, letters are grouped into things called words, which in a certain order ... ' Penelope: 'OK, fine, what about numbers?' Alley: 'We just got numbers back home, so I know all about one, seven, five. All the numbers.' Penelope: 'Can you do *math*, though? What's three plus three?' Alley: 'Easy. It's threethree.' Penelope, to the librarian, with a mathematics book open in front of Alley: 'Can you put on a pot of coffee, Nancy? We're gonna be here a while.'
Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 10th of May, 2020. So first, hey, neat: Little Alley Oop is a Javascript routine! Second, essays in which I talk about this comic, either the daily Alley Oop or the Sunday Little Oop pages, are at this link.

Sam Hurt’s Eyebeam for the 11th uses heaps of mathematical expressions, graphs, charts, and Venn diagrams to represent the concept of “data”. It’s spilled all over to represent “sloppy data”. Usually by the term we mean data that we feel is unreliable. Measurements that are imprecise, or that are unlikely to be reliable. Precision is, roughly, how many significant digits your measurement has. Reliability is, roughly, if you repeated the measurement would you get about the same number?

Nate Fakes’s Break of Day for the 12th is the anthropomorphic numerals joke for the week.

Ryan North’s Dinosaur Comics for the 12th talks about immortality. And what the probability of events means when there are infinitely many opportunities for a thing to happen.

We’re accustomed in probability to thinking of the expectation value. This is the chance that something will happen, given some number N opportunities to happen, if at each opportunity it has the probability p of happening. Let me assume the probability is always the same number. If it’s not, our work gets harder, although it’s basically the same kind of work. But, then, the expectation value, the number of times we’d expect to see the thing happen, is N times p. Which, as Utahraptor points out, we can expect has to be at least 1 for any event, however unlikely, given enough chances. So it should be.

But, then, to take Utahraptor’s example: what is the probability that an immortal being never trips down the stairs? At least not badly enough to do harm? Why should we think that’s zero? It’s not as if there’s a physical law that compels someone to go to stairs and then to fall down them to their death. And, if there’s any nonzero chance of someone not dying this way? Then, if there are enough immortals, there’s someone who will go forever without falling down stairs.

That covers just the one way to die, of course. But the same reasoning holds for every possible way to die. If there’s enough immortals, there’s someone who would not die from falling down stairs and from never being struck by a meteor. And someone who’d never fall down stairs and never be struck by a meteor and never fall off a cliff trying to drop an anvil on a roadrunner. And so on. If there are infinitely many people, there’s at least one who’d avoid all possible accidental causes of death.

God: 'T-Rex let's assume somehow you never die of natural causes. That's still not immortality.' T-Rex: 'Impossible!' T-Rex: 'You're still mortal. The difference is you won't die from your body getting old. Instead everything around you will be trying to kill you. You know. Accidents.' T-rex: 'PRETTY Sure I can avoid tripping down stairs if it means LIVING FOREVER.' Utahraptor: 'Pretty sure I can prove you can't!' T-Rex: 'Pretty sure I can get a book on how to hold the handrail!' Utahraptor: 'Forever is INFINITELY LONG. Say you have a 1 in 10 trillion chance of dying on the stairs. How often can you expect that happens if you life, oh, 10 trillion years?' T-Rex: 'O-once?' Utahraptor: 'And if you live INFINITY YEARS the chance of you dying from it becomes : total certainty. With an infinite natural lifespan the chance you die of ANYTHING rises to 1. Literally the entire universe will kill you if you give it enough time.' T-Rex: 'That means if I live long enough YOU'LL kill me too! Oh man! This friendship just got ... dangerous!
Ryan North’s Dinosaur Comics for the 12th of May, 2020. I often talk about this strip and when I do, Dinosaur Comics appears among the essays at this link.

More. If there’s infinitely many immortals, then there are going to be a second and a third — indeed, an infinite number — of people who happen to be lucky enough to never die from anything. Infinitely many immortals die of accidents, sure, but somehow not all of them. We can’t even say that more immortals die of accidents than don’t.

My point is that probability gets really weird when you try putting infinities into it. Proceed with extreme caution. But the results of basic, incautious, thinking can be quite heady.

Bill Amend’s FoxTrot Classics for the 12th has Paige cramming for a geometry exam. Don’t cram for exams; it really doesn’t work. It’s regular steady relaxed studying that you need. That and rest. There is nothing you do that you do better for being sleep-deprived.

Bob Weber Jr and Jay Stephens’s Oh Brother for the 12th has Lily tease her brother with a story problem. I believe the strip’s a rerun, but it had been gone altogether for more than a year. It’s nice to see it returned anyway.

And while I don’t regularly cover web-only comics here, Norm Feuti has carried on his Gil as a Sunday-only web comic. The strip for the 10th of May has Gil using a calculator for mathematics homework, with a teacher who didn’t say he couldn’t. I’m surprised she hadn’t set a guideline.


This carries me through half a week. I’ll have more mathematically-themed comic strips at this link soon. Thanks for reading.

Reading the Comics, May 7, 2020: Getting to Golf Edition


Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.

Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.

Brewster giving a presentation: 'For my presentation, I couldn't decide what graphs to use.' [ In front of a bar chart ] 'I did a bar chart to find the most-used graphs.' [ In front of a line graph ] 'This line graph shows the growing popularity of bar graphs.' [ Scatter plot ] 'This scatter plot graph shows a pattern of people who don't understand scatter plot graphs.' [ Pie chart ] 'This one shows which graph most reminds us of food.' Audience member: 'Wasn't your presentation supposed to be on not getting distracted?' [ Brewster looks at his bubble chart ] 'And bubble charts really pop!'
Tim Rickard’s Brewster Rockit for the 3rd of May, 2020. It’s been surprisingly long since I last reviewed this strip here. Essays featuring Brewster Rockit are at this link.

The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.

Jef Mallett’s Frazz for the 3rd is a resisting-the-story-problem joke. It’s built not just on wondering the point of story problems at all, but of these story problems during the pandemic. (Which Mallett on the 27th of April, would be taking “some liberties” with the real world. It’s a respectable decision.)

And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.

Students looking at story problems: '... how many more pints will it take to empty Alec's barrel?' '... and Doug waves to Qing four-tenths of the way across, how long is the bridge?' '... 12 per bag and 36 are left on the shelf, how many bags of bagels did Bill Banks buy?' Mrs Olsen, looking over papers: 'Suddenly every story problem answer begins with 'in the greater scheme of things' ... ' Frazz: 'These are interesting times.'
Jef Mallett’s Frazz for the 3rd of May, 2020. Reading the Comics essays with some mention of something in Frazz are gathered at this link.

It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.

Jason Poland’s Robbie and Bobby for the 4th riffs on the difference between rectangles and trapezoids. It’s also a repeat, featured here just five years ago. Amazing how time slips on like that.

Samson’s Dark Side of the Horse for the 4th is another counting-sheep joke. It features one of those shorthands for large numbers which often makes them more manageable.

Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.

Bill Amend’s FoxTrot Classics for the 7th is another golf and mathematics joke. Jason has taken the homonym of ‘fore’ for ‘four’, and then represented ‘four’ in a needlessly complicated way. Amend does understand how nerd minds work. The strip originally ran the 21st of May, 1998.


That’s enough comics for me for today. I should have the rest of last week’s in a post at this link soon. Thank you.

Reading the Comics, March 25, 2020: Regular Old Mathematics Mentions Edition


I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.

Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.

Marcus, signalling a pitch: 'Two ... plus ... two ... minus .. one ... point ... three ... ' Jason, to Peter: 'If teams want to steal our signals, they're welcome to try.' Marcus: 'Can I just use a slash for 'divided by'?'.
Bill Amend’s FoxTrot for the 22nd of March, 2020. Essays mentioning either the new-run, Sunday, strips or the rerun, weekday, FoxTrot strips are gathered at this link.

Jerry Scott and Jim Borgman’s Zits for the 23rd has Jeremy preparing for a calculus test. Could be any subject.

James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself of his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.

Tom Batiuk’s Funky Winkerbean Vintage strip for the 24th, originally run the 13th of May, 1974, is wordplay about acute triangles.

Hector D Cantú and Carlos Castellanos’s Baldo for the 25th has Gracie work out a visual joke about plus signs. Roger Price, name-checked here, is renowned for the comic feature Droodles, extremely minimalist comic panels. He also, along with Get Smart’s Leonard Stern, created Mad Libs.

Man wrapped in flame, standing before God: 'Oh, come on! Grant me that I was within an order of magnitude of believing in the correct number of deities!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th of March, 2020. It is quite common for me to write about this strip. You can see me explaining Saturday Morning Breakfast Cereal at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.


There’s another half-dozen strips from last week that at least mention mathematics. I’ll at least mention them soon, in an essay at this link. Thank you.

Reading the Comics, February 29, 2020: Leap Day Quiet Edition


I can clear out all last week’s mathematically-themed comic strips in one move, it looks like. There were a fair number of strips; it’s just they mostly mention mathematics in passing.

Bill Amend’s FoxTrot for the 23rd — a new strip; it’s still in original production for Sundays — has Jason asking his older sister to double-check a mathematics problem. Double-checking work is reliably useful, as proof against mistakes both stupid and subtle. But that’s true of any field.

Mark Tatulli’s Heart of the City for the 23rd has Heart preparing for an algebra test.

Jim Unger’s Herman for the 23rd has a parent complaining about the weird New Math. The strip is a rerun and I don’t know from when; it hardly matters. The New Math has been a whipping boy for mathematics education since about ten minutes after its creation. And the complaint attaches to every bit of mathematics education reform ever. I am sympathetic to parents, who don’t see why their children should be the test subjects for a new pedagogy. And who don’t want to re-learn mathematics in order to understand what their children are doing. But, still, let someone know you were a mathematics major and they will tell you how much they didn’t understand or like mathematics in school. It’s hard to see why not try teaching it differently.

(If you do go out pretending to be a mathematics major, don’t worry. If someone challenges you on a thing, cite “Euler’s Theorem”, and you’ll have said something on point. And I’ll cover for you.)

Phil Dunlap’s Ink Pen rerun for the 24th has Bixby Rat complain about his mathematics skills.

Father and child duck sitting on the starry sky. Father: 'Hey, Champ, I know you're only 5, but I think it's time I introduce you to the wonders of the universe! See those stars? How many do you think there are?' Child: 'Um ... 12?' Father: 'Actually, there's over 300 sextillion stars! That's a 3 with 23 zeroes after it.' Child: 'And that's more than 12?' Father: 'Maybe I should introduce you to the wonders of math, first.'
Brian Gordon’s Fowl Language for the 25th of February, 2020. This strip previously ran the 5th of February, 2016, which happens to be the only other time I have an essay mentioning this comic. That’s from before I tagged comic strips by title, though. So this essay and any future repetitions that happen to mention Fowl Language should be at this link, although the previous one probably won’t be.

Brian Gordon’s Fowl Language for the 25th has a father trying to explain the vastness of Big Numbers to their kid. Past a certain point none of us really know how big a thing is. We can talk about 300 sextillion stars, or anything else, and reason can tell us things about that number. But do we understand it? Like, can we visualize that many stars the way we can imagine twelve stars? This gets us into the philosophy of mathematics pretty soundly. 300 sextillion is no more imaginary than four is, but I know I feel more confident in my understanding of four. How does that make sense? And can you explain that to your kid?

Vic Lee’s Pardon my Planet for the 28th has an appearance by Albert Einstein. And a blackboard full of symbols. The symbols I can make out are more chemistry than mathematics, but they do exist just to serve as decoration.

Bud Blake’s Tiger rerun for the 28th has Hugo mourning his performance on a mathematics test.

Ruben Bolling’s Super-Fun-Pak Comix for the 28th is an installment of The Uncertainty Principal. This is a repeat, even allowing that Super-Fun-Pak Comix are extracted reruns from Tom The Dancing Bug. As I mention in the essay linked there, the uncertainty principle being referred to here is a famous quantum mechanics result. It tells us there are sets of quantities whose values we can’t, even in principle, measure simultaneously to unlimited precision. A precise measurement of, for example, momentum destroys our ability to be precise about position. This is what makes the joke here. The mathematics of this reflects non-commutative sets of operators.

Dave Blazek’s Loose Parts for the 29th is another with a blackboard full of symbols used to express deep thought on a subject.


