Reading the Comics, August 14, 2022: Not Being Wrong Edition

The handful of comic strips I’ve chosen to write about this week include a couple with characters who want to not be wrong. That’s a common impulse among people learning mathematics, that drive to have the right answer.

Will Henry’s Wallace the Brave for the 8th opens the theme, with Rose excited to go to mathematics camp as a way of learning more ways to be right. I imagine everyone feels this appeal of mathematics, arithmetic particularly. If you follow these knowable rules, and avoid calculation errors, you get results that are correct. Not just coincidentally right, but right for all time. It’s a wonderful sense of security, even when you get past that childhood age where so little is in your control.

Will Henry’s Wallace the Brave for the 8th of August, 2022. This and other essays mentioning Wallace the Brave are at this link.

A thing that creates a problem, if you love this too closely, is that much of mathematics builds on approximations. Things we know not to be right, but which we know are not too far wrong. You expect this from numerical mathematics, yes. But it happens in analytic mathematics too. I remember struggling in high school physics, in the modeling a pendulum’s swing. To do this you have to approximate the sine of the angle the pendulum bob with the angle itself. This approximation is quite good, if the angle is small, as you can see from comparing the sine of 0.01 radians to the number 0.01. But I wanted to know when that difference was accounted for, and it never was.

(An alternative interpretation is to treat the path swung by the end of the pendulum as though it were part of a parabola, instead of the section of circle that it really is. A small arc of parabola looks much like a small arc of circle. But there is a difference, not accounted for.)

Nor would it be. A regular trick in analytic mathematics is to show that the thing you want is approximated well enough by a thing you can calculate. And then show that if one takes a limit of the thing you can calculate you make the error infinitesimally small. This is all rigorous and you can in time come to accept it. I hope Rose someday handles the discovery that we get to right answers through wrong-but-useful ones well.

Charles Schulz’s Peanuts Begins for the 8th of August, 2022. It originally ran the 11th of August, 1954. Essays with some mention of Peanuts (or Peanuts Begins) are at this link.

Charles Schulz’s Peanuts Begins for the 8th is one that I have featured here before. It’s built on Lucy not accepting that the answer to a multiplication can be zero, even if it is zero times zero. It’s also built on the mixture of meanings between “zero” and “nothing” and “not existent”. Lucy’s right that zero times zero has to be something, as in a thing with some value. But we also so often use zero to mean “nothing that exists” makes zero a struggle to learn and to work with.

Dan Thompson’s Brevity for the 12th of August, 2022. My various essays with some mention of Brevity should be at this link.

Dan Thompson’s Brevity for the 12th is an anthropomorphic numerals joke, built on the ancient playground pun about why six is afraid of seven. And a bit of wordplay about odd and even numbers on top of that. For this I again offer the followup joke that I first heard a couple of years ago. Why was it that 7 ate 9? Because 7 knows to eat 3-squared meals a day!

Lincoln Pierce’s Big Nate for the 14th of August, 2022. Essays including some topic mentioned in Big Nate should be at this link.

Lincoln Pierce’s Big Nate for the 14th is a baseball statistics joke. Really a sabermetrics joke. Sabermetrics and other fine-grained sports analysis study at the enormous number of games played, and situations within those games. The goal is to find enough similar situations to make estimates about outcomes. This is through what’s called the “frequentist” interpretation of statistics. That is, if this situation has come up a hundred times before, and it’s led to one particular outcome 85 of those times, then there’s an 85 percent chance of that outcome in this situation.

Baseball is well-posed to set up this sort of analysis. The organized game has always demanded the keeping of box scores, close records of what happened in what order. Other sports can have the same techniques applied, though. It’s not likely that Randy has thrown enough pitches to estimate his chance of giving up a walk-off grand slam. But combine all the little league teams there are, and all the seasons they’ve played? That starts to sound plausible. Doesn’t help the feeling that one was scheduled for a win and then it didn’t happen.

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And that’s enough comics for now. All of my Reading the Comics posts should be at this link, and I hope to have another next week. Thanks for reading.

Reading the Comics, April 11, 2018: Monkeys at Typewriters Edition

This is closing out a busy week’s worth of comic strips mentioning some mathematics theme. Three of these are of extremely slight mathematical content, but I’ll carry on anyway.

