Reading the Comics, June 3, 2020: Subjective Opinions Edition


Thanks for being here for the last week before my All-2020 Mathematics A to Z starts. By the time this posts I should have decided on the A-topic, but I’m still up for B or C topics, if you’d be so kind as to suggest things.

Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.

Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.

Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?

Kid: 'I hate essay assignments. They're so open to subjective grading. It would be a lot simpler if answers would just be right or wrong.' Frazz: 'You're in luck. I understand there's a math test coming up.' Kid: 'What's the lucky part?'
Jef Mallett’s Frazz for the 2nd of June, 2020. Other essays featuring something discussed in Frazz appear at this link.

And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.

Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.

(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.

On the blackboard are written 7 + 0 = 7, 7 - 0 = 7, and 7 x 0 = 0. Wavehead: 'So the takeaway ehre is, if I'm the number 7, avoid multiplication at all costs.'
Mark Anderson’s Andertoons for the 3rd of June, 2020. When I have an essay that features something mentioned in Andertoons the essay’s put up at this link.

Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.

Dan Collins’s Looks Good on Paper for the 3rd has a cook grilling a “Möbius Strip Steak”. It’s a good joke for putting on a mathematics instructor’s door.

Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.


And this wraps up half of the past week’s mathematically-themed comic strips. I hope to have the rest in a Reading the Comics post at this link in a few days. 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, April 7, 2020: April 7, 2020 Edition (Mostly)


I’m again falling behind the comic strips; I haven’t had the writing time I’d like, and that review of last month’s readership has to go somewhere. So let me try to dig my way back to current. The happy news is I get to do one of those single-day Reading the Comics posts, nearly.

Harley Schwadron’s 9 to 5 for the 7th strongly implies that the kid wearing a lemon juicer for his hat has nearly flunked arithmetic. At the least it’s mathematics symbols used to establish this is a school.

Nate Fakes’s Break of Day for the 7th is the anthropomorphic numerals joke for the week.

Jef Mallett’s Frazz for the 7th has kids thinking about numbers whose (English) names rhyme. And that there are surprisingly few of them, considering that at least the smaller whole numbers are some of the most commonly used words in the language. It would be interesting if there’s some deeper reason that they don’t happen to rhyme, but I would expect that it’s just, well, why should the names of 6 and 8 (say) have anything to do with each other?

Evan, to Kevyn: 'Whoa! Only two numbers rhyme with each other! And only a few other words rhyme with them and they're good words. I think that says something.' Devin: 'What are Evan and Kevyn looking so smug about?' Frazz: 'I don't know, Devin.'
Jef Mallett’s Frazz for the 7th of April, 2020. Essays that explore some topic raised in Frazz are at this link.

There are, arguably, gaps in Evan and Kevyn’s reasoning, and on the 8th one of the other kids brings them up. Basically, is there any reason to say that thirteen and nineteen don’t rhyme? Or that twenty-one and forty-one don’t? Evan writes this off as pedantry. But I, admittedly inclined to be a pedant, think there’s a fair question here. How many numbers do we have names for? Is there something different between the name we have for 11 and the name we have for 1100? Or 2011?

There isn’t an objectively right or wrong answer; at most there are answers that are more or less logically consistent, or that are more or less convenient. Finding what those differences are can be interesting, and I think it bad faith to shut down the argument as “pedantry”.

[ Birds aren't partial to fractions. ] Bird at a chalkboard, looking over a figure of a bird over a hand, set equal to a 3 over a bush. Bird: 'Worth 3 in the bush? No, that doesn't add up ... '
Dave Whamond’s Reality Check for the 7th of April, 2020. The essays that address something that appeared in Reality Check are at this link.

Dave Whamond’s Reality Check for the 7th claims “birds aren’t partial to fractions” and shows a bird working out, partially with diagrams, the saying about birds in the hand and what they’re worth in the bush.

The narration box, phrasing the bird as not being “partial to fractions”, intrigues me. I don’t know if the choice is coincidental on Whamond’s part. But there is something called “partial fractions” that you get to learn painfully well in Calculus II. It’s used in integrating functions. It turns out that you often can turn a “rational function”, one whose rule is one polynomial divided by another, into the sum of simpler fractions. The point of that is making the fractions into things easier to integrate. The technique is clever, but it’s hard to learn. And, I must admit, I’m not sure I’ve ever used it to solve a problem of interest to me. But it’s very testable stuff.


And that’s slightly more than one day’s comics. I’ll have some more, wrapping up last week, at this link within a couple days.

Reading the Comics, March 21, 2020: Pragmatic Calculations Edition


There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.

And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …

Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.

But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?

Kid: 'A number is divisible by 3 if the sum of its digits is divisible by 3. But what if the number is so big there's too many digits to add up easily?' Frazz: 'If it's that big, the 1 or 2 left over isn't going to matter much.' Kid: 'Why don't they teach THAT kind of math more in school?' Frazz: 'I guess there's only jobs for so many songwriters, cartoonists, and janitors.'
Jef Mallett’s Frazz for the 20th of March, 2020. Essays featuring some topic raised by Frazz should be gathered at this link.

This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.

There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.

Speaker at a podium: 'If one person kills someone, 50% of the people involved are victims. If 99 people kill someone, 1% of the people involved are victims. The percent of victims is given by V = the limit of K/x as x approachs infinity, where K is people killed and x is the number of people killed. Thus, for sufficiently large x, murder is a victimless crime. So, the bigger we make a war, the more ethical it becomes!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of March, 2020. I have many essays that mention something raised by this comic strip. The many things Saturday Morning Breakfast Cereal has given me to write about are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.

This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)

Mamet: 'I figure I have about 14,000 remaining days of life. So what's the big deal if I want to spend ONE of those days goofing off? That still leaves me with 13,00 days!' Cobb: 'Maybe you could spend a couple of those days learning math.' Mamet: 'Wait, make that 12,000. I'll need one day to PLAN the goof-off day.'
Jeff Corriveau’s Deflocked for the 20th of March, 2020. I’m surprised to learn this is a new tag for me. I’ve discussed the strip, it appears, only twice before, in 2012 and in 2015, before I tagged strips by name. All right. Well, this and future appearances by Deflocked will be at this link.

Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.


This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.

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, September 28, 2019: Laconic Edition


There were more mathematically-themed comic strips last week than I had time to deal with. This is in part because of something Saturday which took several more hours than I had expected. So let me start this week with some of the comics that, last week, mentioned mathematics in a marginal enough way there’s nothing to say about them besides yeah, that’s a comic strip which mentioned mathematics.

Joey Alison Sayers and Jonathan Lemon’s Little Oop — a variation of Alley Oop — for the 22nd has the caveman struggling with mathematics homework. It’s fun that he has an abacus. Also that the strip keeps with the joke from earlier this year about their only dreaming of a number larger than three.

Jef Mallett’s Frazz for the 22nd sees Caulfield stressing out over a mathematics test.

Ralph Dunagin and Dana Summers’s The Middletons for the 24th has more kids stressing out over a mathematics test. Also about how time is represented in numbers.

Mark Parisi’s Off The Mark for the 24th is a bit of animal-themed wordplay on the New Math.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 24th has a parent offering excuses for not helping with mathematics homework.

Eric the Circle for the 27th, by GeoMaker this time, tries putting out a formula for the area of Eric the circle.

Jef Mallett’s Frazz for the 27th has a kid wondering why they need in-person instruction for arithmetic. (I’d agree that rehearsing arithmetic skills is very easy to automate. You can make practice problems pretty near without limit. How much this has to do with mathematics is a point of debate.)

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 27th is a bit of wordplay and numerals humor.

Daniel Beyer’s Long Story Short for the 28th uses arithmetic, the ever-famous 2 + 2 =, as symbol for knowing anything.


With that, I’ve cleared the easy part of comics for the past week. When I get to the comics needing discussion the essay should post here, likely on Monday. And the Fall 2019 A to Z series should post on Tuesday, with ‘I’. Thanks for reading and for your forbearance.

Reading the Comics, September 21, 2019: Prime Numbers and the Rest


This is almost all a post about some comics that don’t need more than a mention. You know, strips that just have someone in class not buying the word problem. These are the rest of last week’s.

Before I get there, though, I want to share something. I ran across an essay by Chris K Caldwell and Yeng Xiong: What Is The Smallest Prime? The topic is about 1, and whether that should be a prime number. Everyone who knows a little about mathematics knows that 1 is generally not considered a prime number. But we’re also a bit stumped to figure out why, since the idea of “a prime number is divisible by 1 and itself” seems to fit this, even if the fit is weird. And we have an explanation for this: 1 used to be thought of as prime, but it made various theorems more clumsy to present. So it was either cut 1 out of the definition or add the equivalent work to everything, and mathematicians went for the solution that was less work. I know that I’ve shared this story around here. (I’m surprised to find I didn’t share it in my Summer 2017 A-to-Z essay about prime numbers.)

The truth is more complicated than that. The truth of anything is always more complicated than its history. Even an excellent history’s. It’s not that the short story has things wrong, precisely. But that that matters are more complicated than that. The history includes things we forget were ever problems, like, the question of whether 1 should be a number. And that the question of whether mathematicians “used to” consider 1 a number is built on the supposition that mathematicians were a lot more uniform in their thinking than they were. Even to the individual: people were inconsistent in what they themselves wrote, because most mathematicians turn out to be people.

It’s an eight-page paper, and not at all technical, so if you’re just interested in the history of whether 1 is a prime number, this is quite readable. It also points out a word ready for resurrection that we could use to mean “1 and the prime numbers”: the incomposites.


So that’s some good reading. Now to the comic strips that you can glance at and agree are comic strips which say “math” somewhere in there. (They’d say “maths” if I read more British comic strips.)

Bob Scott’s Bear With Me for the 16th has Bear trying to help Molly get out of algebra.

Tim Rickard’s Brewster Rockit for the 17th mentions entropy, which is so central to understanding statistical mechanics and information theory. It’s in the popular understanding of entropy, that of it being a thing which makes stuff get worse. But that’s of mathematical importance too.

John Zakour and Scott Roberts’s Maria’s Day for the 18th is about Maria having trouble with a mathematics exam. By the 20th, though, she’s doing better, and she has reasons.

Jef Mallett’s Frazz for the 20th is set during mathematics class.


This wraps up last week’s comic strips. I hope to have my next Reading the Comics post on Sunday. And then tomorrow I get to ‘H’ in the Fall 2019 A to Z essays. Thank you for reading.

Reading the Comics, April 18, 2019: Slow But Not Stopped Week Edition


The first, important, thing is that I have not disappeared or done something worse. I just had one of those weeks where enough was happening that something had to give. I could either write up stuff for my mathematics blog, or I could feel guilty about not writing stuff up for my mathematics blog. Since I didn’t have time to do both, I went with feeling guilty about not writing, instead. I’m hoping this week will give me more writing time, but I am fooling only myself.

Second is that Comics Kingdom has, for all my complaining, gotten less bad in the redesign. Mostly in that the whole comics page loads at once, now, instead of needing me to click to “load more comics” every six strips. Good. The strips still appear in weird random orders, especially strips like Prince Valiant that only run on Sundays, but still. I can take seeing a vintage Boner’s Ark Sunday strip six unnecessary times. The strips are still smaller than they used to be, and they’re not using the decent, three-row format that they used to. And the archives don’t let you look at a week’s worth in one page. But it’s less bad, and isn’t that all we can ever hope for out of the Internet anymore?

And finally, Comic Strip Master Command wanted to make this an easy week for me by not having a lot to write about. It got so light I’ve maybe overcompensated. I’m not sure I have enough to write about here, but, I don’t want to completely vanish either.

Man walking past a street sign for 52 Ludlow Avenue; the 5 falls down and hits him on the head. Woman with him: 'Numbers are hard.'
Dave Whamond’s Reality Check for the 15th of April, 2019. Appearances in these pages of Reality Check should be gathered at this link.

Dave Whamond’s Reality Check for the 15th is … hm. Well, it’s not an anthropomorphic-numerals joke. It is some kind of wordplay, making concrete a common phrase about, and attitude toward, numbers. I could make the fussy difference between numbers and numerals here but I’m not sure anyone has the patience for that.

Man in a cloudscape: 'I made it to heaven!' Angel: 'You sure did! Now you get to do the best stuff! You can design new systems of mathematics! You can attempt to create self-consistent physics systems. Beset of all, try to create a maximally complex reality using the simplest possible constructions!' Man: 'But that sounds terrible.' Angel: 'QUIET! He hears EVERYTHING.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th of April, 2019. I am surprised that this is the first time this strip has drawn a mention this month. Well, this and other Saturday Morning Breakfast Cereal posts are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th touches around mathematics without, I admit, necessarily saying anything specific. The angel(?) welcoming the man to heaven mentions creating new systems of mathematics as some fit job for the heavenly host. The discussion of creating self-consistent physics systems seems mathematical in nature too. I’m not sure whether saying one could “attempt” to create self-consistent physics is meant to imply that our universe’s physics are not self-consistent. To create a “maximally complex reality using the simplest possible constructions” seems like a mathematical challenge as well. There are important fields of mathematics built on optimizing, trying to create the most extreme of one thing subject to some constraints or other.

I think the strip’s premise is the old, partially a joke, concept that God is a mathematician. This would explain why the angel(?) seems to rate doing mathematics or mathematics-related projects as so important. But even then … well, consider. There’s nothing about designing new systems of mathematics that ordinary mortals can’t do. Creating new physics or new realities is beyond us, certainly, but designing the rules for such seems possible. I think I understood this comic better then I had thought about it less. Maybe including it in this column has only made trouble for me.

First chicken: 'What do you want for your birthday?' Second chicken: 'I want everybody to ignore my birthday!' First: 'But if I ignore your birthday I'll be giving the perfect birthday gift, which means I'll be celebrating your birthday, which means I won't be ignoring it!!! AAAAUGH! BIRTHDAY PARADOX!!'
Doug Savage’s Savage Chickens for the 17th of April, 2019. Essays inspired by something from Savage Chickens should be at this link.

Doug Savage’s Savage Chickens for the 17th amuses me by making a strip out of a logic paradox. It’s not quite your “this statement is a lie” paradox, but it feels close to that, to me. To have the first chicken call it “Birthday Paradox” also teases a familiar probability problem. It’s not a true paradox. It merely surprises people who haven’t encountered the problem before. This would be the question of how many people you need to have in a group before there’s a 50 percent (75 percent, 99 percent, whatever you like) chance of at least one pair sharing a birthday.

And I notice on Wikipedia a neat variation of this birthday problem. This generalization considers splitting people into two distinct groups, and how many people you need in each group to have a set chance of a pair, one person from each group, sharing a birthday. Apparently both a 32-person group of 16 women and 16 men, or a 49-person group of 43 women and six men, have a 50% chance of some woman-man pair sharing a birthday. Neat.

Man speaking to a teacher: 'There are two angry parents outside. One's upset that you're teaching multiplication ... the other us upset you're teaching division.' Outside the door are an angry bunny and an angry amoeba.
Mark Parisi’s Off The Mark for the 18th of April, 2019. And essays inspired by Off The Mark should appear at this link.

Mark Parisi’s Off The Mark for the 18th sports a bit of wordplay. It’s built on how multiplication and division also have meanings in biology. … If I’m not mis-reading my dictionary, “multiply” meant any increase in number first, and the arithmetic operation we now call multiplication afterwards. Division, similarly, meant to separate into parts before it meant the mathematical operation as well. So it might be fairer to say that multiplication and division are words that picked up mathematical meaning.


And if you thought this week’s pickings had slender mathematical content? Jef Mallett’s Frazz, for the 19th, just mentioned mathematics homework. Well, there were a couple of quite slight jokes the previous week too, that I never mentioned. Jenny Campbell’s Flo and Friends for the 8th did a Roman numerals joke. The rerun of Richard Thompson’s Richard’s Poor Almanac for the 11th had the Platonic Fir Christmas tree, rendered as a geometric figure. I’ve discussed the connotations of that before.

And there we are. I hope to have some further writing this coming week. But if all else fails my next Reading the Comics essay, like all of them, should be at this link.

Reading the Comics, February 23, 2019: Numerals Edition


It’s happened again: another slow week around here. My supposition is that Comic Strip Master Command was snowed in about a month ago, and I’m seeing the effects only now. There’s obviously no other reason that more comic strips didn’t address my particular narrow interest in one seven-day span.