And that takes care of last week. I’ll be Reading the Comics for their mathematics content next week, too, although the start of the week has been a slow affair so far. We’ll see if that changes any.

Reading the Comics, February 8, 2020: Exams Edition


There were a bunch of comic strips mentioning some kind of mathematical theme last week. I need to clear some out. So I’ll start with some of the marginal mentions. Many of these involve having to deal with exams or quizzes.

Jonathan Mahood’s Bleeker: The Rechargeable Dog from the 3rd started a sequence about the robot dog helping Skip with his homework. This would include flash cards, which weren’t helping, in preparation for a test. Bleeker would go to slightly ridiculous ends, since, after all, you never know when something will click.

Bleeker extends his arms, cupping them together in a square shape. Skip: 'Do you think this will help me figure out the square root of these numbers?' Bleeker: 'We should try everything, Skip.'
Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 8th of February, 2020. Essays that mention something brought up by Bleeker appear at this link.

There are different ways to find square roots. (I can guarantee that Skip wasn’t expected to use this one.) The term ‘root’ derives from an idea that the root of a number is the thing that generates it: 3 is a square root of 9 because multiplying 3’s together gives you 9. ‘Square’ is I have always only assumed because multiplying a number by itself will give you the area of a square with sides of length that number. This is such an obvious word origin, though, that I am reflexively suspicious. Word histories are usually subtle and capricious things.

Bill Watterson’s Calvin and Hobbes for the 3rd began the reprint of a storyling based on a story-problem quiz. Calvin fantasizes solving it in a wonderful spoof of hardboiled detective stories. There is a moment of Tracer Bullet going over exactly what information he has, which is a good first step for any mathematics problem. I assume it’s also helpful for solving real mysteries.

Calvin, narrating as Tracer Bullet, wandering through inky, rain-soaked city streets at night: 'I stepped out into the rainy streets and reviewed the facts. There weren't many. Two saps, Jack and Joe, drive toward each other at 60 and 30 mph. After 10 minutes, they pass. I'm supposed to find out how far apart they started. Questions pour down like the rain. Who ARE these mugs? What are they trying to accomplish? Why was Jack in such a hurry? And what difference does it make where they started from? I had a hunch that, before this was over, I'd be sorry I asked.'
Bill Watterson’s Calvin and Hobbes rerun for the 5th of February, 2020. It originally ran the 7th of February, 1990. Essays inspired by something in Calvin and Hobbes should be at this link. I appreciate this all the more since I got into old-time radio, and could imagine the narration in the cadence of specific shows. This is more remarkable since Watterson’s claimed he didn’t care about the hardboiled detective genre and was just spoofing stuff he had picked up elsewhere. It speaks to Watterson’s writing skills that this spoof-based-on-spoofs still feels funny. Of course, he’s helped by how if anything is off, that’s all right, since it’s in the voice of a seven-year-old.

The strip for the 8th closing the storyline has a nice example of using “billion” as a number so big as to be magical, capable of anything. Big numbers can do strange and contrary-to-intuition things. But they can be reasoned out.

Tony Cochran’s Agnes for the 4th sees the title character figuring she could sell her “personal smartness”. Her best friend Trout wonders if that’s tutoring math or something. (Incidentally, Agnes is one of the small handful of strips to capture what made Calvin and Hobbes great; I recommend giving it a try.)

Bill Amend’s FoxTrot Classics reprint for the 6th mentions that Peter has a mathematics test scheduled, and shows part of his preparation.

Charlie Brown, looking at the problems 7 + 6 = and 4 - 3 = on the board: 'Why do I always get the hardest problems? Let's see. If our team had 7, and we scored a touchdown but failed to convert, we'd have 13. And if par on a hole is four, and you get a birdie, you're one under.' He walks away, having successfully done 7 + 6 = 13 and 4 - 3 = 1.
Charles Schulz’s Peanuts Begins for the 5th of February, 2020. The strip originally ran the 6th of February, 1952. The other strip ran the 9th of February that year. And appearances by Peanuts or Peanuts Begins should be in essays at this link. (Peanuts Begins reprints comics from the 1950s. The ordinary run of Peanuts is reprinting strips, this year, from 1973.)

Charles Schulz’s Peanuts Begins for the 5th sees Charlie Brown working problems on the board. He’s stuck for what to do until he recasts the problem as scoring in football and golf. We may giggle at this, but I support his method. It’s convinced him the questions are worth solving, the most important thing to doing them at all. And it’s gotten him to the correct answers. Casting these questions as sports problems is the building of falsework: it helps one do the task, and then is taken away (or hidden) from the final product. Everyone who does mathematics builds some falsework like this. If we do a particular problem, or kind of problem, often enough we get comfortable enough with the main work that we don’t need the falsework anymore. So it is likely to be for Charlie Brown.

On the 8th is another strip of Charlie Brown doing arithmetic in class. Here he just makes a mistake from having counted in a funny way all morning. This, too, happens to us all.


I will have more Reading the Comics posts at this link, hopefully this week. Incidentally other essays mentioning Agnes are at this link, and essays mentioning FoxTrot, reruns or the new-run Sundays, are here. Thanks for reading.

Reading the Comics, January 4, 2020: The Little Things Edition


Today’s essay is just to mention the comic strips which, last week, said mathematics but in some incidental way. Or some way that I can’t write a reasonable blog entry for.

Gary Larson’s The Far Side reruns for the 30th of December, 2019, included this classic about curiosity killing cats. This 1985 strip rates a mention because a blackboard of mathematical symbols gets used to represent their intellectual inquiries.

Bill Amend’s FoxTrot for the 29th, a Sunday and thus new strip, is some wordplay based on the Disney+ line of entertainment product.

Jim Meddick’s Monty for the 29th has the time-travelling Professor Xemit (get it?) show a Times Square Ball Drop of the future. The ball gets replaced with a “demihypercube”, the idea being that the future will have some more complicated geometry than a mere “ball”. There is no such thing as “a” demihypercube, in the same way there is not “a” pentagon. There is a family of shapes, all called demihypercubes. There’s a variety of ways to represent them. A reasonable one, though, is a roughly spherical shape made of pointy triangles all over. It wouldn’t look absurd. There are probably time ball drops that use something like a demihypercube already.

Ruben Bolling’s Super-Fun-Pak Comix rerun for the 1st of January, 2020 features a Comics For The Elderly speaking of the advantages an abacus has over a spreadsheet.

Neal Rubin and Rod Whigham’s Gil Thorp for the 2nd has one of the student athletes working on calculus. And coach Mimi Thorp is doing the mathematics of studying athlete performance. If this strip makes you curious, too, my other blog should this Sunday recap what’s going on in Gil Thorp.

Also this coming Sunday I should look at more mathematically-themed comic strips. That should appear at this link, unless something urgent commands my attention first. Thank you.

Reading the Comics, December 2, 2019: Laconic Week Edition


You know, I had picked these comic strips out as the ones that, last week, had the most substantial mathematics content. And on preparing this essay I realize there’s still not much. Maybe I could have skipped out on the whole week instead.

Bill Amend’s FoxTrot for the 1st is mostly some wordplay. Jason’s finding ways to represent the counting numbers with square roots. The joke plays more tightly than one might expect. Root beer was, traditionally, made with sassafras root, hence the name. (Most commercial root beers don’t use actual sassafras anymore as the safrole in it is carcinogenic.) The mathematical term root, meanwhile, derives from the idea that the root of a number is the thing which generates it. That 2 is the fourth root of 16, because four 2’s multiplied together is 16. That idea. This draws on the metaphor of the roots of a plant being the thing which lets the plant grow. This isn’t one of those cases where two words have fused together into one set of letters.

Jason, pouring pop: 'Sqrt(9) ounces .. sqrt(16) ounces ... sqrt(81) ounces ... sqrt(144) cold, delicious ounces!' Paige: 'Weirdo.' Jason: 'I take my root beer pouring seriously.'
Bill Amend’s FoxTrot for the 1st of December, 2019. Essays mentioning either the reprint or Sunday-only new issues of FoxTrot appear at this link.

Jef Mallett’s Frazz for the 1st is set up with an exponential growth premise. The kid — I can’t figure out his name — promises to increase the number of push-ups he does each day by ten percent, with exciting forecasts for how many that will be before long. As Frazz observes, it’s not especially realistic. It’s hard to figure someone working themselves up from nothing to 300 push-ups a day in only two months.

Also much else of the kid’s plan doesn’t make sense. On the second day he plans to do 1.1 push-ups? On the third 1.21 push-ups? I suppose we can rationalize that, anyway, by taking about getting a fraction of the way through a push-up. But if we do that, then, I make out by the end of the month that he’d be doing about 15.863 push-ups a day. At the end of two months, at this rate, he’d be at 276.8 push-ups a day. That’s close enough to three hundred that I’d let him round it off. But nobody could be generous enough to round 15.8 up to 90.

Kid: 'I'm going to do one push-up today. And I'm going to keep doing push-ups every day for a month. And: I'm going to increase the number of push-ups by a modest 10 percent each day. Know how many push-ups I'll do on the last day of the month? 90! And if I keep it up one more month, I'll be up to 300 push-ups at a time!' Frazz: 'Well-intended, if not especially realistic.' Kid: 'Also by then, the world will have completely forgotten about this history assignment I'm avoiding right now.' Frazz: 'Realistic, if not especially well-intended.'
Jef Mallett’s Frazz for the 1st of December, 2019. Essays which mention something from Frazz should be at this link.

An alternate interpretation of his plans would be to say that each day he’s doing ten percent more, and round that up. So that, like, on the second day he’d do 1.1 rounded up to 2 push-ups, and on the third day 2.2 rounded up to 3 push-ups, and so on. Then day thirty looks good: he’d be doing 94. But the end of two months is a mess as by then he’d be doing 1,714 push-ups a day. I don’t see a way to fit all these pieces together. I’m curious what the kid thought his calculation was. Or, possibly, what Jef Mallett thought the calculation was.

Kid: 'I'm not gonna be an accountant like you, dad! [Holding guitar] I'll become a musician so I don't have to work a real job!' [In front of computer, in suit.] 'I can just sit with my guitar, optimizing search results and maximizing click velocity and ... ' [ Realizing he's studying spreadsheets, clicks-per-ad-dollar; curses himself ]
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 2nd of December, 2019. There are a lot of essays that get into Saturday Morning Breakfast Cereal, and those essays are gathered here.

Zach Weinersmith’s for the 2nd has a kid rejecting accounting in favor of his art. But, wanting to do that art with optimum efficiency … ends up doing accounting. It’s a common story. A common question after working out that someone can do a thing is how to do it best. Best has many measures, yes. But the logic behind how to find it stays the same. Here I admit my favorite kinds of games tend to have screen after screen of numbers, with the goal being to make some number as great as possible considering. If they ever made Multiple Entry Accounting Simulator none of you would ever hear from me again.


Which may be some time! Between Reading the Comics, A to Z, recap posts, and the occasional bit of filler I’ve just finished slightly over a hundred days in a row posting something. That is, however, at its end. I don’t figure to post anything tomorrow. I may not have anything before Sunday’s Reading the Comics post, at this link. I’ll be letting my typing fingers sleep in instead. Thanks for reading.

Reading the Comics, November 9, 2019: Two Pairs Edition


So finally I get to the mathematically-themed comic strips of last week. There were four strips which group into natural pairings. So let’s use that as the name for this edition.

Vic Lee’s Pardon My Planet for the 3rd puts forth “cookie and cake charts”, as a riff on pie charts. There’s always room for new useful visual representations of data, certainly, although quite a few of the ones we do use are more than two centuries old now. Pie charts, which we trace to William Playfair’s 1801 Statistical Breviary, were brought to the public renown by Florence Nightingale. She wanted her reports on the causes of death in the Crimean War to communicate well, and illustrations helped greatly.

Woman giving a presentation in an office; the pie chart on display is lumpy and odd-shaped. She says: 'This was way hard, but my cookie and cake charts are awesome!'
Vic Lee’s Pardon My Planet for the 3rd of November, 2019. It’s been over two years since the last time I mentioned this strip. But this, and those, appearances of Pardon My Planet are available at this link.