Reza Farazmand’s Poorly Drawn Lines for the 8th has a bear admit the one thing which frightens him still is mathematics. It adds to it a joke showing that he’s not very good at mathematics, by making a mistake with percentages.

Will Henry’s Wallace the Brave for the 8th has Wallace working out an arithmetic problem in class.

Dana Simpson’s Ozy and Millie rerun for the 9th is part of a sequence of Ozy being home-schooled. The joke puts the transient nature of knowledge up against the apparent permanent of arithmetic. The joke does get at one of those fundamental questions in the philosophy of mathematics: is mathematics created or discovered? The expression of mathematics is unmistakably created. There is nothing universal in declaring “six times eight is forty-eight” and if you wish to say there is, then ask someone who speaks only Tamil and not a word of English whether they agree with exactly that proposition.

Dana Simpson’s Ozy and Millie rerun for the 9th of April, 2020. Essays in which I discuss something raised by Ozy and Millie are at this link.

But, grant that while we may have different representations of the concept, it is the case that “eight” exists, right? We get right back into trouble if we follow up by asking, all right, will “eight” fit in my hand? Is “eight” larger than the weather? Is “eight” more or less red than nominalism? I chose nouns that made those questions obviously ridiculous. But if we want to talk about a mathematical construct existing, someone’s going to ask what traits that existence implies. It’s convenient for mathematicians, and good publicity, for us to think that we work on things that exist independently of the accidental facts of the universe. But then we’re stuck when we’re asked how we, stuck in the universe, can have anything to do with a thing that’s not part of it.

Not mentioned in this particular Ozy and Millie strip is that the characters are Buddhist. The (American) pop culture interpretation of Buddhism includes an emphasis on understanding the transient nature of … everything … which would seem to include mathematical knowledge. Still, there is a long history of great mathematical work done by Buddhist scholars; the oldest known manuscript of Indian mathematics is written in a Buddhist Hybrid Sanskrit. The author of that manuscript is unknown, but it’s not as if that were the lone piece of mathematical writing.

My limited understanding is that Indian mathematics used an interesting twist on the problem of the excluded middle. This is a question important to proofs. Can we take every logical proposition as being either true or false? If we can, then we are able to prove statements by contradiction: suppose the reverse of what we want to prove and show that implies nonsense. This is common in western mathematics. But there is a school of thought that we should not do this, and only allow as true statements we have directly proven to be true. My understanding is that at least one school of Indian mathematics allowed proof by contradiction if it proved that a thing did not exist. It would not be used to show that a thing existed. So, for example, it would allow the ordinary proof that the square root of two can’t be a rational number; it would not allow an indirect proof that, say, a kind of mapping must have a fixed point. (It would allow a proof that showed you how to find that point, though.) It’s an interesting division, and a reminder that even what counts as a logical derivation is a matter of custom.

James Swinnerton’s The Troubles of Noah for the 21st of July, 1901, and reprinted the 10th of April, 2020. I don’t seem to have ever discussed this series before, which is not all that surprising. But if I ever do have an essay mentioning the Origins of the Sunday Comics series I will try to put it at this link.

Peter Maresca’s Origins of the Sunday Comics for the 9th reprints The Troubles of Noah, a comic strip drawn by James Swinnerton and originally printed the 21st of July, 1901. And this is really included just because it depicts a monkey at a typewriter, a dozen years before Émile Borel created the perfect image of endless random processes. (Look to the lower right corner, taking dictation from Noah.) There’s also a bonus monkey setting type in the lower left.

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That’s finally taken care of a week. Time to take care of another week! When I have some of last week’s comic strips written up I will post the essay at this link. Thanks for reading.

Reading the Comics, May 8, 2019: Strips With Art I Like Edition

Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.

Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.

Mark Anderson’s Andertoons for the 5th of May, 2019. Essays mentioning Andertoons are at this link and also at nearly every Reading the Comics post, it feels like.

But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.

Juba’s Viivi and Wagner for the 5th of May, 2019. I don’t often have chances to talk about Viivi and Wagner but when I do, it’s here.

Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.