Samson’s Dark Side of the Horse for the 18th is a numerals joke. The mathematics content is slight, I admit, but I’ve always had a fondness for Dark Side of the Horse. (I know it sounds like I have a fondness for every comic strip out there. I don’t quite, but I grant it’s close.) Conflating numerals and letters, and finding words represented by numerals, is an old tradition. It was more compelling in ancient days when letters were used as numerals so that it was impossible not to find neat coincidences. I suppose these days it’s largely confined to typefaces that make it easy to conflate a letter and a numeral. I mean moreso than the usual trouble telling apart 1 and l, 0 and O, or 5 and S. Or to special cases like hexadecimal numbers where, for ease of representation, we use the letters A through F as numerals.

Samson, counting sheep in bed. #198 leaps over the fence; the numbers are written as might appear on a 14-segment LED. #199 follows. The next panel #200, makes the numerals look like the word ZOO, and an elephant marches through, uprooting the fence.
Samson’s Dark Side of the Horse for the 18th of February, 2019. I don’t always discussDark Side of the Horse but when I do, it should appear at this link.

Jef Mallett’s Frazz for the 18th is built on an ancient problem. I remember being frustrated with it. How is “questions 15 to 25” eleven questions when the difference between 15 and 25 is ten? The problem creeps into many fields. Most of the passion has gone out of the argument but around 1999 you could get a good fight going about whether the new millennium was to begin with January 2000 or 2001. The kind of problem is called a ‘fencepost error’. The name implies how often this has complicated someone’s work. Divide a line into ten segments. There are nine cuts on the interior of the line and the two original edges. I’m not sure I could explain to an elementary school student how the cuts and edges of a ten-unit-long strip match up to the questions in this assignment. I might ask how many birthdays someone’s had when they’re nine years old, though. And then flee the encounter.

Kid: 'If 25 minus 10 is 15, how is *doing* questions 15-25 eleven questions? I should get credit for answering one extra question.' Teacher: 'Two extra questions, if you can answer the one you just brought up.'
Jef Mallett’s Frazz for the 18th of February, 2019. Essays with a mention of Frazz appear at this link.

Mark Parisi’s Off The Mark for the 19th is another numerals joke. This one’s also the major joke to make about an ice skater doing a figure eight: write the eight some other way. (I’d have sworn there was an M-G-M Droopy cartoon in which Spike demonstrates his ability to skate a figure 8, and then Droopy upstages him by skating ‘4 + 4’. I seem to be imagining it; the only cartoon where this seems to possibly fit is 1950’s The Chump Champ, and the joke isn’t in that one. If someone knows the cartoon I am thinking of, please let me know.) Here, the robot is supposed to be skating some binary numeral. It’s nothing close to an ‘8’, but perhaps the robot figures it needs to demonstrate some impressive number to stand out.

An ice-staking woman does a figure 8. An ice-skating robot does a figure 1110101101.
Mark Parisi’s Off The Mark for the 19th of February, 2019. When I have discussed Off The Mark I’ve tried to tag it so the essays appear at this link.

Bud Blake’s Tiger for the 21st has Tiger trying to teach his brother arithmetic. Working it out with fingers seems like a decent path to try, given Punkinhead’s age and background. And Punkinhead has a good point: why is the demonstration the easy problem and the homework the hard problem? I haven’t taught in a while, but do know I would do that sort of thing. My rationalization, I think, would be that a hard problem is usually hard because it involves several things. If I want to teach a thing, then I want to highlight just that thing. So I would focus on a problem in which that thing is the only tricky part, and everything else is something the students are so familiar with they don’t notice it. The result is usually an easy problem. There isn’t room for toughness. I’m not sure if that’s a thing I should change, though. Demonstrations of how to work harder problems are worth doing. But I usually think of those as teaching “how to use these several things we already know”. Using a tough problem to show one new thing, plus several already-existing tricky things, seems dangerous. It might be worth it, though.

Tiger, holding up his fingers: 'Look, one plus one is two, two plus two is four. So then what's four plus four?' Punkinhead: 'How come YOU do the easy ones and you save the tough oens for me?'
Bud Blake’s Tiger rerun for the 21st of February, 2019. I have no information about when it first appeared. Essays inspired by Tiger should appear at this link.

This was not a busy week for comic strips. If it had been, I likely wouldn’t have brought in Dark Side of the Horse. Still there were a handful of comics too slight to get a write-up, even so. John Zakour and Scott Roberts’s Maria’s Day on the 19th just mentioned mathematics homework as hard, for example. Eric the Circle for the 22nd has a binary numeral written out. That one was written by ‘urwatuis’. Maybe that would have been a good, third, numeral comic strip to discuss.


That’s all the mathematically-themed comic strips for the week, though. Next Sunday I should have a fresh Reading the Comics post at this link.

Reading the Comics, December 19, 2018: Andertoons Is Back Edition


I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Mrs Olsen: 'How do you know I haven't got my flu shot?' Caulfield: 'Just playing the odds.' Mrs Olsen: 'Maybe I was playing some odds myself. Maybe I got to the pharmacy and remembered that this year's vaccine is 30-40% effective.' Caulfield: 'I'd take those odds.' Mrs Olsen: 'They're not my kind of odds.' Caulfield: 'And what are the odds you bought a lottery ticket on your way out?' (Pause.) Mrs Olsen: 'You are getting under my skin.' Caulfield: 'That's good news. Now there's a 30-40% chance you'll develop a resistance.'
Jef Mallett’s Frazz for the 16th of December, 2018. Other essays discussing topics raised by Frazz are at this link.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Guard, to new prisoner: 'Never mind Professor Phillip. He's always preoccupied with some theory of escape probability.' The cell walls are covered with mathematical scrawls.
Tak Bui’s PC and Pixel for the 16th of December, 2018. This and other essays, when they’re written, inspired by PC and Pixel should be at this link. It’s a new tag, which surprises me.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Amazing Yet Tautological strip: 'Each year America consumes enough EGG SALAD ... ' (Picture of a woman holding up a lumpy pile that context indicates is egg salad.) ' ... to give EACH AMERICAN an annualized national-average serving of the tasty concoction!'
Ruben Bolling’s Super-Fun-Pak Comix for the 18th of December, 2018. Essays based on Super-Fun-Pak Comix are at this link. (Amazing Yet Tautological is one of the features that turns up in Super-Fun-Pak Comix, which is why it doesn’t rate a tag on its own)

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Teacher showing solid geometry to the class. Wavehead: 'I saw a movie where the robot monster came right at me. If you want me to get excited about 3D shapes, you're going to have to do better than that.'
Mark Anderson’s Andertoons for the 19th of December, 2018. The many essays which discuss Andertoons are at this link.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.


Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

Reading the Comics, October 24, 2018: Frazz Really Wants To Be My Friend Edition


It’s another week with several on-topic installments of Frazz. Again, Jef Mallet, you and I live in the same metro area. Wave to me at the farmer’s market or something. I’m kind of able to talk to people in real life, if I can keep in view three different paths to escape and know two bathrooms to hide in. Horrock’s is great for that.

Jef Mallet’s Frazz for the 22nd is a bit of wordplay. It’s built on the association between “negative” and “wrong”. And the confusing fact that multiplying a negative number by a negative number results in a positive number. It sounds like a trick. Still, negative numbers are tricky. The name connotes something that’s gone a bit wrong. It took time to understand what they were and how they should work. This weird multiplication rule follows from that. If we don’t suppose this to be true, then we break other ideas we have about multiplication and comparative sizes and such. Mathematicians needed to get comfortable with negative numbers. For a long time, for example, mathematicians would treat x^2 - 4x + 4 = 0 and x^4 + 4x + 4 = 0 as different kinds of polynomials to solve. Today we see a -4 as no harder than a +4, now that we’re good at multiplying it out. And I have read, but have not seen explained, that there was uncertainty among the philosophers of mathematics about whether we should consider negative numbers, as a group, to be greater than or less than positive numbers. (I have reasons for thinking this a mighty interesting speculation.) There’s reasons to doubt them, is what I have to say.

Mrs Olsen: 'Any questions? Goody. Caulfield.' Caulfield: 'If a negative times a negative is a positive, how come two wrongs don't make a right?' [Later] Frazz: 'Maybe negative isn't the same as wrong.' Caulfield: 'You are not incorrect.'
Jef Mallet’s Frazz for the 22nd of October, 2018. Good thing to learn, really.

Bob Weber Jr and Jay Stephens’s Oh Brother for the 22nd reminds me of my childhood. At some point I was pairing up the counting numbers and the letters of the alphabet, and realized that the alphabet ended while the numbers did not. Something about that offended my young sense of justice. I’m not sure how, anymore. But that it was always possible to find a bigger number than whatever you thought was the biggest caught my imagination.

Bud: 'Lily! Lily! What's the biggest number?' Lily: 'It's the same as the number of times you bug me.' Bud: 'But that's an ongoing, never-ending number.' Lily: 'Exactly!' Bud: 'Thanks for explaining math in practical terms!'
Bob Weber Jr and Jay Stephens’s Oh Brother for the 22nd of October, 2018. This may be a rerun; I don’t know if the strip is still in original production.

There is, surely, a largest finite number that anybody will ever use for something, even if it’s just hyperbole. I’m curious what it will be. Surely we can’t have already used it. A number named Skewes’s Number was famous, for a while, as the largest number actually used in a proof of something. The fame came from Isaac Asimov writing an essay about the number, and why someone might care, and how hard it is just describing how big the number is in a comprehensible way. Wikipedia tells me this number’s far been exceeded by, among other things, something called Rayo’s Number. It’s “the smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less” (plus some technical points to keep you from cheating). Which, all right, but I’d like to know if we think the first digit is a 1, maybe a 2? Somehow I don’t demand that of Skewes, perhaps because I read that Asimov essay when I was at an impressionable age.

Caulfield: 'If a fraction divided by a fraction is just a fraction times a flipped fraction, what happens if you fish with a fly for flying fish?' Mrs Olsen: 'You can't wade that far out in the ocean.' [ Later ] Frazz: 'So, nothing.' Caulfield: 'I don't think you can divide a fraction by a fraction and get zero.'
Jef Mallet’s Frazz for the 23rd of October, 2018. I appreciate when Mrs Olsen is given the chance to show she does know things.

Jef Mallet’s Frazz for the 23rd has Caulfield talk about a fraction divided by a fraction. And particularly he says “a fraction divided by a fraction is just a fraction times a flipped fraction”. This offends me, somehow. This even though that is how I’d calculate the value of the division, if I needed to know that. But it seems to me like automatically going to that process skips recognizing that, say, \frac{2}{5} \div \frac{1}{10} shouldn’t be surprising if it turns out not to be a fraction. Well, Caulfield’s just looking to cause trouble with a string of wordplay. I can think of how to divide a fraction by a fraction and get zero.

One is really the only number there is! All other numbers are simply collections of ones.
Ashleigh Brilliant’s Pot-Shots for the 23rd of October, 2018. This is a rerun, but from 1977; the strip is not in regular production anymore.

Ashleigh Brilliant’s Pot-Shots for the 23rd promises to recapitulate the whole history of mathematics in a single panel. Ambitious bit of work. It’s easy to picture going from the idea of 1 to any of the positive whole numbers, though. It’s so easy it doesn’t even need humans to do it; animals can count, at least a bit. We just carry on to a greater extent than the crows or the raccoons do, so far as we’ve heard. From those, it takes some squinting, but you can think of negative whole numbers. And from that you get zero pretty quickly. You can also get rational numbers. The western mathematical tradition did this by looking at … er … ratios, that something might be to another thing as two is to five. Circumlocutions like that. Getting to irrational numbers is harder. Can be harder. Some irrational numbers beg you to notice them: the square root of two, for example. Square root of three. Numbers that come up from solving polynomial equations. But there are more number than those. Many more numbers. You might suspect the existence of a transcendental number, that isn’t the root of any polynomial that’s decently behaved. But finding one? Or finding that there are more transcendental number than there are real numbers? This takes a certain brilliance to suspect, and to prove out. But we can get there with rational numbers — which we get to from collections of ones — and the idea of cutting sets of numbers into those smaller than and those bigger than something. Ashleigh Brilliant has more truth than, perhaps, he realized when he drew this panel.

Goldfish, in a tank, to its peers: 'This may seem weird, but my research indicates that the universe has the shape of a perfect cuboid.
Niklas Eriksson’s Carpe Diem for the 24th of October, 2018. I’m curious how the ground is accounted for.

Niklas Eriksson’s Carpe Diem for the 24th has goldfish work out the shape of space. A goldfish in this case has the advantage of being able to go nearly everywhere in the space. But working out what the universe must look like, when you can only run local experiments, is a great geometric problem. It’s akin to working out that the Earth must be a sphere, and about how big a sphere, from the surveying job one can do without travelling more than a few hundred kilometers.


If you’re interested in reading the comics, you might want to see Reading the Comics posts. They’re here. More essays mentioning Frazz should be at this link. Essays that discuss ideas brought up by Oh Brother! should be this link. Essays which talk about Frazz — wait. I said that. This and other appearances by Pot Shots should be at this link. And posts which feature Carpe Diem should be at link. Do please stick around for more of my Fall 2018 Mathematics A-To-Z, too. I’m trying to keep up at two essays a week through the end of the year, which is not precisely fall.

Reading the Comics, October 2, 2018: Frazz Loves Mathematics Edition


Jef Mallet’s Frazz did its best to take over my entire Reading-the-Comics bit this week. I won’t disrespect his efforts, especially as I take the viewpoint of the strip to be that arithmetic is a good thing to learn. Meanwhile let me offer another mention of Playful Mathematics Education Blog Carnival #121, hosted here last week. And to point out the Fall 2018 Mathematics A To Z continues this week with the letters ‘E’ and ‘F’. And I’m still looking for topics to discuss for select letters between H and M yet.

Sandra Bell-Lundy’s Between Friends for the 1st is a Venn Diagram joke to start off the week. The form looks wrong, though. This can fool the reader into thinking the cartoonist messed up the illustration. Here’s why. The point of a Venn Diagram is to show the two or more groups of things and identify what they have in common. It is true that any life will have regrets about things done. And regrets about things not done. But what are the things that one both ‘did do’ and ‘didn’t do’? Unless you accept the weasel-wording of “did halfheartedly”, there is nothing that one both did and did not.

[ The Venn Diagram of Living your Best Life ] The balloons are 'Did Do' and 'Didn't Do', with the overlap 'Regret'.
Sandra Bell-Lundy’s Between Friends for the 1st of October, 2018. Regardless of the internal logic of the joke, I like the art touch of the protagonist leaning against one of the bubbles. It makes the joke more interesting than simply being a pair of bubbles would be.

And here is where I will argue Bell-Lundy did this right. The overlap of things one ‘did do’ and ‘didn’t do’ must be empty. Do not be fooled by there being area in common in the overlap. One thing Venn Diagrams help us establish are the different kinds of things we are studying, and to work out whether that kind of thing can have any examples. And if the set of things in your life that you regret is empty — well! Is it not “living your best life”, as the caption advances, to have nothing one regrets doing, and nothing one regrets not doing? Thus I say to you the jury of readers, Sandra Bell-Lundy has correctly used the Venn Diagram form to make a “No Regrets” art.

That said, I can’t explain why the protagonist on the left is slumping and looking depressed. I suppose we have to take that she hasn’t lived her best life, but does have information about what might have been.

Jeff Mallet’s Frazz for the 1st starts a string of mathematics class jokes. Here is one about story problems, particularly ones about pricing apples and groups of apples. I don’t know whether apples are used as story problem examples. They seem like good example objects. They’re reasonably familiar. A person can have up to several dozen of them without it being ridiculously many. (Count a half-bushel of apples sometime.) You can imagine dividing them among people or tasks. You can even imagine halving and quartering them without getting ridiculous. Great set of traits. But the kid has overlooked that if Mrs Olsen wanted the price of an apple she would just look at the price sign.

Frazz: 'Very cool of you to bring treats.' Kid with box of doughnuts: 'Thanks; though it's mostly a story-problem prophylactic. If Mrs Olsen sees I went to the cider mill, she knows I know how much how many apples cost.' Frazz: 'Prophylactic!' Kid: 'If I can intimidate her out of a vocabulary unit, too, so much the better.'
Jeff Mallet’s Frazz for the 1st of October, 2018. I’m sorry, I don’t know the child’s name here.

(Every time I’m at the market I mean to check the apple prices, and I do, and I forget the total on the way out. I mention because I live in the same area as Jef Mallet. So there is a small but not-ridiculous chance he and I have bought apples from the same place. If he has a strip mentioning the place with the free coffee, popcorn, and gelato samples I’ll know to my satisfaction.)