Wayno and Piraro’s Bizarro for the 9th is another pie chart joke. If I weren’t already going on about pie charts this week I probably would have relegated this to the “casual mentions” heap. I love the look of the pie, though.

Woman explaining to a kid: 'It's 30% pumpkin, 24% apple, 19% key lime, 15% cherry, and 12% banana cream.' Label: 'Chart pie.' On the table is a pie divided into five pieces, each a different sort of pie.
Wayno and Piraro’s Bizarro for the 9th of November, 2019. It’s only been about seven months since I last mentioned Bizarro, in this and other essays at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th jokes about stereotypes of mathematics and English classes. Or exams, anyway. There is some stabbing truth in the presentation of English-as-math-class. Many important pieces of mathematics are definitions or axioms. In an introductory class there’s not much you can usefully say about, oh, why we’d define a limit to be this rather than that. The book surely has its reasons and we’ll avoid confusion by trusting in them.

Caption: 'If Mathematics were like English Class' Exam question: 'What is the square root of 64?' Answer; 'Square rooting is a multifaceted process that has been used in myriad times, eras, and epochs. It has its 'roots' in ... ' Caption: 'if English class were like Math Class' Exam question; 'Why did Captain Ahab hunt Moby-Dick?' Answer: 'Book said so. QED.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th of November, 2019. It’s been whole minutes since the most recent essay mentioning Saturday Morning Breakfast Club.

I dislike the stereotype of English as a subject rewarding longwinded essays that avoid the question. It seems at least unfair to what good academic writing strives for. (If you wish to argue about bad English writing, you have your blog for that, but let’s not pretend mathematics lacks fundamentally bad papers.) And writing an essay about why a thing should be true, or interesting, is certainly worthwhile. I’m reminded of a mathematical logic professor I had, who spoke of a student who somehow could not do a traditional proper-looking proof. But could write a short essay explaining why a thing should be true which convinced the professor that the student deserved an A. The professor was sad that the student was taking the course pass-fail.

Question worked out: 'B = 1/3 (bugs encountered per km by a moving vehicle w/1-square-meter forward surface, units bugs/km*m^2); S = 1/3 (forward surface area of Superman, units m^2); D = 5500 (distance from Fortress of Solitude to Metropolis, units km); B * S * D = what superman actually looks like when he saves you. Picture of a horrified woman being mugged as a bug-encrusted Superman declares 'I'm here to help!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of November, 2019. So, uh, my apologies to people who did not need to see Superman with a whomping great mass of dead bugs on him.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th shows off a bit of mathematical modeling. The specific problem is silly, yes. But the approach is dead on: identify the things that affect what you’re interested in, and how they interact. Add to this estimates of the things’ values and you’ll get at least a provisional answer. You can then use that answer to guide the building of a more precise model, if you need one.

This little bugs-on-Superman problem makes note of the units everything’s measured in. Paying attention to the units is often done in dimensional analysis, a great tool for building simple models. I ought to write an essay sequence about that sometime.

Wavehead, looking at the angle the teacher's drawn and labelled 75 degrees; 'What about wind chill?'
Mark Anderson’s Andertoons for the 9th of November, 2019. The Andertoons drought is finally over! The last mention, in August, is at this link, as are other past Andertoons discussions.

Mark Anderson’s Andertoons for the 9th is the Mark Anderson’s Andertoons for the week. This one plays on the use of the same word to measure an angle and a temperature. Degree, etymologically, traces back to “a step”, like you might find in stairs. This, taken to represent a stage of progress, got into English in the 13th century. By the late 14th century “degree” was used to describe this 1/360th slice of a circle. By the 1540s it was a measure of heat. Making the degree the unit of temperature, as on a thermometer, seems to be written down only as far back as the 1720s.


And for a last strip of the week, Gary Wise and Lance Aldrich’s Real Life Adventures for the 7th mentions an advantage of being a cartoonist “instead of an engineer” is how cartooning doesn’t require math. Also I guess this means the regular guy in Real Life Adventures represents one (or both?) of the creators? I guess that makes the name Real Life Adventures make more sense. I just thought he was a generic comic strip male. And, of course, there’s nothing about mathematics that keeps one from being a cartoonist, although I don’t know of any current daily-syndicated cartoonists with strong mathematics backgrounds. Bill Amend, of FoxTrot, and Bud Grade, of The Piranha Club/Ernie, were both physics majors, which is a heavy-mathematics program.


And that covers last week’s comics. Reading the Comics should return Sunday at this link. And tomorrow I hope to get tothe Fall 2019 A to Z’s exploration of the letter ‘U’. Thanks for reading.

Reading the Comics, September 29, 2019: September 29, 2019 Edition


Several of the mathematically-themed comic strips from last week featured the fine art of calculation. So that was set to be my title for this week. Then I realized that all the comics worth some detailed mention were published last Sunday, and I do like essays that are entirely one-day affairs. There are a couple of other comic strips that mentioned mathematics tangentially and I’ll list those later this week.

John Hambrock’s The Brilliant Mind of Edison lee for the 29th has Edison show off an organic computer. This is a person, naturally enough. Everyone can do some arithmetic in their heads, especially if we allow that sometimes approximate answers are often fine. People with good speed and precision have always been wonders, though. The setup may also riff on the ancient joke of mathematicians being ways to turn coffee into theorems. (I would imagine that Hambrock has heard that joke. But it is enough to suppose that he’s aware many adult humans drink coffee.)

Edison: 'Welcome to Edison's Science Sunday. I'm going to show you how to build a simple organic calculator. I'll use a bale of hay, a pot of coffee, and Bob the postman. First, I'll have Bob sit on the hay.' Joules, rat: 'OK, now what?' Edison: 'Bob, what is 46 times 19?' Bob :'874.' Joules: 'You have GOT to be kidding me!' Edison: 'He's a whiz with numbers.' Joules: 'Where does the coffee come in?' Edison: 'It extends Bob's battery life.' Bob: 'Cream and sugar, please.'
John Hambrock’s The Brilliant Mind of Edison lee for the 29th of September, 2019. Essays featuring something mentioned in Edison Lee appear at this link.

John Kovaleski’s Daddy Daze for the 29th sees Paul, the dad, working out the calculations his son (Angus) proposed. It’s a good bit of arithmetic that Paul’s doing in his head. The process of multiplying an insubstantial thing by many, many times until you get something of moderate size happens all the time. Much of integral calculus is based on the idea that we can add together infinitely many infinitesimal numbers, and from that get something understandable on the human scale. Saving nine seconds every other day is useless for actual activities, though. You need a certain fungibility in the thing conserved for the bother to be worth it.

Kid: 'Ba ba'. Dad: 'A brilliant math-related idea?' Kid: 'Ba ba ba ba'. Dad: 'We don't need to wash *all* your toes every time you take a bath since they're not *that* dirty?' 'Ba ba ba ba ba' 'OK, if I've got this. There's 8 space between your 10 toes, each space takes 1.25 seconds to wash. If we wash only one space per bath we save 8.75 seconds each time. Three baths a week, this saves 1365 seconds (22.75 minutes) every year. Gee, what'll we do with all that extra time?' 'Ba ba ba'. 'Play 'This Little Piggy' 107.4 times.'
John Kovaleski’s Daddy Daze for the 29th of September, 2019. This is a new tag. Well, the comic is barely a year old. But this and other essays featuring Daddy Daze should be at this link.

Dan Thompson’s Harley for the 29th gets us into some comic strips not drawn by people named John. The comic has some mathematics in it qualitatively. The observation that you could jump a motorcycle farther, or higher, with more energy, and that you can get energy from rolling downhill. It’s here mostly because of the good fortune that another comic strip did a joke on the same topic, and did it quantitatively. That comic?

Harley, racing on the motorcycle: 'Speeding down this mountain should launch us over Pointy Rock Canyon.' Cat, riding behind: 'How do you figure that?' Harley: 'Math, my friend. Harley + Speed + Ramp = Jump The Canyon. It's so simple, it's genius!' Cat: 'We're going faster than we've ever gone!' Harley: 'I think I heard a sonic boom!' Cat: 'I see the rap!' Harley: 'I see my brilliance!' (They race up the ramp. Final panel, they're floating in space.) Cat: 'Didn't you flunk math in school?' Harley: 'Not the third time.'
Dan Thompson’s Harley for the 29th of September, 2019. This just barely misses being a new tag. This essay and the other time I mentioned Harley are at this link. I’ll keep you up dated if there are more essays to add to this pile.

Bill Amend’s FoxTrot for the 29th. Young prodigies Jason and Marcus are putting serious calculation into their Hot Wheels track and working out the biggest loop-the-loop possible from a starting point. Their calculations are right, of course. Bill Amend, who’d been a physics major, likes putting authentic mathematics and mathematical physics in. The key is making sure the car moves fast enough in the loop that it stays on the track. This means the car experiencing a centrifugal force that’s larger than that of gravity. The centrifugal force on something moving in a circle is proportional to the square of the thing’s speed, and inversely proportional to the radius of the circle. This for a circle in any direction, by the way.

So they need to know, if the car starts at the height A, how fast will it go at the top of the loop, at height B? If the car’s going fast enough at height B to stay on the track, it’s certainly going fast enough to stay on for the rest of the loop.

Diagram on ruled paper showing a track dropping down and circling around, with the conservation-of-energy implications resulting on the conclusion the largest possible loop-the-loop is 4/5 the starting height. Peter: 'I don't think this will work. Your calculations assume no friction.' Jason: 'Peter, please. We're not stupid.' (Jason's friend Marcus is working on the track.) Mom: 'Kids, why is there a Hot Wheels car soaking in a bowl of olive oil?'
Bill Amend’s FoxTrot for the 29th of September, 2019. Essays featuring either the current-run Sunday FoxTrot or the vintage FoxTrot comics from the 90s should be at this link.

The hard part would be figuring the speed at height B. Or it would be hard if we tried calculating the forces, and thus acceleration, of the car along the track. This would be a tedious problem. It would depend on the exact path of the track, for example. And it would be a long integration problem, which is trouble. There aren’t many integrals we can actually calculate directly. Most of the interesting ones we have to do numerically or work on approximations of the actual thing. This is all right, though. We don’t have to do that integral. We can look at potential energy instead. This turns what would be a tedious problem into the first three lines of work. And one of those was “Kinetic Energy = Δ Potential Energy”.

But as Peter observes, this does depend on supposing the track is frictionless. We always do this in basic physics problems. Friction is hard. It does depend on the exact path one follows, for example. And it depends on speed in complicated ways. We can make approximations to allow for friction losses, often based in experiment. Or try to make the problem one that has less friction, as Jason and Marcus are trying to do.

Caption: 'ODDITIONS'. Several people with large numerals as head stand around, reading scripts; the one with a 3 head recites, 'To be or not to be? That is the question.' A 9 leans in, saying, 'Next!'
Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 29th of September, 2019. The occasional essay featuring Mustard and Boloney appears at this link. I feel a bit glad to see this doesn’t seem to be a rerun, or at least it’s not one I’ve discussed before.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 29th is the anthropomorphic numerals joke for the week. This is a slight joke to include here. But there were many comic strips of slight mathematical content. I intend to list them in an essay on Wednesday.

Tuesday I plan to be a day for the Fall 2019 A-to-Z. Again, thank you for reading.

Reading the Comics, September 14, 2019: Friday the 13th Edition


The past week included another Friday the 13th. Several comic strips found that worth mention. So that gives me a theme by which to name this look over the comic strips.

Charles Schulz’s Peanuts rerun for the 12th presents a pretty wordy algebra problem. And Peppermint Patty, in the grips of a math anxiety, freezing up and shutting down. One feels for her. Great long strings of words frighten anyone. The problem seems a bit complicated for kids Peppermint Patty’s and Franklin’s age. But the problem isn’t helping. One might notice, say, that a parent’s age will be some nice multiple of a child’s in a year or two. That in ten years a man’s age will be 14 greater than the combined age of their ages then? What imagination does that inspire?

Francis, reading: 'Problem 5. A man has a daughter and a son. The son is three years older than the daughter. In one year the man will be 6 times as old as the daughter is now, and in ten years he will be 14 years older than the combined ages of his children. What is the man's present age?' Peppermint Patty: 'I'm sorry, we are unable to complete your call. Please check the number and dial again!'
Charles Schulz’s Peanuts rerun for the 12th of September, 2019. It originally ran the 14th of September, 1972. Essays mentioning something inspired by Peanuts should be gathered at this link.