Harry Bliss and Steve Martin’s Bliss for the 6th of May, 2019. Yes, by the way, it’s the Steve Martin you know and love from Looney Tunes: Back In Action and from the 1996 Sergeant Bilko movie. Anyway I haven’t had chance to write about this strip before but this and future appearances of Bliss should be here.

Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.

Will Henry’s Wallace the Brave for the 7th of May, 2019. This is one of the comic strips I’m most excited about, the last several years. Wallace the Brave appears in essays at this link.

Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.

Rick Detorie’s One Big Happy for the 8th of May, 2019. There are two strings of One Big Happy available for daily reading. Appearances by the current or the several-years-old GoComics prints of One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 8th is a bit of wordplay built around geometry, as Ruthie plays teacher. She’s a bit dramatic, but she always has been.

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I’ll read some more comics for later in this week. That essay, and all similar comic strip talk, should appear at this link. Thank you.

Reading the Comics, January 16, 2019: Young People’s Mathematics Edition

Today’s quartet of mathematically-themed comic strips doesn’t have an overwhelming theme. There’s some bits about the mathematics that young people do, so, that’s enough to separate this from any other given day’s comics essay.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is built on a bit of mathematical folklore. As Weinersmith’s mathematician (I don’t remember that we’ve been given her name) mentions, there is a belief that “revolutionary” mathematics is done by young people. That isn’t to say that older mathematicians don’t do great work. But the stereotype is that an older mathematician will produce masterpieces in already-established fields. It’s the young that establish new fields. Indeed, one of mathematics’s most prestigious awards, the Fields Medal, is only awarded to mathematicians under the age of forty. I was cheated of mine. Long story.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th of January, 2019. I have many essays inspired by something said in Saturday Morning Breakfast Cereal. You can find them at this link.

There’s intuitive appeal in the idea that revolutions in thinking are for the young. We think that people get set in their ways as they develop their careers. We have a couple dramatic examples, most notably Évariste Galois, who developed what we now see as foundations of group theory and died at twenty. While the idea is commonly held, I don’t know that it’s actually true. That is, that it holds up to scrutiny. It seems hard to create a definition for “revolutionary mathematics” that could be agreed upon by two people. So it would be difficult to test at what age people do their most breathtaking work, and whether it is what they do when young or when experienced.

Is there harm to believing an unprovable thing? If it makes you give up on trying, yes. My suspicion is that true revolutionary work happens when a well-informed, deep thinker comes to a field that hasn’t been studied in that way before. And when it turns out to be a field well-suited to study that way. That doesn’t require youth. It requires skill in one field, and an understanding that there’s another field ready to be studied that way.

Will Henry’s Wallace the Brave for the 14th of January, 2019. I have only had a few chances to talk about Wallace the Brave so far, but the chances I’ve taken are at this link. (It and Breaking Cat News are the two recently-launched comics I’m most excited by.)

Will Henry’s Wallace the Brave for the 14th is a mathematics anxiety joke. Wallace tries to help by turning an abstract problem into a concrete one. This is often a good way to approach a problem. Even in more advanced mathematics, one can often learn the way to solve a general problem by trying a couple of specific examples. It’s almost as though there’s only a certain amount of abstraction people can deal with, and you need to re-cast problems so they stay within your limits.

Yes, the comments turn to complaining about Common Core. I’m not sure what would help Spud work through this problem (or problems in general). But thinking of alternate problems that estimated or approached what he really wanted might help. If he noticed, for example, that 10 + 12 has to be a little more than 10 + 10, and he found 10 + 10 easy, then he’d be close to a right answer. If he noticed that 10 + 12 had to be 10 + 10 + 2, and he found 10 + 10 easy, then he might find 20 + 2 easy as well. Maybe Spud would be better off thinking of ways to rewrite a problem without changing the result.

Wiley Miller’s Non Sequitur for the 15th of January, 2019. Essays mentioning Non Sequitur should appear at this link.

Wiley Miller’s Non Sequitur for the 15th mentions calculus. It’s more of a probability joke. To speak of a calculated risk is to speak of doing something that’s not certain, but that has enough of a payoff to be worth the cost of failure. But one problem with this attitude is that people are very, very bad at estimating probabilities. We have terrible ideas of how likely losses are and how uncertain rewards can be. But even if we allow that the risks and rewards are calculated right, there’s a problem with things you only do once. Or only can do once. You can get into a good debate about whether there’s even a meaningful idea of probability for things that happen only the one time. Life’s among them.