Jeff Mallet’s Frazz for the 2nd has a complaint about having to show one’s work. But as with apple prices, we don’t really care whether someone has the right answer. We care whether they have the right method for finding an answer. Or, better, whether they have a method that could plausibly find the right answer, and an idea of how to check whether they did get it. This is why it’s worth, for example, working out a rough expected answer before doing a final calculation.

Frazz: 'How was math?' Kid: 'Same old argument: show your work. Do airline passengers care about the flight path? No! They care about the landing. Am I right?' Frazz: 'As right as anyone who's never had to scan for the barf bag while enjoying avoidable turbulence.'
Jeff Mallet’s Frazz for the 2nd of October, 2018. Seriously, why doesn’t every comic have an up-to-date cast roster with pictures? It would make everybody’s life so much easier.

The talk about flight paths reminds me of a story passed around sci.space.history back in the day. The story is about development of the automatic landing computers used for the Apollo Missions. The guidance computers were programmed to get the lunar module from this starting point to a final point on the lunar surface. This turns into a question of polynomial interpolation. That’s coming up with a curve that fits some data points, particularly, the positions and velocities the last couple times those were known plus the intended landing position. You can always find a polynomial that passes smoothly through a finite bunch of data points. That’s not hard. But, allegedly, the guidance computer would project paths where the height above the lunar surface was negative for a while. Numerically, there’s nothing wrong with a negative number. It’s just got some practical problems, as the earliest Apollo missions were before any subway tunnels could be built.

Kid: 'Why do I have to show my work in math when I know the right answer? I don't have to show my work in, say, writing.' Frazz: 'Writing *is* showing your work.' Kid: 'Spelling, then.' Frazz: 'Spelling is to writing what addition and multiplication tables are to algebra.'
Jeff Mallet’s Frazz for the 3rd of October, 2018. Also while writing does show off your work, it doesn’t show all your work, not if you’re writing anything that isn’t really basic. There’s research, there’s thinking, and there’s (often) outlines and drafts and revisions and editing. And, come to think of my own school experience, I didn’t like showing my outlines or drafts either. (In my defense, I am one of those people with a gift for academic narrative, so I can do an essay of up to two thousand words with little to no outline. And while I do better on my second draft, I don’t do enormously better most of the time.)

Jeff Mallet’s Frazz for the 3rd continues the protest against showing one’s work. I do like the analogy of arithmetic skills for mathematics being like spelling skills for writing. You can carry on without these skills, for either mathematics or writing. But knowing them makes your life easier. And enjoying these building-block units foreshadows enjoying the whole. But yeah, addition and multiplication tables can look like tedium if you don’t find something at least a little thrilling in how, say, 9 times 7 is 63.

Executive in a business office, speaking to an underling: 'We need to investigate an ADD cluster in accounting.'
Tim Lachowski’s Get a Life for the 2nd of October, 2018. You know, that guy has an enormous desk. Maybe he’s at a conference table.

Tim Lachowski’s Get a Life for the 2nd is a bit of mathematics wordplay. So that closes the essay out well.

Thanks for reading Reading the Comics. Other comic strip review essays are at this link. More essays with Between Friends should be at this link. Other essays with Frazz in them are at this link. And appearances by Get A Life should be at this link.

Reading the Comics, July 3, 2018: Fine, Jef Mallett Wants My Attention Edition


Three of these essays in a row now that Jef Mallett’s Frazz has done something worth responding to. You know, the guy lives in the same metro area. He could just stop in and visit sometime. There’s a pinball league in town and everything. He could view it as good healthy competition.

Bill Hinds’s Cleats for the 1st is another instance of the monkeys-on-typewriters metaphor. The metaphor goes back at least as far as 1913, when Émile Borel wrote a paper on statistical mechanics and the reversibility problem. Along the way it was worth thinking of the chance of impossibly unlikely events, given enough time to happen. Monkeys at typewriters formed a great image for a generator of text that knows no content or plan. Given enough time, this random process should be able to produce all the finite strings of text, whatever their content. And the metaphor’s caught people’s fancy I guess there’s something charming and Dadaist about monkeys doing office work. Borel started out with a million monkeys typing ten hours a day. Modern audiences sometimes make this an infinite number of monkeys typing without pause. This is a reminder of how bad we’re allowing pre-revolutionary capitalism get.

Kid: 'Mom, Dad, I want to go bungee jumping this summer!' Dad: 'A thousand monkeys working a thousand typewriters would have a better chance of randomly typing the complete works of William Shakespeare over the summer than you have of bungee jumping.' (Awksard pause.) Kid: 'What's a typewriter?' Dad: 'A thousand monkeys randomly TEXTING!'
Bill Hinds’s Cleats rerun for the 1st of July, 2018. It originally ran the 28th of June, 2009. Oh, but you figured that out yourselves, didn’t you? Also, boy, that’s not much of a punch line. Most comics aren’t actually written with disdain for young people and their apps and their podcasts and their emojis and all that. But sometimes one kind of hits it.

Sometimes it’s cut down to a mere thousand monkeys, as in this example. Often it’s Shakespeare, but sometimes it’s other authors who get duplicated. Dickens seems like a popular secondary choice. In joke forms, the number of monkeys and time it would take to duplicate something is held as a measure of the quality of the original work. This comes from people who don’t understand. Suppose the monkeys and typewriters are producing truly random strings of characters. Then the only thing that affects how long it takes them to duplicate some text is the length of the original text. How good the text is doesn’t enter into it.

Jef Mallett’s Frazz for the 1st is about the comfort of knowing about things one does not know. And that’s fine enough. Frazz cites Fermat’s Last Theorem as a thing everyone knows of but doesn’t understand. And that choice confuses me. I’m not sure what there would be to Fermat’s Last Theorem that someone who had heard of it would not understand. The basic statement of it — if you have three positive whole numbers a, b, and c, then there’s no whole number n larger than 2 so that a^n + b^n equals c^n — has it.

Frazz: 'You know what I like? Fermat's last theorem.' Jane: 'Do you even understand it?' Frazz: 'Nope. And neither do you. To paraphrase Mark Twain, we live in a world where too many people don't know what they don't know. With Fermat's Last Theorem, we can all agree on something we don't know.' Jane: 'Nice. Except how many people have ever heard of Fermat's Last Theorem?' Frazz: '2,125,420,566.' Jane: 'You don't know that.' Frazz: 'I know!'
Jef Mallett’s Frazz for the 1st of July, 2018. Frazz’s estimate of how many have heard of Fermat’s Last Theorem seems low to me. But I grew up at a time when the theorem was somewhat famous for being something easy to understand and that had defied four hundred years’ worth of humanity trying to prove. And even then my experience is selected to a particular kind of Western-culture person. Was the theorem ever so interesting to, say, Indian or Chinese mathematicians? (Come to it, was there someone in the South Asian or Chinese or Japanese traditions who ran across the same property but didn’t get famous in Western literature for it?)

But “understanding” is a flexible concept. He might mean that people don’t know why the Theorem is true. Fair enough. Andrew Wiles and Richard Taylor’s proof is a long thing that goes deep into a field of mathematics that even most mathematicians don’t study. Why it should be true can be an interesting question, and one that’s hard to ever satisfyingly answer. What is the difference between a proof that something is true and an explanation for why it’s true? And before you say there’s not one, please consider that many mathematicians do experience a difference between seeing something proved and understanding why something is true.

And Frazz might also mean that nobody knows what use Fermat’s Last Theorem is. This is a fair complaint too. I’m not aware offhand of any interesting results which follow from its truth, nor of anything neat that would come about had it been false. It’s just one of those things that happens to be true, and that we’ve found to be pretty, perhaps because it is easy to ask whether it’s true and hard to answer. I don’t know.

Morrie Turner’s Wee Pals for the 2nd has a kid looking for a square root. We all have peculiar hobbies. His friends speak of it as though it’s a lost physical object. This is a hilarious misunderstanding until it strikes you that we speak about stuff like square roots “existing”. Indeed, the language of mathematics would be trashed if we couldn’t speak about numerical constructs “existing” somewhere to be “found”. But try to put “four” in a box and see what you get. That we mostly have little trouble understanding what we mean by showing some mathematical construct exists, and what we hope to do by looking for it, suggests we roughly know what we mean by the phrases. All right then; what is that, in terms a kid could understand?

Ralph: 'Whatcha doin', Oliver?' Oliver: 'Trying to find the square root of 8,765,510.' Ralph: 'Where did you lose it? Randy and I will help you find it!'
Morrie Turner’s Wee Pals rerun for the 2nd of July, 2018. It originally ran the 2nd of July, 2013. Just saying, it would have been slick if Oliver had been working out something for which 42 was the answer. Why couldn’t he have been looking for the cube root of 74,088 instead?

There are many ways to numerically compute a square root, if you have to do it by hand and it isn’t a perfect square. My preference is for iterative methods, in which you start with a rough guess and try to improve things. One good enough method for we call the Babylonian method, reflecting how old we think it is. Start with your number S whose square root you want. And start with a number x0, a first guess for what the square root is. This can be anything. The great thing about iterative methods is even if you start with a garbage answer, you get to a good answer soon enough. Still, if you have a suspicion of what the square root should be, start there.

Your first iteration, the first guess for a better answer, is to calculate the number x_1 = \frac{1}{2}\left( x_0 + \frac{S}{x_0}\right) . Typically, x1 will be closer to the square root of S than will x0 be. And in any case, we can get closer still. Use x1 to calculate a new number. This is x_2 = \frac{1}{2}\left( x_1 + \frac{S}{x_1}\right) . And then x3 and x4 and x5 and so on. In theory, you never finish; you’re stuck finding an infinitely long sequence of better approximations to the square root. In practice, you finish; you find that you’re close enough to the square root. Well, the square root of a whole number is either a whole number (if it was a perfect square to start) or is an irrational number. You were going to stop on an approximation sooner or later.

The method requires doing division. Long division, too, after the first couple steps. I don’t know a way around that which doesn’t divert into something less pleasant, such as logarithms and exponentials. Or maybe into trigonometric functions. This can be tedious to do by hand. Great thing, though, is if you make a mistake? That’s kind of all right. The next iteration will (usually) correct for it. That’s the glory of iterative methods. They tend to be forgiving of numerical error, whatever its source. Another iteration reduces, or even eliminates, the mistake of the previous iteration.

At the bar. Harley's Friend: 'I've done the math. You won't make it across the canyon without a good ramp, Harley! You need a quadrilateral with exactly one pair of parallel sides. You'll be riding into a trap ... ezoid.' Harley, in jail, to the sheriff: 'Who knew a calculator could go that far up his nose.'
Dan Thompson’s Harley for the 3rd of July, 2018. I don’t know the guy’s name here. The storyline is part of Harley’s annual effort to jump across the canyon and no, it doesn’t go well.

Dan Thompson’s Harley for the 3rd is a shapes joke. Haven’t had a proper anthropomorphic geometric figures joke in a while. This is near enough.


For more of these Reading the Comics posts please follow this link. If you’re only interested in Reading the Cleats strips, please use this link instead. But Cleats is a new tag this essay, so for now, there aren’t others. If you’re hoping to see all my Reading the Comics posts about Frazz, try this link. If you’d like more of my essays which mention Wee Pals, you can use this link. And if you’d like more Reading the Comics posts that mention Harley, use this link. That’s another new tag, but I believe Dan Thompson is still making new examples of the strip. So it may appear again.

Reading the Comics, June 29, 2018: Chuckle and Breakfast Cereal Edition


The last half of last week was not entirely the work of Chuckle Brothers and Saturday Morning Breakfast Cereal. It seemed like it, though. Let’s review.

Patrick Roberts’s Todd the Dinosaur for the 28th is a common sort of fear-of-mathematics joke. In this case the fear of doing arithmetic even when it is about something one would really like to know. I think the question got away from Todd, though. If they just wanted to know whether they had enough money, well, they need twelve dollars and have seven. Subtracting seven from twelve is only needed if they want to know how much more they need. Which they should want to know, but wasn’t part of the setup.

Kid: 'Do we have enough money to go to the movie?' Todd: 'Let's see! You ahve four dollars and I have three dollars. That's seven. The movie is twelve dollars for both of us. So twelve take away seven is ... *GASP* Oh no! I accidentally did math!' Kid: 'So?' Todd: 'This is SUMMER!' Kid: 'I don't even know you!'
Patrick Roberts’s Todd the Dinosaur for the 28th of June, 2018. I’m sorry, I don’t know the kid’s name.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th uses mathematics as the sine qua non of rocket science. As in, well, the stuff that’s hard and takes some real genius to understand. It’s not clear to me that the equations are actually rocket science. There seem to be a shortage of things in exponentials to look quite right to me. But I can’t zoom in on the art, so, who knows just what might be in there.

Professor-type in front of a class labelled Rocket Science 101: 'Doesn't ANYBODY understand this stuff?'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 28th of June, 2018. It originally ran the 16th of July, 2009. Relatable.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a set theory joke. Or a logic joke, anyway. It refers to some of the mathematics/logic work of Bertrand Russell. Among his work was treating seriously the problems of how to describe things defined in reference to themselves. These have long been a source of paradoxes, sometimes for fun, sometimes for fairy-tale logic, and sometimes to challenge our idea of what we mean by definitions of things. Russell made a strong attempt at describing what we mean when we describe a thing by reference to itself. The iconic example here was the “set of all sets not members of themselves”.

Caption: 'Nobody liked Bertrand Russell's scavenger hunts.' Items to find: 'The list of all lists that do not list themselves. (List here).'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of June, 2018. Well, among other things, wouldn’t there be infinitely many such lists? Unless this description were enough to describe them all, by being a description of what to do to get you all of them?

Russell started out by trying to find some way to prove Georg Cantor’s theorems about different-sized infinities wrong. He worked out a theory of types, and what kinds of rules you can set about types of things. Most mathematicians these days prefer to solve the paradox with a particular organization of set theory. But Russell’s type theory still has value, particularly as part of the logic behind lambda calculus. This is an approach to organizing relationships between things that can do wonderful things, including in computer programming. It lets one write code that works extremely efficiently and can never be explained to another person, modified, or debugged ever. I may lack the proper training for the uses I’ve made of it.

News anchor: 'In a cruel, bizarre twist of fate, this week's $1 million winning lotto number 579281703 was shared by exactly one million people. In other news ... ' (The person watching the news has a lottery ticket number 579281703.)
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 29th of June, 2018. It originally ran the 17th of July, 2009. You can tell it’s from so long ago because the TV set is pre-HD.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 29th is a lottery joke. It does happen that more than one person wins a drawing; sometimes three or even four people do, for the larger prizes. The chance that there’s a million winners? Frightfully unlikely unless something significant went wrong with the lottery mechanism.

So what are the chances of a million lottery winners? If I’m not mistaken the only way to do this is to work out a binomial distribution. The binomial distribution is good for cases where you have many attempts at doing a thing, where each thing can either succeed or fail, and the likelihood of success or failure is independent of all the other attempts. In this case each lottery ticket is an attempt; it winning is success and it losing is failure. Each ticket has the same chance of winning or losing, and that chance doesn’t depend on how many wins or losses there are. What is that chance? … Well, if each ticket has one chance in a million of winning, and there are a million tickets out there, the chance of every one of them winning is about one-millionth raised to the millionth power. Which is so close to zero it might as well be nothing. … And yet, for all that it’s impossible, there’s not any particular reason it couldn’t happen. It just won’t.

What I Learned This Year. Kid: 'Um ... you can divide a number by 3 if the sum of its digits can be divided by 3.' [ Later ] Frazz: 'So, what'd you learn this year?' Kid: 'Don't go last on what-I-learned-this-year day.'
Jef Mallet’s Frazz for the 29th of June, 2018. Sorry, again, not sure of this kid’s name. The comic is often so good about casually dropping in character names.

Jef Mallet’s Frazz for the 29th is a less dire take on what-you-learned-this-year. In this case it’s trivia, but it’s a neat sort of trivia. Once you understand how it works you can understand how to make all sorts of silly little divisibility rules. The threes rule — and the nines rule — work by the same principle. Suppose you have a three-digit number. Let me call ‘a’ the digit in the hundreds column, ‘b’ the digit in the tens column, and ‘c’ the digit in the ones column. Then the number is equal to 100\cdot a + 10\cdot b + 1\cdot c . And, well, that’s equal to 99\cdot a + 1\cdot a + 9 \cdot b + 1 \cdot b + 1 \cdot c . Which is 99\cdot a + 9 \cdot b + a + b + c . 99 times any whole number is a multiple of 9, and also of 3. 9 times any whole number is a multiple of 9, and also of 3. So whether the original number is divisible by 9, or by 3, depends on whether a + b + c is. And that’s why adding the digits up tells you whether a number is a whole multiple of three.