Grant Peppermint Patty her fears. The situation isn’t hopeless. It helps to write out just what know, and what we would like to know. At least what we would like to know if we’ve granted the problem worth solving. What we would like is to know the man’s age. That’s some number; let’s call it M. What we know are things about how M relates to his daughter’s and his son’s age, and how those relate to one another. Since we know several things about the daughter’s age and the son’s age it’s worth giving those names too. Let’s say D for the daughter’s age and S for the son’s.

So. We know the son is three years older than the daughter. This we can write as S = D + 3 . We know that in one year, the man will be six times as old as the daughter is now. In one year the man will be M + 1 years old. The daughter’s age now is D; six times that is 6D. So we know that M + 1 = 6D . In ten years the man’s age will be M + 10; the daughter’s age, D + 10; the son’s age, S + 10. In ten years, M + 10 will be 14 plus D + 10 plus S + 10. That is, M + 10 = 14 + D + 10 + S + 10 . Or if you prefer, M + 10 = D + S + 34 . Or even, M = D + S + 24 .

So this is a system of three equation, all linear, in three variables. This is hopeful. We can hope there will be a solution. And there is. There are different ways to find an answer. Since I’m grading this, you can use the one that feels most comfortable to you. The problem still seems a bit advanced for Peppermint Patty and Franklin.

Timmy, reading the news: 'A Stanford University math computer found the largest prime number, grandpa. It's 13 million digits long.' Burl: '13 million digits! That's gonna cost a fortune to print in kids' math books!' Dale: 'It's probably a mistake. Joy? Get me my pocket calculator out of my office supplies caboodle.'
Julie Larson’s The Dinette Set rerun for the 13th of September, 2019. It originally ran the 5th of November, 2008. Essays built on something from The Dinette Set should be gathered at this link.

Julie Larson’s The Dinette Set rerun for the 13th has a bit of talk about a mathematical discovery. The comic is accurate enough for its publication. In 2008 a number known as M43112609 was proven to be prime. The number, 243,112,609 – 1, is some 12,978,189 digits long. It’s still the fifth-largest known prime number (as I write this).

Prime numbers of the form 2N – 1 for some whole number N are known as Mersenne primes. These are named for Marin Mersenne, a 16th century French friar and mathematician. They’re a neat set of numbers. Each Mersenne prime matches some perfect number. Nobody knows whether there are finite or infinitely many Mersenne primes. Every even perfect number has a form that matches to some Mersenne prime. It’s unknown whether there are any odd perfect numbers. As often happens with number theory, the questions are easy to ask but hard to answer. But all the largest known prime numbers are Mersenne primes; they’re of a structure we can test pretty well. At least that electronic computers can test well; the last time the largest known prime was found by mere mechanical computer was 1951. The last time a non-Mersenne was the largest known prime was from 1989 to 1992, and before that, 1951.

Numeral 3, alongside a 1, at a counselor: 'We used to be unlucky, but we turned it around!'
Mark Parisi’s Off The Mark for the 13th of September, 2019. Essays including discussion of Off The Mark should be gathered at this link.

Mark Parisi’s Off The Mark for the 13th starts off the jokes about 13 for this edition. It’s also the anthropomorphic-numerals joke for the week.

Panel of good luck: ladybugs, the number 7, and four-leaf clovers. Panel of bad luck: black cats, the number 13, and serial killers. Panel of neutral luck: hamsters, the number 20, and yams.
Doug Savage’s Savage Chickens for the 13th of September, 2019. Appearances by the Savage Chickens in my essays are at this link.

Doug Savage’s Savage Chickens for the 13th is a joke about the connotations of numbers, with (in the western tradition) 7 lucky and 13 unlucky. And many numbers just lack any particular connotation.

Nervous cat: 'Today is Friday the 13th! Isn't 13 an unlucky number?' Snow: 'It's always beeen a lucky number for me.' (Panel reveals Snow to have 13 kittens.)
T Shepherd’s Snow Sez for the 13th of September, 2019. The occasional appearance by Snow Sez in my essays should be at this link.

T Shepherd’s Snow Sez for the 13th finishes off the unlucky-13 jokes. It observes that whatever a symbol might connote generally, your individual circumstances are more important. There are people for whom 13 is a good omen, or for whom Mondays are magnificent days, or for whom black cats are lucky.


These are all the comics I can write paragraphs about. There were more comics mentioning mathematics last week. Here were some of them:

Brian Walker, Greg Walker, and Chance Browne’s Hi and Lois for the 14th supposes that a “math nerd” can improve Thirsty’s golf game.

Bill Amend’s FoxTrot Classics for the 14th, rerunning a strip from 1997, is a word problem joke. I needed to re-read the panels to see what Paige’s complaint was about.

Greg Evans’s Luann Againn for the 14th, repeating a strip from 1991, is about prioritizing mathematics homework. I can’t disagree with putting off the harder problems. It’s good to have experience, and doing similar but easier problems can help one crack the harder ones.

Jonathan Lemon’s Rabbits Against Magic for the 14th is the Rubik’s Cube joke for the week.


And that’s my comic strips for the week. I plan to have the next Reading the Comics post here on Sunday. The A to Z series resumes tomorrow, all going well. I am seeking topics for the letters I through N, at this post. Thank you for reading, and for offering your thoughts.

Reading the Comics, June 20, 2019: Old Friends Edition


We continue to be in the summer vacation doldrums for mathematically-themed comic strips. But there’ve been a couple coming out. I could break this week’s crop into two essays, for example. All of today’s strips are comics that turn up in my essays a lot. It’s like hanging out with a couple of old friends.

Samson’s Dark Side of the Horse for the 17th uses the motif of arithmetic expressions as “difficult” things. The expressions Samson quotes seem difficult for being syntactically weird: What does the colon under the radical sign mean in \sqrt{9:}33 ? Or they’re difficult for being indirect, using a phrase like “50%” for “half”. But with some charity we can read this as Horace talking about 3:33 am to about 6:30 am. I agree that those are difficult hours.

Horace: 'I've lived through some difficult times. Especially from sqrt{9:}33 AM to 50% past sixish o'clock. Maybe I should get my watch fixed.'
Samson’s Dark Side of the Horse for the 17th of June, 2019. Some of the many essays inspired by Dark Side of the Horse are at this link.

It also puts me in mind of a gift from a few years back. An aunt sent me an Irrational Watch, with a dial that didn’t have the usual counting numbers on it. Instead there were various irrational numbers, like the Golden Ratio or the square root of 50 or the like. Also the Euler-Mascheroni Constant, a number that may or may not be irrational. Nobody knows. It’s likely that it is irrational, but it’s not proven. It’s a good bit of fun, although it does make it a bit harder to use the watch for problems like “how long is it until 4:15?” This isn’t quite what’s going on here — the square root of nine is a noticeably rational number — but it seems in that same spirit.

Mark Anderson’s Andertoons for the 18th sees Wavehead react to the terminology of the “improper fraction”. “Proper” and “improper” as words carry a suggestion of … well, decency. Like there’s something faintly immoral about having an improper fraction. “Proper” and “improper”, as words, attach to many mathematical concepts. Several years ago I wrote that “proper” amounted to “it isn’t boring”. This is a fair way to characterize, like, proper subsets or proper factors or the like. It’s less obvious that \frac{13}{12} is a boring fraction.

The teacher has on the blackboard 1/3 + 3/4 rewritten as 4/12 + 9/12 = 13/12. Wavehead: 'OK, we made it so they had something in common, added them together, and the result is *improper*? I mean, I kinda feel like we just made things worse!'
Mark Anderson’s Andertoons for the 18th of June, 2019. Essays with some mention of a topic from Andertoons are at this link.

I may need to rewrite that old essay. An “improper” form satisfies all the required conditions for the term. But it misses some of the connotation of the term. It’s true that, say, the new process takes “a fraction of the time” of the old, if the old process took one hour and the new process takes fourteen years. But if you tried telling someone that they would assume you misunderstood something. The ordinary English usage of “fraction” carries the connotation of “a fraction between zero and one”, and that’s what makes a “proper fraction”.

In practical terms, improper fractions are fine. I don’t know of any mathematicians who seriously object to them, or avoid using them. The hedging word “seriously” is in there because of a special need. That need is: how big is, say, \frac{75}{14} ? Is it bigger than five? Is it smaller than six? An improper fraction depends on you knowing, in this case, your fourteen-times tables to tell. Switching that to a mixed fraction, 5 + \frac{5}{14} , helps figure out what the number means. That’s as far as we have to worry about the propriety of fractions.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses the form of a Fermi problem for its joke. Fermi problems have a place in mathematical modeling. The idea is to find an estimate for some quantity. We often want to do this. The trick is to build a simple model, and to calculate using a tiny bit of data. The Fermi problem that has someone reached public consciousness is called the Fermi paradox. The question that paradox addresses is, how many technologically advanced species are there in the galaxy? There’s no way to guess. But we can make models and those give us topics to investigate to better understand the problem. (The paradox is that reasonable guesses about the model suggest there should be so many aliens that they’d be a menace to air traffic. Or that the universe should be empty except for us. Both alternatives seem unrealistic.) Such estimates can be quite wrong, of course. I remember a Robert Heinlein essay in which he explained the Soviets were lying about the size of Moscow, his evidence being he didn’t see the ship traffic he expected when he toured the city. I do not remember that he analyzed what he might have reasoned wrong when he republished this in a collection of essays he didn’t seem to realize were funny.

HR interviewer: 'At this company we only want geniuses. So we ask puzzles and judge how well you solve them. Quick! Estimate how many employees we have!' Job applicant: 'Given other companies use empirically validated non-annoying hiring protocols and that engineers have lots of options, I'd estimate your company has exactly one employee.' Interviewer: 'Please don't leave me.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of June, 2019. Anyone who’s been reading these for a couple weeks knows, but, Saturday Morning Breakfast Cereal features in essays at this link. Hey, every essay is somebody’s first.

So the interview question presented is such a Fermi problem. The job applicant, presumably, has not committed to memory the number of employees at the company. But there would be clues. Does the company own the whole building it’s in, or just a floor? Just an office? How large is the building? How large is the parking lot? Are there people walking the hallways? How many desks are in the offices? The question could be answerable. The applicant has a pretty good chain of reasoning too.

Bill Amend’s FoxTrot Classics for the 20th has several mathematical jokes in it. One is the use of excessively many decimal points to indicate intelligence. Grant that someone cares about the hyperbolic cosines of 15.2. There is no need to cite its wrong value to nine digits past the decimal. Decimal points are hypnotic, though, and listing many of them has connotations of relentless, robotic intelligence. That is what Amend went for in the characters here. That and showing how terrible nerds are when they find some petty issue to rage over.

Eugene: 'Lousy camp-issued calculator!' Marcus: 'What's wrong now?' Eugene: 'This thing says the hyperbolic cosine of 15.2 is 0.965016494 when any moron knows this can't be right! What kin of boneheads run this palce? See? It did it again!' Marcus: 'You need to hit the blue button first. Right now you're just getting the regular cosine. ... No need to say 'thank you'. I'm enjoying this silence.' Jason: 'Did you want to borrow mine? Some of us don't need them.'
Bill Amend’s FoxTrot Classics for the 20th of June, 2019. It originally ran the 3rd of July, 1997. Essays based on FoxTrot, either the current-run Sundays, newspaper-rerun 2000s strips, or 90s-run Classics, are at this link.

Eugene is correct about the hyperbolic cosine being wrong, there, though. He’s not wrong to check that. It’s good form to have some idea what a plausible answer should be. It lets one spot errors, for one. No mathematician is too good to avoid making dumb little mistakes. And computing tools will make mistakes too. Fortunately they don’t often, but this strip originally ran a couple years after the discovery of the Pentium FDIV bug. This was a glitch in the way certain Pentium chips handled floating-point division. It was discovered by Dr Thomas Nicely, at Lynchberg College, who found inconsistencies in some calculations when he added Pentium systems to the computers he was using. This Pentium bug may have been on Amend’s mind.

Eugene would have spotted right away that the hyperbolic cosine was wrong, though, and didn’t need nine digits for it. The hyperbolic cosine is a function. Its domain is the real numbers. It range is entirely numbers greater than or equal to one, or less than or equal to minus one. A 0.9 something just can’t happen, not as the hyperbolic cosine for a real number.