Bob Weber Sr’s Moose and Molly for the 16th of January, 2019. I haven’t had the chance to talk about Moose and Molly before. But now I have the tag, and will be putting essays mentioning it at this link.

Bob Weber Sr’s Moose and Molly for the 16th is a homework joke. It does actually depend on being mathematics homework, though, or there’d be no grounds for Moose’s kid to go to the savings and loan clerk who’ll help with “money problems”.

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I think there’s one more batch of comic strips to discuss this week. When I’ve published it, you should find the essay at this link. And then there’ll be Sunday again.

Reading the Comics, October 18, 2018: Quick Half-Week Edition

There were enough mathematically-themed comic strips last week to split across two essays. The first half of them don’t take too much time to explain. Let me show you.

Henry Scarpelli and Craig Boldman’s Archie for the 15th is the pie-chart wordplay joke for the week. I don’t remember there ever being pie at the high school cafeteria, but back when I was in high school I often skipped lunch to hang out in the computer room.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 15th of October, 2018. Oh gads. If Jughead makes this much noise just imagining food then when he really eats he’s got to be one of those people you can hear from the next state over. I have no information about when this strip first ran.

Will Henry’s Wallace the Brave for the 15th alludes to a report on trapezoids. I can’t imagine what about this would be so gold-star-worthy when I’ve surely already written plenty about trapezoids. … Really, that thing trying to classify how many different kinds of trapezoids there are would be my legacy to history if I hadn’t also written about how many grooves are on a record’s side.

Will Henry’s Wallace the Brave for the 15th of October, 2018. All right, the strip is only marginally on topic. It and Breaking Cat News are the syndicated comic strips I’ve been most excited for since Richard Thompson wasn’t able to continue Cul de Sac.

Thaves’s Frank and Ernest for the 17th is, for me, extremely relatable content. I don’t say that my interest in mathematics is entirely because there was this Berenstain Bears book about jobs which made it look like a mathematician’s job was to do sums in an observatory on the Moon. But it didn’t hurt. When I joke about how seven-year-old me wanted to be the astronaut who drew Popeye, understand, that’s not much comic exaggeration.

Thaves’s Frank and Ernest for the 17th of October, 2018. “I’d be getting taped to the doors of mathematics teacher’s classrooms — saaaaaay!”

Justin Thompson’s Mythtickle rerun for the 17th is a timely choice about lotteries and probabilities. Vlad raises a fair point about your chance of being struck by lightning. It seems like that’s got to depend on things like where you are. But it does seem like we know what we mean when we say “the chance you’ll be hit by lightning”. At least I think it means “the probability that a person will be hit by lightning at some point in their life, if we have no information about any environmental facts that might influence this”. So it would be something like the number of people struck by lightning over the course of a year divided by the number of people in the world that year. You might have a different idea of what “the chance you’ll be hit by lightning” means, and it’s worth trying to think what precisely that does mean to you.

Justin Thompson’s Mythtickle rerun for the 17th of October, 2018. Not to step on a joke Thompson left nicely underplayed, but I find funny the premise that of course the clerk in the Transylvanian convenience store is a werewolf. I have no information about when this strip first ran.

Lotteries are one of those subjects that a particular kind of nerd likes to feel all smug about. Pretty sure every lottery comic ever has drawn a comment about a tax on people who can’t do mathematics. This one did too. But then try doing the mathematics. The Mega Millions lottery, in the US, has a jackpot for the first drawing this week estimated at more than a billion dollars. The chance of winning is about one in 300 million. A ticket costs two dollars. So what is the expectation value of playing? You lose two dollars right up front, in the cost of the ticket. What do you get back? A one-in-300-million chance of winning a billion dollars. That is, you can expect to get back a bit more than three dollars. The implication is: you make a profit of dollar on each ticket you buy. There’s something a bit awry here, as you can tell from my decision not to put my entire savings into lottery tickets this week. But I won’t say someone is foolish or wrong if they buy a couple.