This has only proven anything for three-digit numbers. But with that proof in mind, you probably can imagine what the proof looks like for two- or four-digit numbers, and would believe there’s one for five- and for 500-digit numbers. Or, for that matter, the proof for an arbitrarily long number. So I’ll skip actually doing that. You can fiddle with it if you want a bit of fun yourself.

Also maybe it’s me, or the kind of person who gets into mathematics. But I find silly little rules like this endearing. It’s a process easy to understand that anyone can do and it tells you something not obvious from when you start. It feels like getting let in on a magic trick. That seems like the sort of thing that endears people to mathematics.

Michael: 'Grandma broke out the math workbooks!' Gabby: 'She does this every summer!' (They hide behind a tree.) Gabby: 'Says she doesn't want us to forget what we learned during the school year.' Michael: 'She has a point. We do need to keep our homework-avoidance skills sharp.'
Mike Thompson’s Grand Avenue for the 29th of June, 2018. At the risk of taking the art too literally: isn’t that tree kind of short to be that fat? Shouldn’t the leaves start higher up?

Mike Thompson’s Grand Avenue for the 29th is trying to pick its fight with me again. I can appreciate someone wanting to avoid kids losing their mathematical skills over summer. It’s just striking how Thompson has consistently portrayed their grandmother as doing this in a horrible, joy-crushing manner.

Greek: 'Why are you the wisest man, Socrates?' Socrates: 'Because I know one thing: that I know nothing.' Greek: 'That's all you know?' Socrates: 'I mean strictly speaking ... ' Greek: 'What about the infinite universe of analytic statements, like if A = A then A = A?' Socrates: 'Okay yeah That stuff. Just that.' Greek: 'Just ALL of math.' (Pause.) Greek: 'Sorry, did I make you sad?' Socrates: 'I can't be certain, but probably.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th of June, 2018. I am curious if anyone in the philosophy department would offer an idea which Ancient Greek might be chatting with Socrates here. If Weinersmith had anyone in mind I would guess whichever one has Socrates getting a slave to do a geometry proof. But there’s also … I want to say Parmenides, where the elder scholar whips the young Socrates in straight syllogisms. Again, if anyone specific was in mind and it wasn’t just “another Ancient Greek type”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th gets into a philosophy-of-mathematics problem. Also a pure philosophy problem. It’s a problem of what things you can know independently of experience. There are things it seems as though are true, and that seem independent of the person who is aware of them, and what culture that person comes from. All right. Then how can these things be relevant to the specifics of the universe that we happen to be in just now? If ‘2’ is an abstraction that means something independent of our universe, how can there be two books on the table? There’s something we don’t quite understand yet, and it’s taking our philosophers and mathematicians a long while to work out what that is.


And as ever, if you’d like to see more Reading the Comics posts, please look to this page. For essays with Todd the Dinosaur in them, look here. For essays with the Chuckle Brothers, here you go. For some of the many, many essays with Saturday Morning Breakfast Cereal, follow this link. For more talk about Frazz, look here. And for the Grand Avenue comics, try this link please.

Reading the Comics, June 27, 2018: Stitch Day Edition


For a while I thought this essay would include only the mathematically-themed strips which Comic Strip Master Command sent out through to June 26th, which is picking up the nickname Stitch Day (for 6-26, the movie character’s experiment number). And then I decided some from last Sunday weren’t on-point enough (somehow), and there were enough that came later in the week that I couldn’t do a June 26th Only edition. Which is my longwinded way of saying this one doesn’t have a nonsense name. It just has a name that’s only partially on point.

Mike Baldwin’s Cornered for the 26th is the Rubik’s Cube/strange geometry joke for the week. It seems to me I ought to be able to make some link between the number of various ways to arrange a Rubik’s Cube — which pieces can and which ones cannot be neighbors to a red piece, say, no matter how one scrambles the cube — and the networking between people that you can get from an office where people have to see each other. But I’m not sure that I can make that metaphor work. I’m blaming the temperature, both mine (I have a cold) and the weather’s (it’s a heat wave).

Man sitting behind an upside-down desk, to a person standing on a horizontal wall-with-window: 'Hang on --- I've almost got it.' Caption: Rubik's Cubicle.
Mike Baldwin’s Cornered for the 26th of June, 2018. Say what you will; at least it’s not an open-office plan.

Mark Leiknes’s Cow and Boy for the 26th makes literal the trouble some people have with the phrase “110 percent”. Read uncharitably, yes, “110 of a hundred” doesn’t make sense, if 100 percent is all that could conceivably be of the thing. But if we can imagine, say, the number of cars passing a point on the highway being 90 percent of the typical number, surely we can imagine the number of cars also being 110 percent. To give an example of why I can’t side with pedants in objecting to the phrase.

Boy (Billy), playing chess with Cow: 'I hate it when people say they're giving a hundred and ten percent. I mean, how is that even possible? Wouldn't you be trying so hard that your body couldn't contain the extra ten percent of effort and your head would explode?' Cow: 'Check mate!' [ Cow's head explodes. ] Boy: 'OK, but I was only giving it like 35 percent.' Headless Cow: 'Darn.'
Mark Leiknes’s Cow and Boy for the 26th of June, 2018. This strip originally ran the 12th of October, 2011 and it’s not usually so gruesome.

Jef Mallett’s Frazz for the 26th is just itching for a fight. From me and from the Creative Writing department. Yes, mathematics rewards discipline. All activities do. At the risk of making a prescription: if you want to do something well, spend time practicing the boring parts. For arithmetic, that’s times tables and regrouping calculations and factoring and long division. For writing, that’s word choice and sentence structure and figuring how to bring life to describing dull stuff. Do the fun stuff too, yes, but because it is fun. Getting good at the boring stuff makes you an expert. When you discover that the boring stuff is also kinda fun, you will do the fun stuff masterfully.

Student presenting 'What I Learned This Year': 'Writing rewards creativity while math rewards a disciplined pursuit of a single right answer.' Later, Frazz: 'So, what'd you learn this year?' Student: 'Apparently we don't learn how to fudge the numbers until business school.'
Jef Mallett’s Frazz for the 26th of June, 2018. Again I apologize; I don’t know who the student is. Cast lists, cartoonists. Get your cast on your web page.

But to speak of mathematics as pursuing a single right answer — well, perhaps. In an elementary-school problem there is typically just the one right answer, and the hope is that students learn how to get there efficiently. But if the subject is something well-worn, then there are many ways to do any problem. All are legitimate and the worst one can say of a method is maybe it’s not that efficient, or maybe it’s good here but doesn’t generally work. If the subject is on the edge of what mathematics we know, there may be only one way to get there. But there are many things to find, including original ways to understand what we have already found. To not see that mathematics is creative is to not see mathematics. Or, really, any field of human activity.

Horace, reading the newspaper: 'Your horoscope: you will be positively surprised.' A giant + sign drops from the sky, barely missing Horace.
Samson’s Dark Side of the Horse for the 27th of June, 2018. So, how would you rewrite the horoscope to make this work for multiplication? ‘You’re encountering some surprising times’?

Samson’s Dark Side of the Horse for the 27th edges up to being the anthropomorphic numerals joke for the week. I need a good name for this sort of joke about mathematical constructs made tangible, even if they aren’t necessarily characters.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th I hope makes sense if you just know the words “graph” and “drunk”, and maybe “McNugget”. That’s all you truly need to understand why this contains a joke. But there is some good serious mathematical terminology at work here.

Mathematics instructor: 'Here we have a graph which embodies a stochastic process. Now, we perform a random walk on the graph for n steps and --- HEY! [ Curses ] The graph went out for McNuggets!' (The graph looks faintly more like a person, has a basket of McNuggets, and is saying, 'Nuggs nuggs nuggie nuggie nugg WOOH! God you're so hot.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of June, 2018. Can’t be intended, but that graph looks to me like plots of what the constellation Orion is expected to look like after several ten thousand years of stellar movement.

So. A “graph” is a thing that’s turned up in my A To Z serieses. In this context a graph is a collection of points, called “vertices”, and a collection of “edges” that connect vertices. Often the vertices represent something of interest and the edges ways to turn one thing into another. Sometimes the edges are the thing of interest and the vertices are just there to be manipulated in some way by edges. It’s a way to make visual the studying of how stuff is connected, and how things can pass from one to another.

A “stochastic process” is about random variables. Random variables are some property about a system. And you know some things about that variable’s value. You know maybe the range of possible values it could have. You know whether some values are more likely than others. But you do not know what the value is at any particular moment. Consider, say, the temperature outside where you live at a particular time of day. You may have no idea what that is. But you can say, for example, whether today it is more likely to be 90 degrees Fahrenheit or 60 degrees Fahrenheit or 20 degrees Fahrenheit. For a stochastic process we have some kind of index. We can say, for example, which values of temperature are more likely today, the 1st of July, and which ones will be more likely the 1st of August, and which ones will be less likely the 1st of December. Calling it a “process”, to my intuition, makes it sound like we expect something to happen that causes the likelihood of some temperatures to change. And many processes are time-indexed. They study problems where something interesting changes in time, predictable in aggregate but not in detail.

So a graph like this, representing a stochastic process, is a shorthand. Each vertex is a state that something might be in. Each edge is a way to get from one state to another when — something — happens. Doesn’t matter what thing.

A “drunk walk”, or as it’s known to tenderer writers a “random walk”, is a term of art. Not a deep one. It’s meant to evoke the idea of a severely drunk person who yes, can move, but has no control over which way. Thus he wanders around, reaching any point only by luck. Many things look like random walks, in which there is no overall direction, just an unpredictable shuffling around. A drunk walk on this graph would be, well, start at any of the vertices. Then follow edges, chosen randomly. If you start at the uppermost point of the triangle on top, for example, there’s two places to go on the second step: the lower-left or the center-right vertex on the upper triangle. Suppose you go to the center-right vertex. On the next step, you might go right back where you started. You might go to the lower-left vertex on the triangle. You might drop down that bridge to the top of that quadrilateral. And so on, for another step.

Do that some presumably big number of times. Where are you? … Anywhere, of course. But are there vertices you’re more likely to be on? Ones you’re less likely to be on? How does the shape of the graph affect that likelihood? How does how long you spend walking affect that? These tell us things about the process, and are why someone would draw this graph and talk about a random walk on it.


If you’d like to read more of my comic-strip review posts please do! They all should be available at this link, listed in reverse chronological order.

To read more of the individual comics? Here are essays with Cornered in them. These are Cow and Boy comics at this link. Frazz strips are here. Essays including Dark Side of the Horse are here. And Saturday Morning Breakfast Cereal, which is threatening to take over “being the majority of my blog” from Andertoons, I have at that link.

Reading the Comics, June 16, 2018: No Panels Edition


My week got busier than I imagined, but it was in ways worthwhile. I apologize for running late, and for not having an essay I meant to put up here this week. But I should be back to something more normal next week. I keep saying that. Also, for what seems like a rarity, all the strips for this essay are comic strips. No panels. That won’t last, I know.

Johnny Hart’s Back to B.C. for the 14th features arithmetic as a demonstration of The Smartest Man in the World’s credentials. I understand using a bit of arithmetic as a quick check that someone has any intelligence at all. It seems to me that checking “two plus two” is more common than “one plus one”, and either is more common than, say, “one plus two” or “three plus five” or anything. I’m curious why that is, though. Might one plus one just seem too simple? Or is it the bias against odd numbers and feeling that two plus two is somehow more balanced? If only there were some smart person I could ask.

Peter(?) is by a sign reading 'The Smartest Man in the World'. Other Caveman (BC?): 'How much is 2 + 2?' Peter(?): 'Four.' BC: 'What makes day?' Peter: 'The sun.' BC: 'What made people?' (Peter looks frazzled.) BC: 'Here we go again.'
Johnny Hart’s Back to B.C. for the 14th of June, 2018. The strip originally ran the 17th of December, 1960. Thing to remember about Peter(?)’s claim is that at this time there’s like eight people in the world so, you know, yeah.

Jef Mallett’s Frazz for the 14th has a blackboard full of arithmetic as the icon of “doing a lot of school work”. Can’t say it’s age-inappropriate or anything. It’s just an efficient way to show a lot of work that’s kind of tiring to do has been done. … Also somehow one of the commenters didn’t understand the use of ‘flag’ as meaning to lose energy or enthusiasm. Huh.

[ In front of board full of multiplication problems. ] Mrs Olsen: 'Very good. Would you like to do a few more before the bell rings?' Student: 'No, thank you. It's flag day.' [ Later ] Frazz: 'What did that have to do with it?' Student: 'I was beginning to flag.'
Jef Mallett’s Frazz for the 14th of June, 2018. I apologize that I can’t remember this student’s name and I couldn’t find it on a reasonable search. Comic strip About pages need character names.

Jef Mallett’s Frazz for the 15th is a percentages joke, built on confusion between how to go from percentages to fractions and back again. Must say that I had thought 50 percent was tied well enough to one-half in ordinary language (or in phrases like splitting something fifty-fifty) that someone wouldn’t be confused by that. But everyone does miss some obvious things.

Student, to Mrs Olsen: 'If we're just going to forget 60% of this stuff over the summer, why not study only the half of it we'll remember?' [ Later ] Student: 'Annnnnd she doubled our homework.' Frazz: 'What percent of it is math now?'
Jef Mallett’s Frazz for the 15th of June, 2018. I have a similar apology for this student’s name, too. Shall happily accept information on this point.

Mark Pett’s Lucky Cow for the 16th is a probability strip. It is based on what seems obvious, that the fact of any person’s existing is an incredibly unlikely event. We can imagine restarting the universe, and letting it all develop again. And we’re forced to conclude there are so many other ways that galaxies might form and stars might come into being and planets might form and life might develop and evolution might proceed and people might meet and children might be born, and only one way that gets us here. So the chance of any of us existing is impossibly tiny. This is all consistent with the “frequentist” idea of what probability means. In that, we say the probability of a thing happening is all the ways that it could happen divided by all the ways that something could happen. (There are a bunch of technical points to go along with this.)

Clare: 'I need to win the lottery. That would solve all my problems!' Leticia: 'You know, Clare, if you think about it, we've all already *won* the lottery! Each one of us is here because of a long line of happy accidents! Eons ago, our ancestors happened to meet and have children and so on down to our parents! Really, the odds against you or me even being here are *astronomical*!' ... Clare: 'Now I see what they mean when they say winning the lottery can be a curse.'
Mark Pett’s Lucky Cow for the 16th of June, 2018. It originally ran the 20th of August, 2006.

But there are a lot of buried assumptions in there. Many of them seem reasonable. For example: could the universe unfold any differently? It seems obvious that, for example, the radius of the Earth’s orbit around the sun is arbitrary and might be anything in a band that could support life. And, surely, if the year had more or fewer days to it all human history would be different. But then this seems obvious: drop a bunch of short needles across a set of parallel straight lines. The number of needles that cross any of those lines should be arbitrary and unpredictable. Except that it is predictable; there’s a well-known formula that says how many of those needles have to cross those lines. The prediction can be lousy for a handful of needles. For millions of needles, though, it’ll be dead on. The universe won’t make sense any other way.

I can’t go so far as to say that it’s impossible for a universe to exist without me existing and just as I am. That seems egotistical. Even the needle-drop talk has room for variations on the universe. In ten million needle drops, one needle crossing more or less would not be an implausible difference. Ten or a thousand needles falling differently wouldn’t stand out. But, then, after enough needle drops? … If infinitely many needles dropped, I could say exactly what percentage of them crossed lines. (I am speaking so very casually about very difficult technical points. Please pretend I have clear answers for them.) There are deep philosophical questions about the idea of “other universes” that we have to ask if we want to take the subject seriously. But there are deep mathematical questions too.

X figure in a circle: 'DNA tests show I'm related to a Roman beauty by the name of Boderikus Maximus.' Woman: 'Good looking, was she?' X: 'Caesar himself called her a perfect 10.'
Bob Shannon’s Tough Town for the 16th of June, 2018. And the woman here is in nearly every strip and she’s not named either. The About page just talks about Rudolph, “a divorced reindeer working unhappily as a 4th grade teacher” and I think I remember him appearing in the strip back when it started. Oh, I guess that’s him in the title panel on the page, but not in the strip worth mentioning anymore.

Bob Shannon’s Tough Town for the 16th is more or less the anthropomorphized Roman Numerals joke for the week. I don’t know that there’s a strong consensus about why X was used to represent “ten”. Likely it’s impossible to prove any explanation is right. But X has settled into meaning ten, and to serve a host of other uses in typography and in symbols. Some of them are likely connected. Some are probably just coincidence.