And what is the hyperbolic cosine? It’s one of the hyperbolic trigonometric functions. The other trig functions — sine, tangent, arc-sine, and all that — have their shadows too. You’ll see the hyperbolic sine and hyperbolic tangent some. You will never see the hyperbolic arc-cosecant and anyone trying to tell you that you need it is putting you on. They turn up in introductory calculus classes because you can differentiate them, and integrate them, the way you can ordinary trig functions. They look just different enough from regular trig functions to seem interesting for half a class. By the time you’re doing this, your instructor needs that.

The ordinary trig functions come from the unit circle. You can relate the Cartesian coordinates of a point on the circle described by x^2 + y^2 = 1 to the angle made between that point and the center of the circle and the positive x-axis. Hyperbolic trig functions we can relate the Cartesian coordinates of a point on the hyperbola described by x^2 - y^2 = 1 to angles instead. The functions … don’t have a lot of use at the intro-to-calculus level. Again, other than that they let you do some quite testable differentiation and integration problems that don’t look exactly like regular trig functions do. They turn up again if you get far enough into mathematical physics. The hyperbolic cosine does well in describing catenaries, that is, the shape of flexible wires under gravity. And the family of functions turn up in statistical mechanics, often, in the mathematics of heat and of magnetism. But overall, these functions aren’t needed a lot. A good scientific calculator will offer them, certainly. But it’ll be harder to get them.

There is another oddity at work here. The cosine of 15.2 degrees is about 0.965, yes. But mathematicians will usually think of trigonometric functions — regular or hyperbolic — in terms of radians. This is just a different measure of angles. A right angle, 90 degrees, is measured as \frac{1}{2}\pi radians. The use of radians makes a good bit of other work easier. Mathematicians get to accustomed to using radians that to use degrees seems slightly alien. The cosine of 15.2 radians, then, would be about -0.874. Eugene has apparently left his calculator in degree mode, rather than radian mode. If he weren’t so worked up about the hyperbolic cosine being wrong he might have noticed. Perhaps that will be another exciting error to discover down the line.

This strip was part of a several-months-long story Bill Amend did, in which Jason has adventures at Math Camp. I don’t remember the whole story. But I do expect the strip to have several more appearances here this summer.


And that’s about half of last week’s comics. A fresh Reading the Comics post should be at this link later this week. Thank you for reading along.

Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition


I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Offscreen voice: 'So a pizza sliced into fourths has ... ' Paige: '90 degrees per slice.' Voice: 'Correct! And a pizza sliced into sixths has ... ' Page: '60 degrees per slice.' Voice: 'Good! And a pizza sliced into eighths has ... ' Paige: '45 degrees per slice.' Voice: 'Yep! I'd say you're ready for your geometry final, Paige.' Paige: 'Woo-hoo!' Voice, revealed to be Peter: 'Now help me clean up these [ seven pizza ] boxes.' Page: 'I still don't understand why teaching me this required *actual* pizzas.'
Bill Amend’s FoxTrot for the 19th of May, 2019. Essays featuring FoxTrot, either the current (Sunday-only) strips or the 1990s-vintage reruns, should be at this link.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Calculator salesman: 'This little pocket calculator is a real beauty. It's nice and light so you can take it anywhere. It has an eight-digit readout with automatic roundoff. Not only that, but it has a floating decimal which enables you to solve ANY type of problem with it!' Les Moore: 'Amazing! May I try it out?' (To the calculator) 'Hello, pocket calculator? Why do I have so much trouble getting girls to like me?'
Tom Batiuk’s vintage Funky Winkerbean for the 19th of May, 2019. The strip originally ran the 17th of June, 1973. Comics Kingdom is printing both the current Funky Winkerbean strips and early-70s reprints. Essays that mention Funky Winkerbean, old or new, should appear at this link.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 103. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10-308 or as large as 10308, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10-20, to a huge number, like 1020 won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as \frac{1}{3} . A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Matrix-O-Magic: Draw a nine-square grid on a notepad, filling in the numbers 1-9 like this: 2, 9, 4 // 7, 5, 3 // 6, 1, 8 Hand the pad and marker to a friend and tell him to pick any row of three numbers, upward, downward, or diagonal. Tell him to black out any numbers not in his row. Instruct your friend to add up his three randomly chosen numbers. Ask your friend to flip through the rest of the notepad to make sure the pages are blank. All the pages are blank except one. That one bears the number that his numbers added up to: 15. (All the rows/columns/diagonals add to 15; because the other numbers are blacked out your friend won't notice. If asked to do the trick more than once the grid can be made to look different by rotating the order of the numbers left or right, et, 6, 7, 2 // 1, 5, 9 // 8, 3, 4.)
Mac King and Bill King’s Magic in a Minute for the 19th of May, 2019. So far as I know all these panels are new ones, although they do reuse gimmicks now and then. But the arithmetic and logic tricks featured in Magic In A Minute get discussed at this link, when they get mention from me at all.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Wavehead, to classmate, over lunch: 'Did you know that every square is a rhombus, but not every rhombus is a square? I mean, you can't make this stuff up!'
Mark Anderson’s Andertoons for the 20th of May, 2019. Always glad to discuss Andertoons, as you can see from these essays.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.


There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

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.

Reading the Comics, March 2, 2019: Process Edition


There were a handful of comic strips from last week which I didn’t already discuss. Two of them inspire me to write about how we know how to do things. That makes a good theme.

Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th gets into deep territory. How does we could count to a million? Maybe some determined soul has actually done it. But it would take the better part of a month. Things improve some if we allow that anything a computing machine can do, a person could do. This seems reasonable enough. It’s heady to imagine that all the computing done to support, say, a game of Roller Coaster Tycoon could be done by one person working alone with a sheet of paper. Anyway, a computer could show counting up to a million, a billion, a trillion, although then we start asking whether anyone’s checked that it hasn’t skipped some numbers. (Don’t laugh. The New York Times print edition includes an issue number, today at 58,258, at the top of the front page. It’s meant to list the number of published daily editions since the paper started. They mis-counted once, in 1898, and nobody noticed until 1999.)

Dennis, to Margaret: 'How do you know you can count to a million if you've never done it?'
Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th of February, 2019. I’m not quite confident that I have the credits right here, but if I am parsing Wikipedia’s entry correctly Hamilton and Ketcham work on the daily comics and Ron Ferdinand and Ketcham work on the Sunday strips. And I would have thought this was a new tag but it turns out I have several Dennis the Menace-based essays at this link.

Anyway, allow that. Nobody doubts that, if we put enough time and effort into it, we could count up to any positive whole number, or as they say in the trade, “counting number”. But … there is some largest number that we could possibly count to, even if we put every possible resource and all the time left in the universe to that counting. So how do we know we “could” count to a number bigger than that? What does it mean to say we “could” if the circumstances of the universe are such that we literally could not?

Counting up to a number seems uncontroversial enough. If I wanted to prove it I’d say something like “if we can count to the whole number with value N, then we can count to the whole number with value N + 1 by … going one higher.” And “We can count to the whole number 1”, proving that by enunciating as clearly as I can. The induction follows. Fine enough. That’s a nice little induction proof.

But … what if we needed to do more work? What if we needed to do a lot of work? There is a corner of logic which considers infinitely long proofs, or infinitely long statements. They’re not part of the usual deductive logic that any mathematician knows and relies on. We’re used to, at least in principle, being able to go through and check every step of a proof. If that becomes impossible is that still a proof? It’s not my field, so I feel comfortable not saying what’s right and what’s wrong. But it is one of those lectures in your Mathematical Logic course that leaves you hanging your jaw open.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th is a joke about algorithms. These are the processes by which we know how to do a thing. Here, Hansel and Gretel are shown using what’s termed a “greedy algorithm” to follow pebbles back home. This kind of thing reflects trying to find an acceptable solution, in this case, finding a path somewhere. What makes it “greedy” is each step. You’re at a pebble. You can see other pebbles nearby. Which one do you go to? Go to some extreme one; in this case, the nearest. It could instead have been the biggest, or the shiniest, the one at the greatest altitude, the one nearest a water source. Doesn’t matter. You choose your summum bonum and, at each step, take the move that maximizes that.

During the great famine, Hansel and Gretel's mother decided to leave them in the woods. Overhearing the conversation, Hansel had an idea. `I will take these bright pebbles and leave them along our path, then we can follow them home.` Little did they know, their mother overheard *their* conversation. That night she created loops of shiny pebbles at various points in the woods. The following evening she left them in the forest. Gretel: `Just always go to the nearest pebble, keep doing that until you are home.` On the path they encountered a loop which caused them to go in an endless cycle until they passed out from exhaustion. The moral of this story? There are arts far darker than witchcraft. (Shows the wicked stepmother reading Introduction to Algorithms.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of February, 2019. There’s no mistaking this for a new tag. Saturday Morning Breakfast Cereal inspires many discussions at this link.

The wicked mother knows something about this sort of algorithm, one that promises merely a solution and not the best solution. And that is that all these solutions can be broken. You can set up a problem that the algorithm can’t solve. Greedy algorithms are particularly vulnerable to this. They’re called “local maximums”. You find the best answer of the ones nearby, but not the best one you possibly could locate.

Why use an algorithm like this, that can be broken so? That’s because we often want to do problems like finding a path through the woods. There are so many possible paths that it’s hard to find one of the acceptable ones. But there are processes that will, typically, find an acceptable answer. Maybe processes that will let us take an acceptable answer and improve it to a good answer. And this is getting into my field.

Actual persons encountering one of these pebble rings would (probably) notice they were caught in a loop. And what they’d do, then, is suspend the greedy rule: instead of going to the nearest pebble they could find, they’d pick something else. Maybe simply the nearest pebble they hadn’t recently visited. Maybe the second-nearest pebble. Maybe they’d give up and strike out in a random direction, trusting they’ll find some more pebbles. This can lead them out of the local maximum they don’t want toward the “global maximum”, the path home, that they do. There’s no reason they can’t get trapped again — this is why the wicked mother made many loops — and no reason they might not get caught in a loop of loops again. Every algorithm like this can get broken by some problem, after all. But sometimes taking the not-the-best steps can lead you to a better solution. That’s the insight at the heart of “Metropolis-Hastings” algorithms, which was my field before I just read comic strips all the time.

Father Figure Eight. A big 8, wearing ice skates and holding a tiny 8's hand, says, 'Son, I'll show you how to skate in the shape of a right-side-up infinity symbol!'
Dan Thompson’s Brevity for the 28th of February, 2019. This is another strip that’s inspired a host of essays. Brevity panels get shown off at this link.

Dan Thompson’s Brevity for the 28th is a nice simple anthropomorphic figures joke. It would’ve been a good match for the strips I talked about Sunday. I’m just normally reluctant to sort these comic strips other than by publication date.


And there were some comic strips I didn’t think worth making paragraphs about. Chris Giarrusso’s G-Man Webcomics for the 25th of February mentioned negative numbers and built a joke on the … negative … connotations of that word. (And inaugurates a tag for that comic strip. This fact will certainly come back to baffle me some later day.) Art Sansom and Chip Sansom’s The Born Loser for the 2nd of March has a bad mathematics report card. Tony Rubino and Gary Markstein’s Daddy’s Home for the 2nd has geometry be the subject parents don’t understand. Bill Amend’s FoxTrot Classics for the 2nd has a mathematics-anxiety dream.


And this closes out my mathematics comics for the week. Come Sunday I should have a fresh post with more comics, and I thank you for considering reading that.

Reading the Comics, January 13, 2019: January 13, 2019 Edition


I admit I’m including a fairly marginal strip in this, just so I can have the fun of another single-day edition. What can I say? I can be easily swayed by silly things. Also, somehow, all four strips today have circumstances where one might mistake them for reruns. Let’s watch.

Bill Amend’s FoxTrot for the 13th is wordplay, mashing up ‘cell division’ with ‘long division’. As you might expect from Bill Amend — who loves sneaking legitimate mathematics and physics in where it’s not needed — Paige’s long cell division is a legitimate one. If you’d like a bit of recreational mathematics fun, you can figure out which microscopic organisms correspond to which numerals. The answer is also the Featured Comment on the page, at least as I write this. So if you need an answer, or you want to avoid having the answer spoiled, know what’s there.