Mike Baldwin’s Cornered for the 18th of October, 2018. What is that grit on the teacher’s desk, to the reader’s right of the pen?

Mike Baldwin’s Cornered for the 18th is a bit of mathematics-circling wordplay, featuring the blackboard full of equations. The blackboard doesn’t have any real content on it, but it is a good visual shorthand. And it does make me notice that rounding a quantity off is, in a way, making it simpler. If we are only a little interested in the count of the thing, “two thousand forty” or even “two thousand” may be more useful than the exact 2,038. The loss of precision may be worth it for the ease with which the rounded-off version is remembered and communicated.

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If you’d like to see more Reading the Comics posts then try this link. Other essays which mention Archie should be at this link. Topics raised by Wallace the Brave should be at this link. Frank and Ernest is the subject of essays at this link. Topics brought up by Mythtickle are at this link. It’s a new tag, though, and I’m not sure there’ll ever be another use of it. And this and other essays mentioning Cornered are at this link. And do please stick around for more of my Fall 2018 Mathematics A-To-Z, coming twice a week through the rest of the year, I hope.

Reading the Comics, February 20, 2018: Bob the Squirrel Edition

So one comic strip was technically on point all this week, without ever quite giving me a specific thing to talk about. And I came to conclude there was another comic strip I could drop from my consideration. Which all were they? Read on.

Frank Page’s Bob the Squirrel for the 18th of February isn’t really about the Rubik’s Cube. It’s just something to occupy Bob’s mind until a deeper mystery emerges. Rubik’s Cubes, meanwhile, are everyone’s favorite group theory pastime, although I’m not sure how many people have learned group theory starting from that point. Where flies come from in the middle of winter I don’t know. We’ve been dealing with box elder bugs ourselves. (We’ve been scooping them up and tossing them outside where they can hopefully find the trees they should be using instead.)

Bob the Squirrel went on, during the week, to start a sequence about Lauren needing a geometry tutor. The story hasn’t done much that geometry-specific — Saturday’s was the most approximately on point — but it’s a comic strip I like. Squirrel fans might agree. (The strip for the 22nd has most tickled me.)

Allison Barrows’s PreTeena rerun for the 19th has a student teacher starting off her experience with a story problem. Your classic time-estimation problem.

Jack Pullan’s Boomerangs rerun for the 20th is one that mentions entropy and that I’ve already talked about at least twice before. These were times in January 2017 and also in November 2013. Given that the strip’s no longer in production and that I’m clearly on at least my third go-round I suppose I’ll retire it from my daily read. I’m curious why, if it was about 14 months between the last appearance and this appearance of this strip, why I didn’t have it at all in 2015 or 2016. Maybe I missed it, or it came a week there was enough to write about that I didn’t need to include a marginal strip.

Christopher Grady’s Lunarbaboon for the 20th is intended to be a heartwarming little story of encouragement and warm feelings. (Most Lunarbaboon strips are intended to be a heartwarming little story of encouragement and warm feelings.) That it’s mathematics the kid struggles with is incidental to the story setup. But it does make it easy to picture a kid struggling and a couple kind words offering some motivation, or at least better feelings.

Richard Thompson’s Richard’s Poor Almanac for the 20th is a casual mention of sudoku and a publication error that would supposedly have made it impossible. If the numbers were transposed consistently — everything that ought to have been a ‘2’ printed as a ‘5’, and everything that ought to have been ‘5’ printed as ‘2’ — the problem would be exactly as solvable. This is why you can sometimes see sudoku-type puzzles that use symbols or letters or other characters. But if, say, the third and the second rows were transposed then there’s a chance the incorrect puzzle would be solvable. Transposing a bunch of squares, like, the top three rows with the bottom three rows, wouldn’t make the puzzle unsolvable. This serves as a reminder that if you make enough mistakes you can still turn out all right, a comforting message for our times. Also I know I’ve featured Richard’s Poor Almanac several times over, but I’m a Richard Thompson fan so I’m not dropping that from my feed.

Will Henry’s Wallace the Brave — to be newspaper-syndicated from the 26th of March, by the way, and I’m glad for that as Wallace and I share the same favorite pinball game — just mentions mathematics as a subject Wallace isn’t thinking enough about. I’m also fond of the Loch Ness Monster, so, all the better.