If you’d like more of these Reading the Comics posts, you can find them in reverse chronological order at this link. If you’re interested in the comics mentioned particularly here, this page has the B.C. comics (both new and vintage). Frazz is on this page. The Lucky Cow strips are on this page. And Tough Town strips are here.

Reading the Comics, June 4, 2018: Weezer’s Africa Edition


Once again the name of this Reading the Comics edition has nothing to do with any of the strips. I’m just aware that Weezer’s cover of Africa is quite popular right now and who am I to deny people things they want? (I like the cover, but it’s not different enough for me to feel satisfied by it. I tend to like covers that highlight something minor in the original, or that go in a strange direction. Shifting a peppy song into a minor key doesn’t count anymore. But bear in mind, I’m barely competent at listening to music. Please now enjoy my eight hours of early electronica in which various beeps and whistles are passed off as music.)

Samson’s Dark Side of the Horse for the 3rd is the Roman numerals joke for the week. And a welcome return for Dark Side of the Horse. It feels like it’s been gone a while. I wouldn’t try counting by Roman numerals to lull myself to sleep; it seems like too much fussy detail work. But I suppose if you’ve gotten good at it, it’s easy.

Horace, counting sheep jumping over the fence: MCDXCVII; MCDXCIX and the sheep falls over the fence; MD and a sheep with a medical bag runs up to tend the fallen sheep.
Samson’s Dark Side of the Horse for the 3rd of June, 2018. Have to say that’s an adorable medical sheep in the third panel.

Jef Mallett’s Frazz for the 3rd builds on removing statistics from their context. It’s a common problem. It’s possible to measure so very many things. Without a clear idea of what we should expect as normal the measurement doesn’t tell us much. And it can be hard to know what the right context for something even is. Let me deconstruct Caulfield’s example. We’re supposed to reflect on and consider that 40% of all weekdays are Monday and Friday too. But it’s not only weekdays that people work. Even someone working a Sunday might take a sick day. Monday and Friday are a bit over 28% of the whole week. But more people do work Monday-to-Friday than do Saturdays and Sundays, so the Sunday sick day is surely rarer than the Monday. So even if we grant Caulfield’s premise, what does it tell us?

Caulfield: 'Did you know 40% of all sick days are taken on Mondays and Fridays?' Three panels of silence. Caulfield: 'Think about it. ... Did you know 60% of some comic strips is filler?' Frazz: 'If the cartoonist can still make it funny and get outside on the first nice day of spring, I'm cool.'
Jef Mallett’s Frazz for the 3rd of June, 2018. So Jef Mallett lives in the same metro area I do, which means I could in principle use this to figure out how far ahead of deadline he wrote this strip. Except that’s a fraud since we never had a first nice day of spring this year. We just had a duplicate of March for all of April and the first three weeks of May, and then had a week of late July before settling into early summer. Just so you know.

Jason Chatfield’s Ginger Meggs for the 3rd is a bit of why-learn-mathematics propaganda. Megg’s father has a good answer. But it does shift the question back one step. Also I see in the top row that Meggs has one of those comic-strip special editions where the name of the book is printed on the back cover instead. (I’m also skeptical of the photo and text layout on the newspaper Megg’s father is reading. But I don’t know the graphic design style of Australian, as opposed to United States, newspapers.)

Ginger Meggs: 'Dad, do I really need to know how to do maths?' Dad: 'Well, of course you need to know how to do mathematics, Ginger! Think about it! Without maths, you could never become an accountant!' (Ginger and his dog stand there stunned for a panel. Next panel, they're gone. Next panel after that ... ) Mom: 'I suppose you know you just blew it.'
Jason Chatfield’s Ginger Meggs for the 3rd of June, 2018. So … I guess Ginger Megg’s father is an accountant? I’m assuming because it makes the joke land better?

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd may belong on some philosopher’s Reading the Comics blog instead. No matter. There’s some mathematical-enough talk going on here. There’s often many ways to approach the same problem. For example, approaching a system as a handful of items. Or as a huge number of them. Or as infinitely many things. Or as a continuum of things. There are advantages each way. A handful of things, for example, we can often model as interactions between pairs of things. We can model a continuum as a fluid. A vast number of things can let one’s computer numerically approximate a fluid. Or infinitely many particles if that’s more convenient.

Professor: 'Monists believe there is no distinction between mind and body.' (Writes 1/1.) 'Dualists believe mind and body are, in some sense, separate aspects of being.' (Writes 1/2.) 'There's a lively debate here, but the important thing to notice is that both are talking about the same human beings. This proves that you can add 1 to the quantity of aspects of being without altering the being itself.' (Writes 1/3, 1/4, 1/5, 1/6, ... ) 'By induction, you can be a monist, dualist, triplist, quadruplist, and so on. There are literally infinite permitted philosophies in ontology-space! Personally, I am a 10-to-the-27th-powerist, in that I believe every one of the atoms in my body is meaningfully distinct.' Student: 'You've taken a difficult philosophy problem and reduced it to a tractable but pointless math problem.' Professor: 'You may also be interested in my work on free will!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd of June, 2018. Also I’m not sure where the professor figures he’s going with this but my understanding is it’s rather key to our understanding of quantum mechanics that, say, every atom of Carbon-12 in our bodies is the same as every other atom. At least apart from accidental properties like which compound it might happen to be in at the moment and where it is in that compound. That is, if you swapped two of the same isotope there’d be no way to tell you had.

To describe all these different models as sharing an “ontology-space” is good mathematical jargon too. In this context the “-space” would mean the collection of all these things that are built by the same plan but with different values of whichever parameter matters.

Julian writes E = mc^2 on a blackboard. He tells Suzy, 'That's Einstein's theory.' Suzy: 'It's real cute, Julian!'
Bud Blake’s Tiger for the 6th of August, 1965. It was rerun the 4th of June, 2018. I confess I’m not sure exactly what the joke is. If it’s not that Suzy has no idea what’s being written but wants to say something nice about Julian’s work … all right, and I guess that’s an unremarkable attitude for a cartoonist to express in 1965, but it’s a weak joke.

Bud Blake’s Tiger for the 6th of August, 1965 features Einstein’s famous equation. I suppose it’s showing how well-informed Julian is, that he knows and can present such a big result. There is beauty in mathematics (and physics). Mathematicians (and physicists) find the subject beautiful to start with, and try to find attractive results. I’m curious what the lay reader makes of mathematical symbols, though, just as pieces of art. I remember as a child finding this beauty in a table of integrals in the front of one of my mother’s old college textbooks. All those parallel rows of integral symbols drew me in though nothing I’d seen in mathematics had prepared me to even read it. I still find that beautiful, but I can’t swear that I would even if I hadn’t formed that impression early in life. Are lay and professional readers’ views of mathematical-expression beauty similar? How are they different?

Reading the Comics, March 13, 2018: One Of My Assumptions Is Shaken Edition


I learn, from reading not-yet-dead Usenet group rec.arts.comics.strips, that Rick Stromoski is apparently ending the comic Soup To Nutz. This is sad enough. But worse, GoComics.com has removed all but the current day’s strip from its archives. I had trusted that GoComics.com links were reliable in a way that Comics Kingdom and Creators.com weren’t. Now I learn that maybe I need to include images of the comics I review and discuss here lest my essays become unintelligible in the future? That’s not a good sign. I can do it, mind you. I just haven’t got started. You’ll know when I swing into action.

Norm Feuti, of Retail, still draws Sunday strips for Gil. They’re to start appearing on GoComics.com soon, and I can talk about them from my regular sources after that. But for now I follow the strip on Twitter. And last Sunday he posted this one.

It’s sort of a protesting-the-problem question. It’s also a reaction a lot of people have to “explain how you found the answer” questions. In a sense, yeah, the division shows how the answer was found. But what’s wanted — and what’s actually worth learning — is to explain why you did this calculation. Why, in this case, 216 divided by 8? Why not 216 times 8? Why not 8 divided by 216? Why not 216 minus 8? “How you found your answer” is probably a hard question to make interesting on arithmetic, unfortunately. If you’re doing a long sheet of problems practicing division, it’s not hard to guess that dividing is the answer. And that it’s the big number divided by the small. It can be good training to do blocks of problems that use the same approach, for the same reason it can be good training to focus on any exercise a while. But this does cheat someone of the chance to think about why one does this rather than that.

Patrick Roberts’s Todd the Dinosaur for the 11th has mathematics as the thing Todd’s trying to get out of doing. (I suppose someone could try to argue the Y2K bug was an offshoot of mathematics, on the grounds that computer science has so much to do with mathematics. I wouldn’t want to try defending that, though.) I grant that most fraction-to-decimal conversion problems hit that sweet spot of being dull, tedious, and seemingly pointless. There’s some fun decimal expansions of fractions. The sevenths and the elevenths and 1/243 have charm to them. There’s some kid who’ll become a mathematician because at the right age she was told about \frac{1}{8991} . 3/16th? Eh.

Teacher: 'Who would like to come up here and work this converting-fractions-to-decimals problem on the board? Let's see ... how about you, Todd?' Todd: 'Look out! Y2K! AAAGH! This is terrible! Just terrible! It finally caught up with us! Goodbye, electricity! Goodbye, civilized society!' Todd: 'Nice try, Todd. Y2K never happened!' Todd: 'Uh, yeah, I knew that. I was just saying' that Y2K is the answer to that problem on the board!' Teacher: 'Also a nice try. Now get up here!'
Patrick Roberts’s Todd the Dinosaur for the 11th of March, 2018. I’m not sure that the loss of electricity would actually keep someone from doing chalkboard work, especially if there’s as many windows as we see here to let light in. I mean, yes, there’d be problems after school, but just during school? The end of civilization is not the cure-all people present it as being.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I don’t remember seeing a spinny wheel like this used to introduce probability. It’s a good prop, though. I would believe in a class having it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 11th is built on the Travelling Salesman Problem. It’s one of the famous unsolved and hard problems of mathematics. Weinersmith’s joke is a nice gag about one way to “solve” the problem, that of making it irrelevant. But even if we didn’t need to get to a collection of places efficiently mathematicians would still like to know good ways to do it. It turns out that finding the shortest (quickest, cheapest, easiest, whatever) route connecting a bunch of places is great problem. You can phrase enormously many problems about doing something as well as possible as a Travelling Salesman Problem. It’s easy conceptually to find the answer: try out all the possibilities and pick the best one. But if there’s more than a handful of cities, there are so many possible routes there’s no checking them all, not before you die of old age. We can do very well finding approximate answers, including by my specialization of Monte Carlo methods. In those you take a guess at an answer. Then make, randomly, a change. You’ll either have made things better or worse. If you’ve made it better, keep the change. If you’ve made it worse, usually you reject the change but sometimes you keep it. And repeat. In surprisingly little time you’ll get a really good answer. Maybe not the best possible, but a great answer for how straightforward setting it up was.

Dan Thompson’s Brevity for the 12th is a Rubik’s Cube joke. There’s not a lot of mathematics to that. But I do admire how Thompson was careful enough to draw a Rubik’s Cube that actually looks like the real article; it’s not just an isometric cube with thick lines partitioning it. Look at the corners of each colored sub-cube. I may be the only reader to notice this but I’m glad Thompson did the work.

Mason Mastroianni’s The Wizard of Id for the 12th gets Sir Rodney in trouble with the King for doing arithmetic. I haven’t read the comments on GoComics.com. I’d like to enter “three” as my guess for how many comments one would have to read before finding the “weapons of math instruction” joke in there.

Jef Mallett’s Frazz for the 13th has mathematics homework given as the thing lost by the time change. It’s just a cameo mention.

Steve Moore’s In The Bleachers for the 13th features a story problem as a test of mental acuity. When the boxer can’t work out what the heck the trains-leaving-Penn-Station problem even means he’s ruled unfit to keep boxing. The question is baffling, though. As put, the second train won’t ever overtake the first. The question: did Moore just slip up? If the first train were going 30 miles per hour and the second 40 there would be a perfectly good, solvable question in this. Or was Moore slipping in an extra joke, making the referee’s question one that sounds like it was given wrong? Don’t know, so I’ll suppose the second.

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.

Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition


Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.

Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.

John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.

Quincy: 'Someday I'm gonna write a book, Gran.' Grandmom: 'Wonderful. Will you dedicate it to me?' Quincy: 'Sure. In fact, if you want, I'll dedicate this math homework to you.'
Ted Shearer’s Quincy for the 16th of December, 1978 and reprinted the 9th of February, 2018. I’m not sure just what mathematics homework Quincy could be doing to inspire him to write a book, but then, it’s not like my mind doesn’t drift while doing mathematics either. And book-writing’s a common enough daydream that most people are too sensible to act on.

Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.

Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.

Redeye: 'C'mon, Pokey. Time for your lessons. Okay, what do you get when you divide 5,967,342 by 973 ... ?' Pokey: 'A headache!'
Gordon Bess’s Redeye for the 30th of September, 1971 and reprinted the 10th of February, 2018. I realized I didn’t know the father’s name and looked it up, and Wikipedia revealed to me that he’s named Redeye. You know, like the comic strip implies right there in the title. Look, I just read the comics, I can’t be expected to think about the comics too.

Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.

Reading the Comics, January 20, 2018: Increased Workload Edition


It wasn’t much of an increased workload, really. I mean, none of the comics required that much explanation. But Comic Strip Master Command donated enough topics to me last week that I have a second essay for the week. And here it is; sorry there’s no pictures.

Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons we’ve been waiting for. It returns to fractions and their frustrations for its comic point.

Jef Mallet’s Frazz for the 17th talks about story problems, although not to the extent of actually giving one as an example. It’s more about motivating word-problem work.

Mike Thompson’s Grand Avenue for the 17th is an algebra joke. I’d call it a cousin to the joke about mathematics’s ‘x’ not coming back and we can’t say ‘y’. On the 18th was one mentioning mathematics, although in a joke structure that could have been any subject.

Lorrie Ransom’s The Daily Drawing for the 18th is another name-drop of mathematics. I guess it’s easier to use mathematics as the frame for saying something’s just a “problem”. I don’t think of, say, identifying the themes of a story as a problem in the way that finding the roots of a quadratic is.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 18th is an anthropomorphic-geometric-figures joke that I’m all but sure is a rerun I’ve shared here before. I’ll try to remember to check before posting this.

Mikael Wulff and Anders Morgenthaler’s WuMo for the 20th gives us a return of the pie chart joke that seems like it’s been absent a while. Worth including? Eh, why not.

Reading the Comics, January 16, 2017: Better Workflow Edition


So one little secret of my Reading the Comics posts is I haven’t been writing them in a way that makes sense to me. To me, I should take each day’s sufficiently relevant comics, describe them in a paragraph or two, and then have a nice pile of text all ready for the posting Sunday and, if need be, later. I haven’t been doing that. I’ve let links pile up until Friday or Saturday, and then try to process them all, and if you’ve ever wondered why the first comic of the week gets 400 words about some subtlety while the last gets “this is a comic that exists”, there you go. This time around, let me try doing each day’s strips per day and see how that messes things up.

Jef Mallett’s Frazz for the 14th of January is another iteration of the “when will we ever use mathematics” complaint. The answer of “you’ll use it on the test” is unsatisfactory. But somehow, the answer of “you’ll use it to think deeply about something you had never considered before” also doesn’t satisfy. Anyway I’d like to see the idea that education is job-training abolished; I think it should be about making a person conversant with the history of human thought. That can’t be done perfectly, and we might ask whether factoring 32 is that important a piece, but it should certainly be striven for.

Ham’s Life on Earth for the 14th is a Gary Larsonesque riff on that great moment of calculus and physics history, Newton’s supposition that gravity has to follow a universally true law. I’m not sure this would have made my cut if I reviewed a week’s worth of strips at a time. Hm.

Mason Mastroianni’s B.C. for the 15th is a joke about story problem construction, and how the numbers in a story problem might be obvious nonsense. It’s also a cheap shot at animal hoarders, I suppose, but that falls outside my territory here.

Anthony Blades’s Bewley rerun for the 15th riffs on the natural number sense we all have. And we do have a number sense, remarkably. We might not be able to work out 9 times 6 instantly. But asked to pick from a list of possible values, we’re more likely to think that 58 is credible than that 78 or 38 are. It’s quite imprecise, but isn’t it amazing that it’s there at all?