A long division problem, with microbes representing the digits. Science teacher: 'Paige, about your diagram of cell division ... ' Paige: 'Did I get the math wrong?'
Bill Amend’s FoxTrot for the 13th of January, 2019. Essays discussing topics raised by FoxTrot, whether new (Sunday strips) or rerun (the weekdays), should be at this link.

Greg Evans’s Luann Againn for the 13th is the strip of most marginal relevance here. Part of Luann’s awful ay is a mathematics test. The given problems are nothing particularly meaningful. There is the sequence ‘mc2’ in the problem, although written as m^c 2 . There’s also a mention of ‘googleplex’, which when the strip was first published in 1991 was nothing more than a misspelling of the quite large number. (‘Googol’ is the number; ‘Google’ a curious misspelling. Or perhaps a reversion. The name was coined in 1938 by Milton Sirotta. Sirotta was seven years old at the time. I accept that it is at least possible Sirotta was thinking of the then-very-popular serial-comic strip Barney Google, and that his uncle Edward Kasner, who brought the name to mathematics, wrote it down wrong.) And that carries with it the connotation that big numbers are harder than small numbers. This is … kind of true. At least, long numbers are more tedious than short numbers. But you don’t really do different work, dividing 1428 by 7, than you do dividing 147 by 7. It’s just longer. “Hard” is a flexible idea.

Panels showing a day in Luann's life: she gets dressed and made up. Then misses the bus and has to run to school, steps in gum, slides into base at gym class, sweats a mathematics test, gets food spilled on her at lunch, and walks in the rain back home. Brad looks over the mess: 'Jeez, Luann, no wonder you don't have any boyfriends. Lookit how you go to school!'
Greg Evans’s Luann Againn for the 13th of January, 2019. It originally ran the 13th of January, 1991. Essays discussing topics raised by Luann, whether new (current day) or rerun (1991 vintage), should be at this link.

Mac King and Bill King’s Magic in a Minute for the 13th felt like a rerun to me. It took a bit of work to find, but yeah, it was. The strip itself, as presented, is new. But the same neat little modular-arithmetic coincidence was used the 31st of July, 2016.

Hickory-Trickery-Clock. From a picture of a standard analog watch, here's what you do: think of any number, one through twelve. Place your fingertip on the number 12 of the clock. Spell the number you thought of, moving one number clockwise for each letter; eg, if you thought 'one', move three spaces, stopping at the 3. Now spell out the number you're touching, advancing the numbers by the same rule. And now do this one more time. You will have reached ... 1:00.
Mac King and Bill King’s Magic in a Minute for the 13th of January, 2019. Essays discussing topics raised by Magic In A Minute, whether new or re-drawn magic, should be at this link.

Mathematics on clock faces is often used as a way to introduce modular arithmetic, a variation on arithmetic with only finitely many integers. This can help, if you’re familiar with clock faces. Like regular arithmetic, modular arithmetic can form a group and a ring. Clock faces won’t give you a group or ring, not unless you replace the number before ‘1’ with a ‘0’. To be a group, you need a collection of items, and a binary operation on the items. This operation we often think of as either addition or multiplication, depending on what makes sense for the problem. To be a ring, you need two binary operations, which interact by a distributive law. So the operations are often matched to addition and multiplication. Modular arithmetic is fun, yes. It’s also useful, not just as a way to do something like arithmetic that’s different. Many schemes for setting up checksums, quick and easy tests against data entry errors, rely on modular arithmetic on the data. And many schemes for generating ‘random’ numbers are built on finding multiplicative inverses in modular arithmetic. This isn’t truly random, of course. But you can look at a string of digits and not see any clear patterns. This is often as close to random as you need.

Avis: 'My niece Jasmine is one of those Millennials.' Nick: 'Ah yes, Generation Y.' Avis: 'Y? Why? I'd like to know! Why can't they read cursive? Why can't they do simple multiplication? Why can't they parallel park? Why can't they talk to each other? Why are they always complaining?' Nick: 'Avis, complaining is hardly limited to millennials.' (Avis's questions are illustrated with young adults trying to read cursive or to multiply 3 x 6 or such.)
Rick DeTorie’s One Big Happy for the 13th of January, 2019. Essays discussing topics raised by One Big Happy, whether new (on Creators.com) or rerun (on GoComics.com), should be at this link.

Rick DeTorie’s One Big Happy for the 13th is mostly a bunch of complaints the old always have against the young. Well, the complaint about parallel parking I haven’t seen before. But the rest are common enough. Featured in it is a complaint that the young can’t do arithmetic. I’m not sure there was ever a time that the older generation thought the young were well-trained in arithmetic. Nor that there was ever a time that the current educational vogue wasn’t blamed for destroying a generation’s ability to calculate. I’m sure there are better and worse ways to teach calculation. But I suspect any teaching method will fall short of addressing a couple issues. One is that people over-rate their own competence and under-rate other’s competence. So the older generation will see itself as having got the best possible arithmetic education and anything that’s different is a falling away. And another is that people get worse at stuff they don’t think is enjoyable or don’t have to do a lot. If you haven’t got a use for the fact, or an appreciation for the beauty in it, three times six is a bit of trivia, and not one that inspires much conversation when shared.


There’s more comics with something of a mathematical theme that got published last week. When I get to them the essays should be at this link.

My 2018 Mathematics A To Z: Jokes


For today’s entry, Iva Sallay, of Find The Factors, gave me an irresistible topic. I did not resist.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Jokes.

What’s purple and commutes?
An Abelian grape.

Whatever else you say about mathematics we are human. We tell jokes. I will tell some here. You may not understand the words in them. That’s all right. From the Abelian grape there, you gather this is some manner of wordplay. A pun, particularly. It’s built on a technical term. “Abelian groups” come from (not high school) Algebra. In an Abelian group, the group multiplication commutes. That is, if ‘a’ and ‘b’ are any things in the group, then their product “ab” is the same as “ba’. That is, the group works like ordinary addition on numbers does. We say “Abelian” in honor of Niels Henrik Abel, who taught us some fascinating stuff about polynomials. Puns are a common kind of humor. So common, they’re almost base. Even a good pun earns less laughter than groans.

But mathematicians make many puns. A typical page of mathematics jokes has a whole section of puns. “What’s yellow and equivalent to the Axiom of Choice? Zorn’s Lemon.” “What’s nonorientable and lives in the sea?” “Möbius Dick.” “One day Jesus said to his disciples, `The Kingdom of Heaven is like 3x2 + 8x – 9′. Thomas looked very confused and asked peter, `What does the teacher mean?’ Peter replied, `Don’t worry. It’s just another one of his parabolas’.” And there are many jokes built on how it is impossible to tell the difference between the sounds of “π” and “pie”.

It shouldn’t surprise that mathematicians make so many puns. Mathematics trains people to know definitions. To think about precisely what we mean. Puns ignore definitions. They build nonsense out of the ways that sounds interact. Mathematicians practice how to make things interact, even if they don’t know or care what the underlying things are. If you’ve gotten used to proving things about aba^{-1}b^{-1} , without knowing what ‘a’ or ‘b’ are, it’s difficult to avoid turning “poles on the half-plane” (which matters in some mathematical physics) to a story about Polish people on an aircraft.

Popeye's lousy tutor: 'Today I am going to test you at mental multiplication. Quick, how much is 6 1/2 times 656? Quick!' Popeye: '4,264.' 'Right!' 'Blow me down! Anybody what can guess like that don't need no edjacation!'
Elzie Segar’s Thimble Theater from the 14th of September, 1929. Rerun on ComicsKingdom on the 26th of February, 2016. That’s Bernice, the magical Whiffle Hen, as the strange birdlike creature in the last panel there.

If there’s a flaw to this kind of humor it’s that these jokes may sound juvenile. One of the first things that strikes kids as funny is that a thing might have several meanings. Or might sound like another thing. “Why do mathematicians like parks? Because of all the natural logs!”

Jokes can be built tightly around definitions. “What do you get if you cross a mosquito with a mountain climber? Nothing; you can’t cross a vector with a scalar.” “There are 10 kinds of people in the world, those who understand binary mathematics and those who don’t.” “Life is complex; it has real and imaginary parts.”

Paige: 'I keep forgetting ... what's the cosine of 60 degrees?' Jason: 'Well, let's see. If I recall correctly ... 1 - (pi/3)^2/2! + (pi/3)^4/4! - (pi/3)^6/6! + (pi/3)^8/8! - (pi/3)^10/10! + (pi/3)^12/12! - (and this goes on a while, up to (pi/3)^32/32! - ... )' Paige: 'In case you've forgotten, I'm not paying you by the hour.' Jason: '1/2'.
Bill Amend’s FoxTrot Classics for the 23rd of May, 2018. It originally ran the 29th of May, 1996.

There are more sophisticated jokes. Many of them are self-deprecating. “A mathematician is a device for turning coffee into theorems.” “An introvert mathematician looks at her shoes while talking to you. An extrovert mathematician looks at your shoes.” “A mathematics professor is someone who talks in someone else’s sleep”. “Two people are adrift in a hot air balloon. Finally they see someone and shout down, `Where are we?’ The person looks up, and studies them, watching the balloon drift away. Finally, when they are barely in shouting range, the person on the ground shouts back, `You are in a balloon!’ The first passenger curses their luck at running across a mathematician. `How do you know that was a mathematician?’ `Because her answer took a long time, was perfectly correct, and absolutely useless!”’ These have the form of being about mathematicians. But they’re not really. It would be the same joke to say “a poet is a device for turning coffee into couplets”, the sleep-talker anyone who teachers, or have the hot-air balloonists discover a lawyer or a consultant.

Some of these jokes get more specific, with mathematics harder to extract from the story. The tale of the nervous flyer who, before going to the conference, sends a postcard that she has a proof of the Riemann hypothesis. She arrives and admits she has no such thing, of course. But she sends that word ahead of every conference. She knows if she died in a plane crash after that, she’d be famous forever, and God would never give her that. (I wonder if Ian Randal Strock’s little joke of a story about Pierre de Fermat was an adaptation of this joke.) You could recast the joke for physicists uniting gravity and quantum mechanics. But I can’t imagine a way to make this joke about an ISO 9000 consultant.

'If it's a hunnert miles to th' city an' a train is travelin' thurty miles an hour is due t'arrive at 5:00 pm --- what time does th' train leave Hootin' Holler, Jughaid?' 'I dunno, Miz Prunelly, but you better go now jest t'be on th' safe side!!'
John Rose’s Barney Google and Snuffy Smith for the 12th of February, 2016.

A dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

This joke is very mathematical. I know of no important results actually based on spherical cows. But the attitude that tries to make spheres of cows comes from observing mathematicians. To describe any real-world process is to make a model of that thing. A model is a simplification of the real thing. You suppose that things behave more predictably than the real thing. You trust the error made by this supposition is small enough for your needs. A cow is complicated, all those pointy ends and weird contours. A sphere is easy. And, besides, cows are funny. “Spherical cow” is a funny string of sounds, at least in English.

The spherical cows approach parodying the work mathematicians do. Many mathematical jokes are burlesques of deductive logic. Or not even burlesques. Charles Dodgson, known to humans as Lewis Carroll, wrote this in Symbolic Logic:

“No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
∴ This party of tourists need not run.”

[ Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premises to be true, I should be quite clear that I needn’t run — and I should walk!

And you will reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum! Ha! I must think that over a bit!” ]

The punch line is diffused by the text being so educational. And by being written in the 19th century, when it was bad form to excise any word from any writing. But you can recognize the joke, and why it should be a joke.

Not every mathematical-reasoning joke features some manner of cattle. Some are legitimate:

Claim. There are no uninteresting whole numbers.
Proof. Suppose there is a smalled uninteresting whole number. Call it N. That N is uninteresting is an interesting fact. Therefore N is not an uninteresting whole number.

Three mathematicians step up to the bar. The bartender asks, “you all want a beer?” The first mathematician says, “I don’t know.” The second mathematician says, “I don’t know.” The third says, “Yes”.

Some mock reasoning uses nonsense methods to get a true conclusion. It’s the fun of watching Mister Magoo walk unharmed through a construction site to find the department store exchange counter:

5095 / 1019 = 5095 / 1019 = 505 / 101 = 55 / 11 = 5

This one includes the thrill of division by zero.