Bill Amend’s FoxTrot Classics for the 15th is a story problem joke, in this case, creating one with a strong motivation for its solution to be found. The strip originally ran the 22nd of January, 1996.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th is maybe marginal to include, too. It’s about the kinds of logic puzzles that mathematicians grow up reading and like to pass around. And the way you can fake out someone by presenting a problem with too obvious a solution. It’s not just professors who’ll be stymied by having the answer look too obvious, by the way. Everyone’s similarly vulnerable. To see anything, including an abstract thing like the answer to a puzzle, you need some idea of what you are looking at. If you don’t think the answer could be something that simple, you won’t see it there.

Paw: 'It's four o'clock ... what time are we going to eat?' Maw :'About five.' Paw: 'Good! That gives me two hours to work with Pokey on his arithmeteic.'
Gordon Bess’s Redeye for the 6th of September, 1971. That’s the sort of punch line that really brings out the comically-anachronistic Old West theme.

Gordon Bess’s Redeye for the 6th of September, 1971, was reprinted the 17th. It’s about the fun of teaching a subject you aren’t all that good on yourself. The mathematics is a name-drop here, but the joke wouldn’t make sense if it were about social studies.

Popeye: 'King, they's one thing I wants to know. How much is a pezozee?' King Blozo: 'Why bring that up?' Popeye: 'Yer men hired me to help lick yer emeny at a thousing pezozees a week - tha's why I'd like to know what is a pezozee.' Blozo: 'A pezozee is two pazookas.' Popeye: 'What's a pazooky?' Blozo: 'A pazooka is two pazinkas.' Popeye: 'What's a pazinky?' Blozo: 'A pazinka is two pazoonies.' Popeye: 'What's a pazeenya?' Blozo: 'Phooey! I wish you would quit following me! A pazooney is two pazeenyas.' Popeye: 'what's a pazeenya?' Blozo: 'Two pazimees.' Popeye: 'Hey! What's a pazimee worth?' Blozo: 'Absolutely nothing!' Popeye: 'Blow me down, I'm glad I ain't gettin' paid in pazimees!'
Elzie Segar’s Thimble Theatre for the 10th of August, 1931. Not listed: the rate of exchange for paczki, which reappeared this week.

Elzie Segar’s Thimble Theatre for the 10th of August, 1931, was also reprinted the 17th. It’s an old gag, even back when it was first run. But I suppose there’s some numerical-conversion mathematics to wring out of it. Given the rate of exchange, a pezozee would seem to be 24 pazimees. I’m not sure we need so many units in-between the pazimee and the pezozee, but perhaps King Blozo’s land set its units in a time when fractions were less familiar to the public. The punch line depends on the pazimee being worth nothing and, taken literally, that has sad implications for the pezozee too. If you take the King as speaking roughly, though, sixteen times a small amount is … at least a less small amount. It wouldn’t take many doublings to go from an infinitesimally tiny sum to a respectable one.

And it turns out there were enough comic strips I need to split this into two segments. So I should schedule that to appear. It’s already written and everything.

Reading the Comics, January 3, 2018: Explaining Things Edition


There were a good number of mathematically-themed comic strips in the syndicated comics last week. Those from the first part of the week gave me topics I could really sink my rhetorical teeth into, too. So I’m going to lop those off into the first essay for last week and circle around to the other comics later on.

Jef Mallett’s Frazz started a week of calendar talk on the 31st of December. I’ve usually counted that as mathematical enough to mention here. The 1st of January as we know it derives, as best I can figure, from the 1st of January as Julius Caesar established for 45 BCE. This was the first Roman calendar to run basically automatically. Its length was quite close to the solar year’s length. It had leap days added according to a rule that should have been easy enough to understand (one day every fourth year). Before then the Roman calendar year was far enough off the solar year that they had to be kept in synch by interventions. Mostly, by that time, adding a short extra month to put things more nearly right. This had gotten all confusingly messed up and Caesar took the chance to set things right, running 46 BCE to 445 days long.

But why 445 and not, say, 443 or 457? And I find on research that my recollection might not be right. That is, I recall that the plan was to set the 1st of January, Reformed, to the first new moon after the winter solstice. A choice that makes sense only for that one year, but, where to set the 1st is literally arbitrary. While that apparently passes astronomical muster (the new moon as seen from Rome then would be just after midnight the 2nd of January, but hitting the night of 1/2 January is good enough), there’s apparently dispute about whether that was the objective. It might have been to set the winter solstice to the 25th of December. Or it might have been that the extra days matched neatly the length of two intercalated months that by rights should have gone into earlier years. It’s a good reminder of the difficulty of reading motivation.

Brian Fies’s The Last Mechanical Monster for the 1st of January, 2018, continues his story about the mad scientist from the Fleischer studios’ first Superman cartoon, back in 1941. In this panel he’s describing how he realized, over the course of his long prison sentence, that his intelligence was fading with age. He uses the ability to do arithmetic in his head as proof of that. These types never try naming, like, rulers of the Byzantine Empire. Anyway, to calculate the cube root of 50,653 in his head? As he used to be able to do? … guh. It’s not the sort of mental arithmetic that I find fun.

But I could think of a couple ways to do it. The one I’d use is based on a technique called Newton-Raphson iteration that can often be used to find where a function’s value is zero. Raphson here is Joseph Raphson, a late 17th century English mathematician known for the Newton-Raphson method. Newton is that falling-apples fellow. It’s an iterative scheme because you start with a guess about what the answer would be, and do calculations to make the answer better. I don’t say this is the best method, but it’s the one that demands me remember the least stuff to re-generate the algorithm. And it’ll work for any positive number ‘A’ and any root, to the ‘n’-th power.

So you want the n-th root of ‘A’. Start with your current guess about what this root is. (If you have no idea, try ‘1’ or ‘A’.) Call that guess ‘x’. Then work out this number:

\frac{1}{n}\left( (n - 1) \cdot x + \frac{A}{x^{n - 1}} \right)

Ta-da! You have, probably, now a better guess of the n-th root of ‘A’. If you want a better guess yet, take the result you just got and call that ‘x’, and go back calculating that again. Stop when you feel like your answer is good enough. This is going to be tedious but, hey, if you’re serving a prison term of the length of US copyright you’ve got time. (It’s possible with this sort of iterator to get a worse approximation, although I don’t think that happens with n-th root process. Most of the time, a couple more iterations will get you back on track.)

But that’s work. Can we think instead? Now, most n-th roots of whole numbers aren’t going to be whole numbers. Most integers aren’t perfect powers of some other integer. If you think 50,653 is a perfect cube of something, though, you can say some things about it. For one, it’s going to have to be a two-digit number. 103 is 1,000; 1003 is 1,000,000. The second digit has to be a 7. 73 is 343. The cube of any number ending in 7 has to end in 3. There’s not another number from 1 to 9 with a cube that ends in 3. That’s one of those things you learn from playing with arithmetic. (A number ending in 1 cubes to something ending in 1. A number ending in 2 cubes to something ending in 8. And so on.)

So the cube root has to be one of 17, 27, 37, 47, 57, 67, 77, 87, or 97. Again, if 50,653 is a perfect cube. And we can do better than saying it’s merely one of those nine possibilities. 40 times 40 times 40 is 64,000. This means, first, that 47 and up are definitely too large. But it also means that 40 is just a little more than the cube root of 50,653. So, if 50,653 is a perfect cube, then it’s most likely going to be the cube of 37.

Bill Watterson’s Calvin and Hobbes rerun for the 2nd is a great sequence of Hobbes explaining arithmetic to Calvin. There is nothing which could be added to Hobbes’s explanation of 3 + 8 which would make it better. I will modify Hobbes’s explanation of what the numerator. It’s ridiculous to think it’s Latin for “number eighter”. The reality is possibly more ridiculous, as it means “a numberer”. Apparently it derives from “numeratus”, meaning, “to number”. The “denominator” comes from “de nomen”, as in “name”. So, you know, “the thing that’s named”. Which does show the terms mean something. A poet could turn “numerator over denominator” into “the number of parts of the thing we name”, or something near enough that.

Hobbes continues the next day, introducing Calvin to imaginary numbers. The term “imaginary numbers” tells us their history: they looked, when first noticed in formulas for finding roots of third- and fourth-degree polynomials, like obvious nonsense. But if you carry on, following the rules as best you can, that nonsense would often shake out and you’d get back to normal numbers again. And as generations of mathematicians grew up realizing these acted like numbers we started to ask: well, how is an imaginary number any less real than, oh, the square root of six?

Hobbes’s particular examples of imaginary numbers — “eleventenn” and “thirty-twelve” — are great-sounding compositions. They put me in mind, as many of Watterson’s best words do, of a 1960s Peanuts in which Charlie Brown is trying to help Sally practice arithmetic. (I can’t find it online, as that meme with edited text about Sally Brown and the sixty grapefruits confounds my web searches.) She offers suggestions like “eleventy-Q” and asks if she’s close, which Charlie Brown admits is hard to say.

Cherry Trail: 'Good morning, honey! Where's Dad?' Mark Trail: 'He's out on the porch reading the paper!' Cherry: 'Rusty sure is excited about our upcoming trip to Mexico!' Mark: 'Did you get everything worked out with the school?' Cherry: 'Rusty will need to do some math assignments, but he'll get credit for his other subjects since it's an educational trip!'
James Allen’s Mark Trail for the 3rd of January, 2018. James Allen has changed many things about the comic strip since Jack Elrod’s retirement, as I have observed over on the other blog. There are less ruthlessly linear stories. There’s no more odd word balloon placement implying that giant squirrels are talking about the poachers. Mark Trail sometimes has internal thoughts. I’m glad that he does still choose to over-emphasize declarations like “[Your Dad]’s out on the porch reading the paper!” There are some traditions.

And finally, James Allen’s Mark Trail for the 3rd just mentions mathematics as the subject that Rusty Trail is going to have to do some work on instead of allowing the experience of a family trip to Mexico to count. This is of extremely marginal relevance, but it lets me include a picture of a comic strip, and I always like getting to do that.

Reading the Comics, December 30, 2017: Looking To 2018 Edition


The last full week of 2017 was also a slow one for mathematically-themed comic strips. You can tell by how many bits of marginally relevant stuff I include. In this case, it also includes a couple that just mention the current or the upcoming year. So you’ve been warned.

Mac King and Bill King’s Magic in a Minute activity for the 24th is a logic puzzle. I’m not sure there’s deep mathematics to it, but it’s some fun to reason out.

John Graziano’s Ripley’s Believe It Or Not for the 24th mentions the bit of recreational group theory that normal people know, the Rubik’s Cube. The group theory comes in from rotations: you can take rows or columns on the cube and turn them, a quarter or a half or a three-quarters turn. Which rows you turn, and which ways you turn them, form a group. So it’s a toy that inspires deep questions. Who wouldn’t like to know in how few moves a cube could be solved? We know there are at least some puzzles that take 18 moves to solve. (You can calculate the number of different cube arrangements there are, and how many arrangements you could make by shuffling a cube around with 17 moves. There’s more possible arrangements than there are ones you can get to in 17 moves; therefore, there must be at least one arrangement that takes 18 moves to solve.) A 2010 computer-assisted proof by Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge showed that at most 20 face turns are needed for every possible cube to be solved. I don’t know if there’s been any success figuring out whether 19 or even 18 is necessarily enough.

Griffith: 'Here we are, Zippy, back in the land of our childhood.' Zippy: 'Are we still in the ninth grade?' Griffith: 'Kind of ... although I still can't remember a thing about algebra.' Zippy: 'So many spitballs and paper airplanes ago!!' Griffith: 'Why did I act up so much in school, Zippy? Was it a Freudian thing?' Zippy: 'It was a cry for kelp.' Griffith: 'Don't you mean a cry for help? I don't think kelp was even a word I knew back in the 50s.' Zippy: 'Seaweed is the fifth dimension!'
Bill Griffith’s Zippy the Pinhead for the 26th of December, 2017. This is not as strongly a memoir or autobiographical strip as Griffith will sometimes do, which is a shame. Those are always captivating. I have fun reading Zippy the Pinhead and understand why people wouldn’t. But the memoir strips I recommend even to people who don’t care for the usual fare.

Bill Griffith’s Zippy the Pinhead for the 26th just mentions algebra as a thing that Griffith can’t really remember, even in one of his frequent nostalgic fugues. I don’t know that Zippy’s line about the fifth dimension is meant to refer to geometry. It might refer to the band, but that would be a bit odd. Yes, I know, Zippy the Pinhead always speaks oddly, but in these nostalgic fugue strips he usually provides some narrative counterpoint.

Larry Wright’s Motley Classics for the 26th originally ran in 1986. I mention this because it makes the odd dialogue of getting “a new math program” a touch less odd. I confess I’m not sure what the kid even got. An educational game? Something for numerical computing? The coal-fired, gear-driven version of Mathematica that existed in the 1980s? It’s a mystery, it is.

Ryan Pagelow’s Buni for the 27th is really a calendar joke. It seems to qualify as an anthropomorphic numerals joke, though. It’s not a rare sentiment either.

Jef Mallett’s Frazz for the 29th is similarly a calendar joke. It does play on 2017 being a prime number, a fact that doesn’t really mean much besides reassuring us that it’s not a leap year. I’m not sure just what’s meant by saying it won’t repeat for another 2017 years, at least that wouldn’t be just as true for (say) 2015 or 2019. But as Frazz points out, we do cling to anything that floats in times like these.

Reading the Comics, November 18, 2017: Story Problems and Equation Blackboards Edition


It was a normal-paced week at Comic Strip Master Command. It was also one of those weeks that didn’t have anything from Comics Kingdom or Creators.Com. So I’m afraid you’ll all just have to click the links for strips you want to actually see. Sorry.

Bill Amend’s FoxTrot for the 12th has Jason and Marcus creating “mathic novels”. They, being a couple of mathematically-gifted smart people, credit mathematics knowledge with smartness. A “chiliagon” is a thousand-sided regular polygon that’s mostly of philosophical interest. A regular polygon with a thousand equal sides and a thousand equal angles looks like a circle. There’s really no way to draw one so that the human eye could see the whole figure and tell it apart from a circle. But if you can understand the idea of a regular polygon it seems like you can imagine a chilagon and see how that’s not a circle. So there’s some really easy geometry things that can’t be visualized, or at least not truly visualized, and just have to be reasoned with.

Rick Detorie’s One Big Happy for the 12th is a story-problem-subversion joke. The joke’s good enough as it is, but the supposition of the problem is that the driving does cover fifty miles in an hour. This may not be the speed the car travels at the whole time of the problem. Mister Green is maybe speeding to make up for all the time spent travelling slower.

Brandon Sheffield and Dami Lee’s Hot Comics for Cool People for the 13th uses a blackboard full of equations to represent the deep thinking being done on a silly subject.

Shannon Wheeler’s Too Much Coffee Man for the 15th also uses a blackboard full of equations to represent the deep thinking being done on a less silly subject. It’s a really good-looking blackboard full of equations, by the way. Beyond the appearance of our old friend E = mc2 there’s a lot of stuff that looks like legitimate quantum mechanics symbols there. They’re at least not obvious nonsense, as best I can tell without the ability to zoom the image in. I wonder if Wheeler didn’t find a textbook and use some problems from it for the feeling of authenticity.

Samson’s Dark Side of the Horse for the 16th is a story-problem subversion joke.

Jef Mallett’s Frazz for the 18th talks about making a bet on the World Series, which wrapped up a couple weeks ago. It raises the question: can you bet on an already known outcome? Well, sure, you can bet on anything you like, given a willing partner. But there does seem to be something fundamentally different between betting on something whose outcome isn’t in principle knowable, such as the winner of the next World Series, and betting on something that could be known but happens not to be, such as the winner of the last. We see this expressed in questions like “is it true the 13th of a month is more likely to be Friday than any other day of the week?” If you know which month and year is under discussion the chance the 13th is Friday is either 1 or 0. But we mean something more like, if we don’t know what month and year it is, what’s the chance this is a month with a Friday the 13th? Something like this is at work in this World Series bet. (The Astros won the recently completed World Series.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is also featured on some underemployed philosopher’s “Reading the Comics” WordPress blog and fair enough. Utilitarianism exists in an odd triple point, somewhere on the borders of ethics, economics, and mathematics. The idea that one could quantize the good or the utility or the happiness of society, and study how actions affect it, is a strong one. It fits very well the modern mindset that holds everything can be quantified even if we don’t know how to do it well just yet. And it appeals strongly to a mathematically-minded person since it sounds like pure reason. It’s not, of course, any more than any ethical scheme can be. But it sounds like the ethics a Vulcan would come up with and that appeals to a certain kind of person. (The comic is built on one of the implications of utilitarianism that makes it seem like the idea’s gone off the rails.)

There’s some mathematics symbols on The Utilitarian’s costume. The capital U on his face is probably too obvious to need explanation. The \sum u on his chest relies on some mathematical convention. For maybe a half-millennium now mathematicians have been using the capital sigma to mean “take a sum of things”. The things are whatever the expression after that symbol is. Usually, the Sigma will have something below and above which carries meaning. It says what the index is for the thing after the symbol, and what the bounds of the index are. Here, it’s not set. This is common enough, though, if this is understood from context. Or if it’s obvious. The small ‘u’ to the right suggests the utility of whatever’s thought about. (“Utility” being the name for the thing measured and maximized; it might be happiness, it might be general well-being, it might be the number of people alive.) So the symbols would suggest “take the sum of all the relevant utilities”. Which is the calculation that would be done in this case.

Reading the Comics, November 8, 2017: Uses Of Mathematics Edition


Was there an uptick in mathematics-themed comic strips in the syndicated comics this past week? It depends how tight a definition of “theme” you use. I have enough to write about that I’m splitting the week’s load. And I’ve got a follow-up to that Wronski post the other day, so I’m feeling nice and full of content right now. So here goes.

Zach Weinersmith’s Saturday Morning Breakfast Cereal posted the 5th gets my week off to an annoying start. Science and mathematics and engineering people have a tendency to be smug about their subjects. And to see aptitude or interest in their subjects as virtue, or at least intelligence. (If they see a distinction between virtue and intelligence.) To presume that an interest in the field I like is a demonstration of intelligence is a pretty nasty and arrogant move.

And yes, I also dislike the attitude that school should be about training people. Teaching should be about letting people be literate with the great thoughts people have had. Mathematics has a privileged spot here. The field, as we’ve developed it, seems to build on human aptitudes for number and space. It’s easy to find useful sides to it. Doesn’t mean it’s vocational training.

Lincoln Peirce’s Big Nate on the 6th discovered mathematics puzzles. And this gave him the desire to create a new mathematical puzzle that he would use to get rich. Good luck with that. Coming up with interesting enough recreational mathematics puzzles is hard. Presenting it in a way that people will buy is another, possibly greater, challenge. It takes luck and timing and presentation, just as getting a hit song does. Sudoku, for example, spent five years in the Dell Magazine puzzle books before getting a foothold in Japanese newspapers. And then twenty years there before being noticed in the English-speaking puzzle world. Big Nate’s teacher tries to encourage him, although that doesn’t go as Mr Staples might have hoped. (The storyline continues to the 11th. Spoiler: Nate does not invent the next great recreational mathematics puzzle.)

Jef Mallett’s Frazz for the 7th start out in a mathematics class, at least. I suppose the mathematical content doesn’t matter, though. Mallett’s making a point about questions that, I confess, I’m not sure I get. I’ll leave it for wiser heads to understand.

Mike Thompson’s Grand Avenue for the 8th is a subverted word-problem joke. And I suppose a reminder about the need for word problems to parse as things people would do, or might be interested in. I can’t go along with characterizing buying twelve candy bars “gluttonous” though. Not if they’re in a pack of twelve or something like that. I may be unfair to Grand Avenue. Mind, until a few years ago I was large enough my main method of getting around was “being rolled by Oompa-Loompas”, so I could be a poor judge.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 8th does a rounding joke. It’s not much, but I’ve included appearances of this joke before and it seems unfair to skip it this time.

Reading the Comics, September 24, 2017: September 24, 2017 Edition


Comic Strip Master Command sent a nice little flood of comics this week, probably to make sure that I transitioned from the A To Z project to normal activity without feeling too lost. I’m going to cut the strips not quite in half because I’m always delighted when I can make a post that’s just a single day’s mathematically-themed comics. Last Sunday, the 24th of September, was such a busy day. I’m cheating a little on what counts as noteworthy enough to talk about here. But people like comic strips, and good on them for liking them.

Norm Feuti’s Gil for the 24th sees Gil discover and try to apply some higher mathematics. There’s probably a good discussion about what we mean by division to explain why Gil’s experiment didn’t pan out. I would pin it down to eliding the difference between “dividing in half” and “dividing by a half”, which is a hard one. Terms that seem almost alike but mean such different things are probably the hardest part of mathematics.

Gil, eating cookies and doing mathematics. 'Dividing fractions. 1/2 divided by 1/2', which he works out to be 1. 'One half divided in half equals one? Wait a minute. If these calculations are correct, then that means ... ' And he takes a half-cookie and snaps it in half, to his disappointment. 'Humph. what's the point of this advanced math if it only works on paper?'
Norm Feuti’s Gil for the 24th of September, 2017, didn’t appear on Gocomics.com or Comics Kingdom, my usual haunts for these comics. But I started reading the strip when it was on Comics Kingdom, and keep reading its reruns. Feuti has continued the comic strip on his own web site, and posts it on Twitter. So it’s quite easy to pick the strip back up, if you have a Twitter account or can read RSS from it. I assume you can read RSS from it.

Russell Myers’s Broom Hilda looks like my padding. But the last panel of the middle row gets my eye. The squirrels talk about how on the equinox night and day “can never be of identical length, due to the angular size of the sun and atmospheric refraction”. This is true enough for the equinox. While any spot on the Earth might see twelve hours facing the sun and twelve hours facing away, the fact the sun isn’t a point, and that the atmosphere carries light around to the “dark” side of the planet, means daylight lasts a little longer than night.

Ah, but. This gets my mathematical modelling interest going. Because it is true that, at least away from the equator, there’s times of year that day is way shorter than night. And there’s times of year that day is way longer than night. Shouldn’t there be some time in the middle when day is exactly equal to night?

The easy argument for is built on the Intermediate Value Theorem. Let me define a function, with domain each of the days of the year. The range is real numbers. It’s defined to be the length of day minus the length of night. Let me say it’s in minutes, but it doesn’t change things if you argue that it’s seconds, or milliseconds, or hours, if you keep parts of hours in also. So, like, 12.015 hours or something. At the height of winter, this function is definitely negative; night is longer than day. At the height of summer, this function is definitely positive; night is shorter than day. So therefore there must be some time, between the height of winter and the height of summer, when the function is zero. And therefore there must be some day, even if it isn’t the equinox, when night and day are the same length

There’s a flaw here and I leave that to classroom discussions to work out. I’m also surprised to learn that my onetime colleague Dr Helmer Aslaksen’s grand page of mathematical astronomy and calendar essays doesn’t seem to have anything about length of day calculations. But go read that anyway; you’re sure to find something fascinating.

Mike Baldwin’s Cornered features an old-fashioned adding machine being used to drown an audience in calculations. Which makes for a curious pairing with …

Bill Amend’s FoxTrot, and its representation of “math hipsters”. I hate to encourage Jason or Marcus in being deliberately difficult. But there are arguments to make for avoiding digital calculators in favor of old-fashioned — let’s call them analog — calculators. One is that people understand tactile operations better, or at least sooner, than they do digital ones. The slide rule changes multiplication and division into combining or removing lengths of things, and we probably have an instinctive understanding of lengths. So this should train people into anticipating what a result is likely to be. This encourages sanity checks, verifying that an answer could plausibly be right. And since a calculation takes effort, it encourages people to think out how to arrange the calculation to require less work. This should make it less vulnerable to accidents.

I suspect that many of these benefits are what you get in the ideal case, though. Slide rules, and abacuses, are no less vulnerable to accidents than anything else is. And if you are skilled enough with the abacus you have no trouble multiplying 18 by 7, you probably would not find multiplying 17 by 8 any harder, and wouldn’t notice if you mistook one for the other.

Jef Mallett’s Frazz asserts that numbers are cool but the real insight is comparisons. And we can argue that comparisons are more basic than numbers. We can talk about one thing being bigger than another even if we don’t have a precise idea of numbers, or how to measure them. See every mathematics blog introducing the idea of different sizes of infinity.

Bill Whitehead’s Free Range features Albert Einstein, universal symbol for really deep thinking about mathematics and physics and stuff. And even a blackboard full of equations for the title panel. I’m not sure whether the joke is a simple absent-minded-professor joke, or whether it’s a relabelled joke about Werner Heisenberg. Absent-minded-professor jokes are not mathematical enough for me, so let me point once again to American Cornball. They’re the first subject in Christopher Miller’s encyclopedia of comic topics. So I’ll carry on as if the Werner Heisenberg joke were the one meant.

Heisenberg is famous, outside World War II history, for the Uncertainty Principle. This is one of the core parts of quantum mechanics, under which there’s a limit to how precisely one can know both the position and momentum of a thing. To identify, with absolutely zero error, where something is requires losing all information about what its momentum might be, and vice-versa. You see the application of this to a traffic cop’s question about knowing how fast someone was going. This makes some neat mathematics because all the information about something is bundled up in a quantity called the Psi function. To make a measurement is to modify the Psi function by having an “operator” work on it. An operator is what we call a function that has domains and ranges of other functions. To measure both position and momentum is equivalent to working on Psi with one operator and then another. But these operators don’t commute. You get different results in measuring momentum and then position than you do measuring position and then momentum. And so we can’t know both of these with infinite precision.

There are pairs of operators that do commute. They’re not necessarily ones we care about, though. Like, the total energy commutes with the square of the angular momentum. So, you know, if you need to measure with infinite precision the energy and the angular momentum of something you can do it. If you had measuring tools that were perfect. You don’t, but you could imagine having them, and in that case, good. Underlying physics wouldn’t spoil your work.

Probably the panel was an absent-minded professor joke.

Reading the Comics, September 22, 2017: Doughnut-Cutting Edition


The back half of last week’s mathematically themed comic strips aren’t all that deep. They make up for it by being numerous. This is how calculus works, so, good job, Comic Strip Master Command. Here’s what I have for you.

Mark Anderson’s Andertoons for the 20th marks its long-awaited return to these Reading The Comics posts. It’s of the traditional form of the student misunderstanding the teacher’s explanations. Arithmetic edition.

Marty Links’s Emmy Lou for the 20th was a rerun from the 22nd of September, 1976. It’s just a name-drop. It’s not like it matters for the joke which textbook was lost. I just include it because, what the heck, might as well.

Jef Mallett’s Frazz for the 21st uses the form of a story problem. It’s a trick question anyway; there’s really no way the Doppler effect is going to make an ice cream truck’s song unrecognizable, not even at highway speeds. Too distant to hear, that’s a possibility. Also I don’t know how strictly regional this is but the ice cream trucks around here have gone in for interrupting the music every couple seconds with some comical sound effect, like a “boing” or something. I don’t know what this hopes to achieve besides altering the timeline of when the ice cream seller goes mad.

Mark Litzler’s Joe Vanilla for the 21st I already snuck in here last week, in talking about ‘x’. The variable does seem like a good starting point. And, yeah, hypothesis block is kind of a thing. There’s nothing quite like staring at a problem that should be interesting and having no idea where to start. This happens even beyond grade school and the story problems you do then. What to do about it? There’s never one thing. Study it a good while, read about related problems a while. Maybe work on something that seems less obscure a while. It’s very much like writer’s block.

Ryan North’s Dinosaur Comics rerun for the 22nd straddles the borders between mathematics, economics, and psychology. It’s a problem about making forecasts about other people’s behavior. It’s a mystery of game theory. I don’t know a proper analysis for this game. I expect it depends on how many rounds you get to play: if you have a sense of what people typically do, you can make a good guess of what they will do. If everyone gets a single shot to play, all kinds of crazy things might happen.

Jef Mallet’s Frazz gets in again on the 22nd with some mathematics gibberish-talk, including some tossing around of the commutative property. Among other mistakes Caulfield was making here, going from “less is more to therefore more is less” isn’t commutation. Commutation is about binary operations, where you match a pair of things to a single thing. The operation commutes if it never matters what the order of the pair of things is. It doesn’t commute if it ever matters, even a single time, what the order is. Commutativity gets introduced in arithmetic where there are some good examples of the thing. Addition and multiplication commute. Subtraction and division don’t. From there it gets forgotten until maybe eventually it turns up in matrix multiplication, which doesn’t commute. And then it gets forgotten once more until maybe group theory. There, whether operations commute or not is as important a divide as the one between vertebrates and invertebrates. But I understand kids not getting why they should care about commuting. Early on it seems like a longwinded way to say what’s obvious about addition.

Michael Cavna’s Warped for the 22nd is the Venn Diagram joke for this round of comics.

Hugo: 'There's three of us and I have four doughnuts, it won't divide ... so I'll have to eat the extra one!' Punkinhead: 'Wait, Hugo, I can solve it, I'll go get my brother.'
Bud Blake’s Tiger rerun for the 23rd of September, 2017. Do have to wonder what’s going through Julian’s head. On the one hand, he’s getting one doughnut, come what may. On the other, he’s really not needed for the joke since it would play just as well with three doughnuts to split between Hugo and Punkinhead. I suppose cutting a doughnut in thirds is more unthinkable than cutting a doughnut in half, but neither one’s an easy thing for me to imagine.

Bud Blake’s Tiger rerun for the 23rd starts with a real-world example of your classic story problem. I like the joke in it, and I also like Hugo’s look of betrayal and anger in the second panel. A spot of expressive art will do so good for a joke.

Reading the Comics, July 22, 2017: Counter-mudgeon Edition


I’m not sure there is an overarching theme to the past week’s gifts from Comic Strip Master Command. If there is, it’s that I feel like some strips are making cranky points and I want to argue against their cases. I’m not sure what the opposite of a curmudgeon is. So I shall dub myself, pending a better idea, a counter-mudgeon. This won’t last, as it’s not really a good name, but there must be a better one somewhere. We’ll see it, now that I’ve said I don’t know what it is.

Rabbits at a chalkboard. 'The result is not at all what we expected, Von Thump. According to our calculations, parallel universes may exist, and we may also be able to link them with our own by wormholes that, in strictly mathematical terms, end up in a black top hat.'
Niklas Eriksson’s Carpe Diem for the 17th of July, 2017. First, if anyone isn’t thinking of that Pixar short then I’m not sure we can really understand each other. Second, ‘von Thump’ is a fine name for a bunny scientist and if it wasn’t ever used in the rich lore of Usenet group alt.devilbunnies I shall be disappointed. Third, Eriksson made an understandable but unfortunate mistake in composing this panel. While both rabbits are wearing glasses, they’re facing away from the viewer. It’s always correct to draw animals wearing eyeglasses, or to photograph them so. But we should get to see them in full eyeglass pelage. You’d think they would teach that in Cartoonist School or something.

Niklas Eriksson’s Carpe Diem for the 17th features the blackboard full of equations as icon for serious, deep mathematical work. It also features rabbits, although probably not for their role in shaping mathematical thinking. Rabbits and their breeding were used in the simple toy model that gave us Fibonacci numbers, famously. And the population of Arctic hares gives those of us who’ve reached differential equations a great problem to do. The ecosystem in which Arctic hares live can be modelled very simply, as hares and a generic predator. We can model how the populations of both grow with simple equations that nevertheless give us surprises. In a rich, diverse ecosystem we see a lot of population stability: one year where an animal is a little more fecund than usual doesn’t matter much. In the sparse ecosystem of the Arctic, and the one we’re building worldwide, small changes can have matter enormously. We can even produce deterministic chaos, in which if we knew exactly how many hares and predators there were, and exactly how many of them would be born and exactly how many would die, we could predict future populations. But the tiny difference between our attainable estimate and the reality, even if it’s as small as one hare too many or too few in our model, makes our predictions worthless. It’s thrilling stuff.

Vic Lee’s Pardon My Planet for the 17th reads, to me, as a word problem joke. The talk about how much change Marian should get back from Blake could be any kind of minor hassle in the real world where one friend covers the cost of something for another but expects to be repaid. But counting how many more nickels one person has than another? That’s of interest to kids and to story-problem authors. Who else worries about that count?

Fortune teller: 'All of your money problems will soon be solved, including how many more nickels Beth has than Jonathan, and how much change Marian should get back from Blake.'
Vic Lee’s Pardon My Planet for the 17th of July, 2017. I am surprised she had no questions about how many dimes Jonathan must have, although perhaps that will follow obviously from knowing the Beth nickel situation.

Jef Mallet’s Frazz for the 17th straddles that triple point joining mathematics, philosophy, and economics. It seems sensible, in an age that embraces the idea that everything can be measured, to try to quantify happiness. And it seems sensible, in age that embraces the idea that we can model and extrapolate and act on reasonable projections, to try to see what might improve our happiness. This is so even if it’s as simple as identifying what we should or shouldn’t be happy about. Caulfield is circling around the discovery of utilitarianism. It’s a philosophy that (for my money) is better-suited to problems like how ought the city arrange its bus lines than matters too integral to life. But it, too, can bring comfort.