The Venn Diagram of Grocery Shopping. Overlap 'have teenagers', 'haven't grocery shopped in two weeks', and 'grocery shopping on an empty stomach' and you get 'will need to go back in two days', 'bought entire bakery aisle', and 'bought two of everything'. Where they all overlap, 'need to take out second mortgage'.
Terri Libenson’s Pajama Diaries for the 16th of November, 2016. I was never one for buying too much of the bakery aisle, myself, but then I also haven’t got teenagers. And I did go through so much of my life figuring there was no reason I shouldn’t eat another bagel again.

Venn Diagrams are not by themselves jokes (most of the time). But they are a great structure for jokes. And easy to draw, which is great for us who want to be funny but don’t feel sure about their drafting abilities.

And then there are personality jokes. Mathematics encourages people to think obsessively. Obsessive people are often funny people. Alexander Grothendieck was one of the candidates for “greatest 20th century mathematician”. His reputation is that he worked so well on abstract problems that he was incompetent at practical ones. The story goes that he was demonstrating something about prime numbers and his audience begged him to speak about a specific number, that they could follow an example. And that he grumbled a bit and, finally, said, “57”. It’s not a prime number. But if you speak of “Grothendieck’s prime”, many will recognize what you mean, and grin.

There are more outstanding, preposterous personalities. Paul Erdös was prolific, and a restless traveller. The stories go that he would show up at some poor mathematician’s door and stay with them several months. And then co-author a paper with the elevator operator. (Erdös is also credited as the originator of the “coffee into theorems” quip above.) John von Neumann was supposedly presented with this problem:

Two trains are on the same track, 60 miles apart, heading toward each other, each travelling 30 miles per hour. A fly travels 60 miles per hour, leaving one engine flying toward the other. When it reaches the other engine it turns around immediately and flies back to the other engine. This is repeated until the two trains crash. How far does the fly travel before the crash?

The first, hard way to do this is to realize how far the fly travels is a series. The fly starts at, let’s say, the left engine and flies to the right. Add to that the distance from the right to the left train now. Then left to the right again. Right to left. This is a bunch of calculations. Most people give up on that and realize the problem is easier. The trains will crash in one hour. The fly travels 60 miles per hour for an hour. It’ll fly 60 miles total. John von Neumann, say witnesses, had the answer instantly. He recognized the trick? “I summed the series.”

Henry is frustrated with his arithmetic, until he goes to the pool hall and counts off numbers on those score chips.
Don Trachte’s Henry for the 6th of September, 2015.

The personalities can be known more remotely, from a handful of facts about who they were or what they did. “Cantor did it diagonally.” Georg Cantor is famous for great thinking about infinitely large sets. His “diagonal proof” shows the set of real numbers must be larger than the set of rational numbers. “Fermat tried to do it in the margin but couldn’t fit it in.” “Galois did it on the night before.” (Évariste Galois wrote out important pieces of group theory the night before a duel. It went badly for him. French politics of the 1830s.) Every field has its celebrities. Mathematicians learn just enough about theirs to know a couple of jokes.

Anthropomorphic 3/5: 'Honey, what's wrong?' Anthropomorphic 1/4: 'Sour son is leaving the faith! He said he's converting to decimals!'
Scott Hilburn’s The Argyle Sweater for the 9th of May, 2018. I like the shout-out to Archimedes in the background art, too. Archimedes, though, didn’t use fractions in the way we’d recognize them. He’d write out a number as a combination of ratios of some reference number. So he might estimate the length of something being as to the length of something else as 19 is to 7, or something like that. This seems like a longwinded and cumbersome way to write out numbers, or much of anything, and makes one appreciate his indefatigability as much as his insight.

The jokes can attach to a generic mathematician personality. “How can you possibly visualize something that happens in a 12-dimensional space?” “Easy, first visualize it in an N-dimensional space, and then let N go to 12.” Three statisticians go hunting. They spot a deer. One shoots, missing it on the left. The second shoots, missing it on the right. The third leaps up, shouting, “We’ve hit it!” An engineer and a mathematician are sleeping in a hotel room when the fire alarm goes off. The engineer ties the bedsheets into a rope and shimmies out of the room. The mathematician looks at this, unties the bedsheets, sets them back on the bed, declares, “this is a problem already solved” and goes back to sleep. (Engineers and mathematicians pair up a lot in mathematics jokes. I assume in engineering jokes too, but that the engineers make wrong assumptions about who the joke is on. If there’s a third person in the party, she’s a physicist.)

Do I have a favorite mathematics joke? I suppose I must. There are jokes I like better than others, and there are — I assume — finitely many different mathematics jokes. So I must have a favorite. What is it? I don’t know. It must vary with the day and my mood and the last thing I thought about. I know a bit of doggerel keeps popping into my head, unbidden. Let me close by giving it to you.

Integral z-squared dz
From 1 to the cube root of 3
   Times the cosine
   Of three π over nine
Equals log of the cube root of e.

This may not strike you as very funny. I’m not sure it strikes me as very funny. But it keeps showing up, all the time. That has to add up.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. Also, now and then, I talk about comic strips here. You might like that too.

Reading the Comics, May 29, 2018: Finding Reruns Edition


There were a bunch of mathematically-themed comic strips this past week. A lot of them are ones I’d seen before. One of them is a bit risque and I’ve put that behind a cut. This saves me the effort of thinking up a good nonsense name to give this edition, so there’s that going for me too.

Bill Amend’s FoxTrot Classics for the 24th of May ought to have run last Sunday, but I wasn’t able to make time to write about it. It’s part of a sequence of Jason tutoring Paige in geometry. She’s struggling with the areas of common shapes which is relatable. Many of these area formulas could be kept straight by thinking back to rectangles. The size of the area is equal to the length of the base times the length of the height. From that you could probably reason right away the area of a trapezoid. It would have the same area as a rectangle with a base of length the mean length of the trapezoid’s different-length sides. The parallelogram works like the rectangle, length of the base times the length of the height. That you can convince yourself of by imagining the parallelogram. Then imagine slicing a right triangle off one of its sides. Move that around to the other side. Put it together right and you have a rectangle. Already know the area of a rectangle. The triangle, then, you can get by imagining two triangles of the same size and shape. Rotate one of the triangles 180 degrees. Slide it over, so the two triangles touch. Do this right and you have a parallelogram and so you know the area. The triangle’s half the area of that parallelogram.

Paige: 'OK, let me see if I've got these area formulas memorized. For a triangle, it's 1/2 bh. For a trapezoid, it's 1/2 (a + b)h. And for a circle, it's pi r^2.' Jason: 'Yes! Yes! Yes! You got them all right! You're going to ace this test! I'm going to make $10!' Paige: 'I always get confused --- does h stand for my height or the triangle's? ... Just kidding.' Jason: 'WILL YOU QUIT TOYING WITH ME?!'
Bill Amend’s FoxTrot Classics for the 24th of May, 2018. It originally ran the 30th of May, 1996.

The circle, I don’t know. I think just remember that if someone says “pi” they’re almost certainly going to follow it with either “r squared” or “day”. One of those suggests an area; the other doesn’t. Best I can do.

Jeri: 'Arrrrhh'. Teena: 'Sup?' Jeri: 'I'm having issues with this math issue. It's the way they phrase these word things. They're like trick questions. I can never figure them out.' Teena: 'Here, let me see what you're having trouble with. ... 'Sarah sits next to Stephen, who is very good at algebra. This causes Sarah, who has issues with BOTH Steven AND algebra, to feel bad But if Sarah moves next to someone else, Steven will feel bad. How do you protect Sarah's and Steven's self-esteem?' Jeri: 'I'm not that comfortable with my answer.' Teena: 'Which is?' Jeri: 'Eleven.' Teena: 'It must be very interesting in your school.'
Allison Barrows’s PreTeena rerun for the 27th of May, 2018. It originally ran the 15th of February, 2004.

Allison Barrows’s PreTeena rerun for the 27th discusses self-esteem as though it were a good thing that children ought to have. This is part of the strip’s work to help build up the Old Person Complaining membership that every comics section community group relies on. But. There is mathematics in Jeri’s homework. Not mathematics in the sense of something particular to calculate. There’s just nothing to do there. But it is mathematics, and useful mathematics, to work out the logic of how to satisfy multiple requirements. Or, if it’s impossible to satisfy them all at once, then to come as near satisfying them as possible. These kinds of problems are considered optimization or logistics problems. Most interesting real-world examples are impossibly hard, or at least become impossibly hard before you realize it. You can make a career out of doing as best as possible in the circumstances.

Charles Schulz’s Peanuts rerun for the 27th features an extended discussion by Lucy about the nature of … well, she explicitly talks about “nothing”. Is she talking about zero? Probably; you have to get fairly into mathematics or philosophy to start worrying about the difference between the number zero and the idea of nothing. In Algebra, mathematicians learn to work with systems of things that work like numbers enough that you can add and subtract and multiply them together, without committing to the idea that they’re working with numbers. They will have something that works like zero, though, a “nothing” that can be added to or subtracted from anything without changing it. And for which multiplication turns something into that “nothing”.

Charlie Brown, at Lucy's Psychiatric Help 5 cents booth: 'I always wanted to go up to that little red-haired girl and talk to her, but I just couldn't. I couldn't start a conversation because I was such a nothing and she was something. If she had wanted to talk to me, it would have been easy because someone who is really something can just go right up to someone who is nothing, and just talk.' Lucy: 'I think your problem is mathematical, Charlie Brown.' Charlie Brown: 'Mathematical?' Lucy: 'If you add nothing and something, what do you get?' Charlie Brown: 'Something, I guess.' Lucy: 'Right ... now, if you subtract nothing from something, what do you get?' Charlie Brown: 'Something.' Lucy: 'Very good ... now, if you multiply something by nothing, what do you get?' Charlie Brown: 'Nothing.' Lucy: 'Five cents, please.' Charlie Brown: 'When you're a nothing, you have a hard time understanding anything!'
Charles Schulz’s Peanuts rerun for the 27th of May, 2018. It originally ran the 30th of May, 1971. This strip originally ran during a time when, in-continuity, the Little Red-Haired Girl had moved away and Charlie Brown was coping with having never spoken to her. At some point she moved back, possibly because Schulz felt he had done everything he could with that or possibly because he forgot she had moved away.

I’m with Charlie Brown in not understanding where Lucy was going with all this, though. Maybe she lost the thread herself.

Mark Anderson’sAndertoons for the 28th is Mark Anderson’sAndertoons for the week. Wavehead’s worried about the verbs of both squaring and rounding numbers. Will say it’s a pair of words with contrary alternate meanings that I hadn’t noticed before. I have always taken the use of “square” to reflect, well, if you had a square with sides of size 4, then you’d have a square with area of size 16. The link seems obvious and logical. So on reflection that’s probably not at all where English gets it from. I mean, not to brag or anything but I’ve been speaking English all my life. If I’ve learned anything about it, it’s that the origin is probably something daft like “while Tisquantum [Squanto] was in England he impressed locals with his ability to do arithmetic and his trick of multiplying one number by itself got nicknamed squantuming, which got shortened to squaning to better fit the meter in a music-hall song about him, and a textbook writer in 1704 thought that was a mistake and `corrected’ it to squaring and everyone copied that”. I’m not even going to venture a guess about the etymology of “rounding”.

On the board: 2^2 = 4, 3^2 = 9, 4^2 = 16. Wavehead: 'Wait, we're squaring numbers now? We just figured out how to round them!'
Mark Anderson’sAndertoons for the 28th of May, 2018. But why would the examples be written out before the students were told what the were doing?

Marguerite Dabaie and Tom Hart’s Ali’s House for the 28th sets up a homework-help session over algebra. Can’t say where exactly Maisa is going wrong. Her saying “x equals 30 but the train equals” looks like trouble to me. It’s often good practice to start by writing out what are the things in the problem that seem important. And what symbol one wants each to mean. And what one knows about the relationship between these things. It helps clarify why someone would want to do that instead of something else. This is a new comic strip tag and I don’t think I’ve ever had cause to discuss it before.