Corey Pandolph’s Barkeater Lake rerun for the 20th features some mischievous arithmetic. I’m amused. It turns out that people do have enough of a number sense that very few people would let “17 plus 79 is 4,178” pass without comment. People might not be able to say exactly what it is, on a glance. If you answered that 17 plus 79 was 95, or 102, most people would need to stop and think about whether either was right. But they’re likely to know without thinking that it can’t be, say, 56 or 206. This, I understand, is so even for people who aren’t good at arithmetic. There is something amazing that we can do this sort of arithmetic so well, considering that there’s little obvious in the natural world that would need the human animal to add 17 and 79. There are things about how animals understand numbers which we don’t know yet.

Alex Hallatt’s Human Cull for the 21st seems almost a direct response to the Barkeater Lake rerun. Somehow “making change” is treated as the highest calling of mathematics. I suppose it has a fair claim to the title of mathematics most often done. Still, I can’t get behind Hallatt’s crankiness here, and not just because Human Cull is one of the most needlessly curmudgeonly strips I regularly read. For one, store clerks don’t need to do mathematics. The cash registers do all the mathematics that clerks might need to do, and do it very well. The machines are cheap, fast, and reliable. Not using them is an affectation. I’ll grant it gives some charm to antiques shops and boutiques where they write your receipt out by hand, but that’s for atmosphere, not reliability. And it is useful the clerk having a rough idea what the change should be. But that’s just to avoid the risk of mistakes getting through. No matter how mathematically skilled the clerk is, there’ll sometimes be a price entered wrong, or the customer’s money counted wrong, or a one-dollar bill put in the five-dollar bill’s tray, or a clerk picking up two nickels when three would have been more appropriate. We should have empathy for the people doing this work.

Reading the Comics, May 2, 2017: Puzzle Week


If there was a theme this week, it was puzzles. So many strips had little puzzles to work out. You’ll see. Thank you.

Bill Amend’s FoxTrot for the 30th of April tries to address my loss of Jumble panels. Thank you, whoever at Comic Strip Master Command passed along word of my troubles. I won’t spoil your fun. As sometimes happens with a Jumble you can work out the joke punchline without doing any of the earlier ones. 64 in binary would be written 1000000. And from this you know what fits in all the circles of the unscrambled numbers. This reduces a lot of the scrambling you have to do: just test whether 341 or 431 is a prime number. Check whether 8802, 8208, or 2808 is divisible by 117. The integer cubed you just have to keep trying possibilities. But only one combination is the cube of an integer. The factorial of 12, just, ugh. At least the circles let you know you’ve done your calculations right.

Steve McGarry’s activity feature Kidtown for the 30th plays with numbers some. And a puzzle that’ll let you check how well you can recognize multiples of four that are somewhere near one another. You can use diagonals too; that’s important to remember.

Mac King and Bill King’s Magic in a Minute feature for the 30th is also a celebration of numerals. Enjoy the brain teaser about why the encoding makes sense. I don’t believe the hype about NASA engineers needing days to solve a puzzle kids got in minutes. But if it’s believable, is it really hype?

Marty Links’s Emmy Lou from the 29th of October, 1963 was rerun the 2nd of May. It’s a reminder that mathematics teachers of the early 60s also needed something to tape to their doors.

Mel Henze’s Gentle Creatures rerun for the 2nd of May is another example of the conflating of “can do arithmetic” with “intelligence”.

Mark Litzler’s Joe Vanilla for the 2nd name-drops the Null Hypothesis. I’m not sure what Litzler is going for exactly. The Null Hypothesis, though, comes to us from statistics and from inference testing. It turns up everywhere when we sample stuff. It turns up in medicine, in manufacturing, in psychology, in economics. Everywhere we might see something too complicated to run the sorts of unambiguous and highly repeatable tests that physics and chemistry can do — things that are about immediately practical questions — we get to testing inferences. What we want to know is, is this data set something that could plausibly happen by chance? Or is it too far out of the ordinary to be mere luck? The Null Hypothesis is the explanation that nothing’s going on. If your sample is weird in some way, well, everything is weird. What’s special about your sample? You hope to find data that will let you reject the Null Hypothesis, showing that the data you have is so extreme it just can’t plausibly be chance. Or to conclude that you fail to reject the Null Hypothesis, showing that the data is not so extreme that it couldn’t be chance. We don’t accept the Null Hypothesis. We just allow that more data might come in sometime later.

I don’t know what Litzler is going for with this. I feel like I’m missing a reference and I’ll defer to a finance blogger’s Reading the Comics post.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 3rd is another in the string of jokes using arithmetic as source of indisputably true facts. And once again it’s “2 + 2 = 5”. Somehow one plus one never rates in this use.

Aaron Johnson’s W T Duck rerun for the 3rd is the Venn Diagram joke for this week. It’s got some punch to it, too.

Je Mallett’s Frazz for the 5th took me some time to puzzle out. I’ll allow it.

Reading the Comics, April 18, 2017: Give Me Some Word Problems Edition


I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.

Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.

Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.

(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)

J C Duffy’s Lug Nuts for the 16th is a Pi Day joke that ran late.

Jef Mallett’s Frazz for the 17th starts a little thread about obsolete references in story problems. It’s continued on the 18th. I’m sympathetic in principle to both sides of the story problem debate.

Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, $2.25 per pound had to be divided by three.

But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.

And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.

So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?

Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.

Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.

Lee: 'I'm developing a new theory about avocado intelligence.' Joules: 'You can't be serious.' Lee: 'Avocado, what is the square root of 8,649?' Avocado: 'That's easy. It's 92?' Lee: 'Wrong. It's 93.' Joules: 'See? It's just a dumb piece of fruit.' Lee: 'I honestly thought I was on to something.'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of April, 2017. Before you ask what exactly the old theory of avocado intelligence was remember that Edison Lee’s lab partner there is a talking rat. Just saying.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.

Reading the Comics, March 4, 2017: Frazz, Christmas Trees, and Weddings Edition


It was another of those curious weeks when Comic Strip Master Command didn’t send quite enough comics my way. Among those they did send were a couple of strips in pairs. I can work with that.

Samson’s Dark Side Of The Horse for the 26th is the Roman Numerals joke for this essay. I apologize to Horace for being so late in writing about Roman Numerals but I did have to wait for Cecil Adams to publish first.

In Jef Mallett’s Frazz for the 26th Caulfield ponders what we know about Pythagoras. It’s hard to say much about the historical figure: he built a cult that sounds outright daft around himself. But it’s hard to say how much of their craziness was actually their craziness, how much was just that any ancient society had a lot of what seems nutty to us, and how much was jokes (or deliberate slander) directed against some weirdos. What does seem certain is that Pythagoras’s followers attributed many of their discoveries to him. And what’s certain is that the Pythagorean Theorem was known, at least a thing that could be used to measure things, long before Pythagoras was on the scene. I’m not sure if it was proved as a theorem or whether it was just known that making triangles with the right relative lengths meant you had a right triangle.

Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.

Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.

Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.

Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.

'Has Carl given you any reason not to trust him?' 'No, not yet. But he might.' 'Fi ... you seek 100% certainty in people, but that doesn't exist. In the end,' and Dethany is drawn as her face on a pi symbol, 'we're *all* irrational numbers.'
Bill Holbrook’s On The Fastrack for the 3rd of March, 2017. Fi’s dress isn’t one of those … kinds with the complicated pattern of holes in it. She got it torn while trying to escape the wedding and falling into the basement.

Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.

That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.

'Fi ... humans are complicated. Like the irrational number pi, we can go on forever. You never get to the bottom of us! But right now, upstairs, there are two variables who *want* you in their lives. Assign values to them.' Carl, Fi's fiancee, is drawn as his face with a y; his kid as a face on an x.
Bill Holbrook’s On The Fastrack for the 4th of March, 2017. I feel bad that I completely forgot Carl had a kid and that the face on the x doesn’t help me remember anything.

Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.

Reading the Comics, February 23, 2017: The Week At Once Edition


For the first time in ages there aren’t enough mathematically-themed comic strips to justify my cutting the week’s roundup in two. No, I have no idea what I’m going to write about for Thursday. Let’s find out together.

Jenny Campbell’s Flo and Friends for the 19th faintly irritates me. Flo wants to make sure her granddaughter understands that just because it takes people on average 14 minutes to fall asleep doesn’t mean that anyone actually does, by listing all sorts of reasons that a person might need more than fourteen minutes to sleep. It makes me think of a behavior John Allen Paulos notes in Innumeracy, wherein the statistically wise points out that someone has, say, a one-in-a-hundred-million chance of being killed by a terrorist (or whatever) and is answered, “ah, but what if you’re that one?” That is, it’s a response that has the form of wisdom without the substance. I notice Flo doesn’t mention the many reasons someone might fall asleep in less than fourteen minutes.

But there is something wise in there nevertheless. For most stuff, the average is the most common value. By “the average” I mean the arithmetic mean, because that is what anyone means by “the average” unless they’re being difficult. (Mathematicians acknowledge the existence of an average called the mode, which is the most common value (or values), and that’s most common by definition.) But just because something is the most common result does not mean that it must be common. Toss a coin fairly a hundred times and it’s most likely to come up tails 50 times. But you shouldn’t be surprised if it actually turns up tails 51 or 49 or 45 times. This doesn’t make 50 a poor estimate for the average number of times something will happen. It just means that it’s not a guarantee.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 19th shows off an unusually dynamic camera angle. It’s in service for a class of problem you get in freshman calculus: find the longest pole that can fit around a corner. Oh, a box-spring mattress up a stairwell is a little different, what with box-spring mattresses being three-dimensional objects. It’s the same kind of problem. I want to say the most astounding furniture-moving event I’ve ever seen was when I moved a fold-out couch down one and a half flights of stairs single-handed. But that overlooks the caged mouse we had one winter, who moved a Chinese finger-trap full of crinkle paper up the tight curved plastic to his nest by sheer determination. The trap was far longer than could possibly be curved around the tube. We have no idea how he managed it.

J R Faulkner’s Promises, Promises for the 20th jokes that one could use Roman numerals to obscure calculations. So you could. Roman numerals are terrible things for doing arithmetic, at least past addition and subtraction. This is why accountants and mathematicians abandoned them pretty soon after learning there were alternatives.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. Probably anything would do for the blackboard problem, but something geometry reads very well.

Jef Mallett’s Frazz for the 21st makes some comedy out of the sort of arithmetic error we all make. It’s so easy to pair up, like, 7 and 3 make 10 and 8 and 2 make 10. It takes a moment, or experience, to realize 78 and 32 will not make 100. Forgive casual mistakes.

Bud Fisher’s Mutt and Jeff rerun for the 22nd is a similar-in-tone joke built on arithmetic errors. It’s got the form of vaudeville-style sketch compressed way down, which is probably why the third panel could be made into a satisfying final panel too.

'How did you do on the math test?' 'Terrible.' 'Will your mom be mad?' 'Maybe. But at least she'll know I didn't cheat!'
Bud Blake’s Tiger for the 23rd of February, 2017. I want to blame the colorists for making Hugo’s baby tooth look so weird in the second and third panels, but the coloring is such a faint thing at that point I can’t. I’m sorry to bring it to your attention if you didn’t notice and weren’t bothered by it before.

Bud Blake’s Tiger rerun for the 23rd just name-drops mathematics; it could be any subject. But I need some kind of picture around here, don’t I?

Mike Baldwin’s Cornered for the 23rd is the anthropomorphic numerals joke for the week.

Reading the Comics, January 16, 2017: Numerals Edition


Comic Strip Master Command decreed that last week should be busy again. So I’m splitting its strips into two essays. It’s a week that feels like it had more anthropomorphic numerals jokes than usual, but see if I actually count these things.

2 asks 4: 'Six, six, six, can't you think of anything but six?'
Mike Peters’s Mother Goose and Grimm for the 15th of January, 2017. I understand that sometimes you just have to use the idea you have instead of waiting for something that can best use the space available, but really, a whole Sunday strip for a single panel? And a panel that’s almost a barren stage?

Mike Peters’s Mother Goose and Grimm for the 15th I figured would be the anthropomorphic numerals joke for the week. Shows what I know. It is an easy joke, but I do appreciate the touch of craft involved in picking the numerals. The joke is just faintly dirty if the numbers don’t add to six. If they were a pair of 3’s, there’d be the unwanted connotations of a pair of twins talking about all this. A 6 and a 0 would make at least one character weirdly obsessed. So it has to be a 4 and a 2, or a 5 and a 1. I imagine Peters knew this instinctively, at this point in his career. It’s one of the things you learn in becoming an expert.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 15th is mostly physical comedy, with a touch of — I’m not sure what to call this kind of joke. The one where a little arithmetic error results in bodily harm. In this sort of joke it’s almost always something not being carried that’s the error. I suppose that’s a matter of word economy. “Forgot to carry the (number)” is short, and everybody’s done it. And even if they don’t remember making this error, the phrasing clarifies to people that it’s a little arithmetic mistake. I think in practice mistaking a plus for a minus (or vice-versa) is the more common arithmetic error. But it’s harder to describe that clearly and concisely.

Jef Mallett’s Frazz for the 15th puzzled me. I hadn’t heard this thing the kid says about how if you can “spew ten random lines from a classic movie” to convince people you’ve seen it. (I don’t know the kid’s name; it happens.) I suppose that it would be convincing, though. I certainly know a couple lines from movies I haven’t seen, what with living in pop culture and all that. But ten would be taxing for all but the most over-saturated movies, like any of the Indiana Jones films. (There I’m helped by having played the 90s pinball machine a lot.) Anyway, knowing ten random mathematics things isn’t convincing, especially since you can generate new mathematical things at will just by changing a number. But I would probably be convinced that someone who could describe what’s interesting about ten fields of mathematics had a decent understanding of the subject. That requires remembering more stuff, but then, mathematics is a bigger subject than even a long movie is.

In Bill Holbrook’s On The Fastrack for the 16th Fi speaks of tallying the pluses and minuses of her life. Trying to make life into something that can be counted is an old decision-making technique. I think Benjamin Franklin explained how he found it so useful. It’s not a bad approach if a choice is hard. The challenging part is how to weight each consideration. Getting into fractions seems rather fussy to me, but some things are just like that. There is the connotation here that a fraction is a positive number smaller than 1. But the mathematically-trained (such as Fi) would be comfortable with fractions larger than 1. Or also smaller than zero. “Fraction” is no more bounded than “real number”. So, there’s the room for more sweetness here than might appear to the casual reader.

'In a couple of weeks I'm getting married, so I'm taking stock of my life, adding up the pluses and minuses that factor into my goals.' 'Am I a positive or a negative integer?' 'You're a fraction.' 'How presumptuous of me.'
Bill Holbrook’s On The Fastrack for the 16th of January, 2017. Were I in Dethany’s position I would have asked about being a positive or negative number, but then that would leave Holbrook without a third panel. Dethany knows what her author needs most.

Scott Hilburn’s The Argyle Sweater for the 16th is the next anthropomorphic numerals joke for this week. I’m glad Hilburn want to be in my pages more. 5’s concern about figuring out x might be misplaced. We use variables for several purposes. One of them is as a name to give a number whose value we don’t know but wish to work out, and that’s how we first see them in high school algebra. But a variable might also be a number whose value we don’t particularly care about and will never try to work out. This could be because the variable is a parameter, with a value that’s fixed for a problem but not what we’re interested in. We don’t typically use ‘x’ for that, though; usually parameter are something earlier in the alphabet. That’s merely convention, but it is convention that dates back to René Descartes. Alternatively, we might use ‘x’ as a dummy variable. A dummy variable serves the same role that falsework on a building or a reference for an artistic sketch does. We use dummy variables to organize and carry out work, but we don’t care what its values are and we don’t even see the dummy variable in the final result. A dummy variable can be any name, but ‘x’ and ‘t’ are popular choices.

Terry LaBan and Patty LaBan’s Edge City rerun for the 16th plays on the idea that mathematics people talk in algebra. Funny enough, although, “the opposing defense is a variable of 6”? That’s an idiosyncratic use of “variable”. I’m going to suppose that Charles is just messing with Len’s head because, really, it’s fun doing a bit of that.

Reading the Comics, June 13, 2012


Because there weren’t many math-themed comic strips, that’s why I went so long without an update in my roster of comic strips that mention math subjects. After Mike Peters’s Mother Goose and Grimm put in the start of a binomial expression the comics pages — through King Features Syndicate and gocomics.com — decided to drop the whole subject pretty completely for the rest of May. It picked up a little in June.

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