Maisa: 'Can you help me with my homework?' Sahib: 'Dad promised me a hamburger.' Maisa: 'You see - x equals 30 but the train equals ... ' Sahib: 'Dad never makes hamburgers ... mutter mutter mutter.' Maisa: 'Look, I really need help with this.' Sahib: 'My brain isn't set on pay attention my brain is set on burger!'
Marguerite Dabaie and Tom Hart’s Ali’s House for the 28th of May, 2018. Relatable.

Hilary Price’s Rhymes With Orange for the 29th is a Rubik’s Cube joke. I’ve counted that as mathematical enough, usually. The different ways that you can rotate parts of the cube form a group. This is something like what I mentioned in the Peanuts discussion. The different rotations you can do can be added to or subtracted from each other, the way numbers can. (Multiplication I’m wary about.)

Rubik's Headquarters. It's a three-by-three wireframe with tiny offices inside. Person looking in: 'My corner office ... gone! I hate when they do a management shuffle.' [ Title panel, Other person: 'Rumor has it they're going to an open-concept model.' ]
Hilary Price’s Rhymes With Orange for the 29th of May, 2018. Pity whoever gets the center office, bottom layer.

And now here’s the strip that is unsuitable for reading at work, owing to the appearance of an undressed woman.

Continue reading “Reading the Comics, May 29, 2018: Finding Reruns Edition”

Reading the Comics, May 23, 2018: Nice Warm Gymnasium Edition


I haven’t got any good ideas for the title for this collection of mathematically-themed comic strips. But I was reading the Complete Peanuts for 1999-2000 and just ran across one where Rerun talked about consoling his basketball by bringing it to a nice warm gymnasium somewhere. So that’s where that pile of words came from.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for this installment. It has Wavehead suggest a name for the subtraction of fractions. It’s not by itself an absurd idea. Many mathematical operations get specialized names, even though we see them as specific cases of some more general operation. This may reflect the accidents of history. We have different names for addition and subtraction, though we eventually come to see them as the same operation.

On the board, 3/5 - 1/4. Wavehead, to teacher: 'You should call it sub-*fraction*. You can use that --- that's a freebie.'
Mark Anderson’s Andertoons for the 21st of May, 2018. I’m not sure the girl in class needs to be quite so horrified by this suggestion. On the other hand, she sees a lot of this kind of stuff in class.

In calculus we get introduced to Maclaurin Series. These are polynomials that approximate more complicated functions. They’re the best possible approximations for a region around 0 in the domain. They’re special cases of the Taylor Series. Those are polynomials that approximate more complicated functions. But you get to pick where in the domain they should be the best approximation. Maclaurin series are nothing but a Taylor series; we keep the names separate anyway, for the reasons. And slightly baffling ones; James Gregory and Brook Taylor studied Taylor series before Colin Maclaurin did Maclaurin series. But at least Taylor worked on Taylor series, and Maclaurin on Macularin series. So for a wonder mathematicians named these things for appropriate people. (Ignoring that Indian mathematicians were poking around this territory centuries before the Europeans were. I don’t know whether English mathematicians of the 18th century could be expected to know of Indian work in the field, in fairness.)

In numerical calculus, we have a scheme for approximating integrals known as the trapezoid rule. It approximates the areas under curves by approximating a curve as a trapezoid. (Any questions?) But this is one of the Runge-Kutta methods. Nobody calls it that except to show they know neat stuff about Runge-Kutta methods. The special names serve to pick out particularly interesting or useful cases of a more generally used thing. Wavehead’s coinage probably won’t go anywhere, but it doesn’t hurt to ask.

Skippy: 'Look at 'im. The meanest kid on the block. He's got a grudge on the school teacher 'cause she made him stop copyin' answers out of his arithmetic. So he tore out the front of the book an' says 'What good is it without the last part?'
Percy Crosby’s Skippy for the 22nd of May, 2018. It was originally run, looks like, the 12th of February, 1931.

Percy Crosby’s Skippy for the 22nd I admit I don’t quite understand. It mentions arithmetic anyway. I think it’s a joke about a textbook like this being good only if it’s got the questions and the answers. But it’s the rare Skippy that’s as baffling to me as most circa-1930 humor comics are.

Lecturer presenting a blackboard full of equations, titled, 'Mathematical Proof that God does not exist'. In the audience is God.
Ham’s Life on Earth for the 23rd of May, 2018. How did the lecturer get stuff on the top of the board there?

Ham’s Life on Earth for the 23rd presents the blackboard full of symbols as an attempt to prove something challenging. In this case, to say something about the existence of God. It’s tempting to suppose that we could say something about the existence or nonexistence of God using nothing but logic. And there are mathematics fields that are very close to pure logic. But our scary friends in the philosophy department have been working on the ontological argument for a long while. They’ve found a lot of arguments that seem good, and that fall short for reasons that seem good. I’ll defer to their experience, and suppose that any mathematics-based proof to have the same problems.

Paige: 'I keep forgetting ... what's the cosine of 60 degrees?' Jason: 'Well, let's see. If I recall correctly ... 1 - (pi/3)^2/2! + (pi/3)^4/4! - (pi/3)^6/6! + (pi/3)^8/8! - (pi/3)^10/10! + (pi/3)^12/12! - (and this goes on a while, up to (pi/3)^32/32! - ... )' Paige: 'In case you've forgotten, I'm not paying you by the hour.' Jason: '1/2'.
Bill Amend’s FoxTrot Classics for the 23rd of May, 2018. It originally ran the 29th of May, 1996.

Bill Amend’s FoxTrot Classics for the 23rd deploys a Maclaurin series. If you want to calculate the cosine of an angle, and you know the angle in radians, you can find the value by adding up the terms in an infinitely long series. So if θ is the angle, measured in radians, then its cosine will be:

\cos\left(\theta\right) = \sum_{k = 0}^{\infty} \left(-1\right)^k \frac{\theta^k}{k!}

60 degrees is \frac{\pi}{3} in radians and you see from the comic how to turn this series into a thing to calculate. The series does, yes, go on forever. But since the terms alternate in sign — positive then negative then positive then negative — you have a break. Suppose all you want is the answer to within an error margin. Then you can stop adding up terms once you’ve gotten to a term that’s smaller than your error margin. So if you want the answer to within, say, 0.001, you can stop as soon as you find a term with absolute value less than 0.001.

For high school trig, though, this is all overkill. There’s five really interesting angles you’d be expected to know anything about. They’re 0, 30, 45, 60, and 90 degrees. And you need to know about reflections of those across the horizontal and vertical axes. Those give you, like, -30 degrees or 135 degrees. Those reflections don’t change the magnitude of the cosines or sines. They might change the plus-or-minus sign is all. And there’s only three pairs of numbers that turn up for these five interesting angles. There’s 0 and 1. There’s \frac{1}{2} and \frac{\sqrt{3}}{2} . There’s \frac{1}{\sqrt{2}} and \frac{1}{\sqrt{2}} . Three things to memorize, plus a bit of orienteering, to know whether the cosine or the sine should be the larger size and whether they should positive or negative. And then you’ve got them all.

You might get asked for, like, the sine of 15 degrees. But that’s someone testing whether you know the angle-addition or angle-subtraction formulas. Or the half-angle and double-angle formulas. Nobody would expect you to know the cosine of 15 degrees. The cosine of 30 degrees, though? Sure. It’s \frac{\sqrt{3}}{2} .

Michael: 'It's near the end of the school year. You should ease up on the homework. I've learned more than enough this year.' Teacher: 'Oh, sure. How does a 50-percent cut sound?' Michael: 'Why cut it by just one-third?' Teacher: 'You're not helping your case.'
Mike Thompson’s Grand Avenue for the 23rd of May, 2018. I don’t know why the kid and the teacher are dressed the same. I’m honestly not sure if they’re related.

Mike Thompson’s Grand Avenue for the 23rd is your basic confused-student joke. People often have trouble going from percentages to decimals to fractions and back again. Me, I have trouble in going from percentage chances to odds, as in, “two to one odds” or something like that. (Well, “one to one odds” I feel confident in, and “two to one” also. But, say, “seven to five odds” I can’t feel sure I understand, other than that the second choice is a perceived to be a bit more likely than the first.)

… You know, this would have parsed as the Maclaurin Series Edition, wouldn’t it? Well, if only I were able to throw away words I’ve already written and replace them with better words before publishing, huh?

Reading the Comics, March 5, 2018: If It’s Even Mathematics Edition


Many of the strips from the first half of last week are ones that just barely touch on mathematical content. I’m not sure how relevant they all are. I hope you like encountering them anyway.

Bill Griffith’s Zippy the Pinhead for the 4th of March offers “an infinite number of mathematicians walk into a bar” as a joke’s setup. Mathematics popularizers have a small set of jokes about infinite numbers of mathematicians, often arriving at hotels. They’re used to talk about how we now understand infinitely large sets. There’s often counter-intuitive or just plain weird results that follow. And presenting it as a joke works surprisingly well in introducing the ideas. There’s a kind of joke that is essentially a tall tale, spinning out an initial premise to as far and as absurd a consequence as you can get. In structure, that’s not much different to a proof, a discussion of the consequences of an idea. It’s a shame that it’s hard to make jokes or anecdotes about more fields of mathematics. Somehow infinitely large groups of people are funnier than, say, upper-bounded nondecreasing sequences.

['A Pinhead Walks Into A Bar' Jokes That Are Only Funny To Other Pinheads.'] An Atheist, a Vegan, and a Crossfitter walk into a bar. Zippy: 'PUNCHLINE!' A gorilla in a tuxedo walks into a bar. Zippy: 'PUNCHLINE!' An infinite number of mathematicians walk into a bar. Zippy: 'PUNCHLINE!' An amnesiac walks into a bar. Zippy: '( Empty word balloon )'.
Bill Griffith’s Zippy the Pinhead for the 4th of March, 2018. You know, it’s kind of a peculiar thing that Zippy the Pinhead is a syndicated daily newspaper comic strip, isn’t it? I’m glad we live in a world strange enough for this to be the case.

Mike Baldwin’s Cornered for the 4th has a bit of fraction-based wordplay. I’m not sure how mathematical this is, but I grinned.

Bill Amend’s FoxTrot for the 4th has Jason try to make a “universal” loot box that consists of zeroes and ones. As he says, accumulate enough and put them in the right order and you have any digital prize imaginable. Implementation is, as joked, the problem. Assembling ones and zeroes at random isn’t likely to turn up anything you might care about in a reasonable time. (It’s the monkeys-at-typewriters problem.) If you know how to assemble ones and zeroes to get what you want, well, what do you need Jason’s boxes for? As with most clever ideas by computer-oriented boys it shouldn’t really be listened to.

Mark Pett’s Lucky Cow rerun for the 4th has Neil make an order-of-magnitude error estimating what animal power can do. We’ve all made them. They’re particularly easy to make when switching the unit measure. Trying to go from meters to kilometers and multiplying the distance by a thousand, say. Which is annoying since often it’s easiest to estimate the order of magnitude of something first. I can’t find easily an estimate of how many calories a hamster eats over the course of the day. That seems like it would give an idea of how much energy a hamster could possibly be expected to provide, and so work out whether the estimate of four million hamsters to power a car is itself plausible. If someone has information, I’d take it.

Jonathan Lemon’s Rabbits Against Magic for the 4th is a Rubik’s Cube joke. Also a random processes joke. If a blender could turn the faces of a cube, and could turn them randomly, and could run the right period of time … well, yeah, it could unscramble a cube. But see the previous talk about Jason Fox and the delivery of ones and zeroes.

Mark Tatulli’s Lio for the 5th is a solid geometry joke. I’ve put more thought into whether and where to put hyphens in the last three words of that sentence than is worth it.

Steve Sicula’s Home and Away rerun for the 6th has the father and son happily doing some mathematics. It’s in the service of better gambling on sports. But at least they know why they would like to do these calculations.

Reading the Comics, February 26, 2018: Possible Reruns Edition


Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.

Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.

Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 25th straddles the line between Pi Day jokes and architecture jokes. I think this is a rerun, but am not sure.

Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)

Mark Anderson’s Andertoons for the 26th is a reassuring island of normal calm in these trying times. It’s a student-at-the-blackboard problem.

Morrie Turner’s Wee Pals rerun for the 26th just mentions arithmetic as the sort of homework someone would need help with. This is another one of those reruns I’d have thought has come up here before, but hasn’t.