I’m not surprised that this seems to be the first time I’ve had Lunarbabboon tagged. I am surprised that Bob the Squirrel seems not to have been tagged here before. Maybe I didn’t give the tag suggested-completion enough time to figure out what to do with ‘bob the’. We’ve been having odd little net glitches that mostly pass quickly, but that kill any sort of client-side Javascript-based page rendering. You know, like every web page does anymore because somehow “the web server puts together a bunch of stuff and transmits that to the reader” is too inefficient a system.

Reading the Comics, September 19, 2017: Visualization Edition

Comic Strip Master Command apparently doesn’t want me talking about the chances of Friday’s Showcase Showdown. They sent me enough of a flood of mathematically-themed strips that I don’t know when I’ll have the time to talk about the probability of that episode. (The three contestants spinning the wheel all tied, each spinning $1.00. And then in the spin-off, two of the three contestants also spun $1.00. And this after what was already a perfect show, in which the contestants won all six of the pricing games.) Well, I’ll do what comic strips I can this time, and carry on the last week of the Summer 2017 A To Z project, and we’ll see if I can say anything timely for Thursday or Saturday or so.

Jim Scancarelli’s Gasoline Alley for the 17th is a joke about the student embarrassing the teacher. It uses mathematics vocabulary for the specifics. And it does depict one of those moments that never stops, as you learn mathematics. There’s always more vocabulary. There’s good reasons to have so much vocabulary. Having names for things seems to make them easier to work with. We can bundle together ideas about what a thing is like, and what it may do, under a name. I suppose the trouble is that we’ve accepted a convention that we should define terms before we use them. It’s nice, like having the dramatis personae listed at the start of the play. But having that list isn’t the same as saying why anyone should care. I don’t know how to balance the need to make clear up front what one means and the need to not bury someone under a heap of similar-sounding names.

Mac King and Bill King’s Magic in a Minute for the 17th is another puzzle drawn from arithmetic. Look at it now if you want to have the fun of working it out, as I can’t think of anything to say about it that doesn’t spoil how the trick is done. The top commenter does have a suggestion about how to do the problem by breaking one of the unstated assumptions in the problem. This is the kind of puzzle created for people who want to motivate talking about parity or equivalence classes. It’s neat when you can say something of substance about a problem using simple information, though.

Terri Libenson’s Pajama Diaries for the 18th of September, 2017. When I first read this I assumed that of course the base of the triangle had length 4 and the second leg, at a 45-degree angle to that, had length 2, and I wondered if those numbers could be consistent for a triangle to exist. Of course they could, though. There is a bit of fun to be had working out whether a particular triangle could exist from knowing its side lengths, though.

Terri Libenson’s Pajama Diaries for the 18th uses trigonometry as the marker for deep thinking. It comes complete with a coherent equation, too. It gives the area of a triangle with two legs that meet at a 45 degree angle. I admit I am uncomfortable with promoting the idea that people who are autistic have some super-reasoning powers. (Also with the pop-culture idea that someone who spots things others don’t is probably at least a bit autistic.) I understand wanting to think someone’s troubles have some compensation. But people are who they are; it’s not like they need to observe some “balance”.

Lee Falk and Wilson McCoy’s The Phantom for the 10th of August, 1950 was rerun Monday. It’s a side bit of joking about between stories. And it uses knowledge of mathematics — and an interest in relativity — as signifier of civilization. I can only hope King Hano does better learning tensors on his own than I do.

Lee Falk and Wilson McCoy’s The Phantom for the 10th of August, 1950 and rerun the 18th of September, 2017. For my money, just reading a mathematics book doesn’t take. I need to take notes, as if it were in class. I don’t quite copy the book, but it comes close.

Mike Thompson’s Grand Avenue for the 18th goes back to classrooms and stuff for clever answers that subvert the teacher. And I notice, per the title given this edition, that the teacher’s trying to make the abstractness of three minus two tangible, by giving it an example. Which pairs it with …

Will Henry’s Wallace the Brace for the 18th, wherein Wallace asserts that arithmetic is easier if you visualize real things. I agree it seems to help with stuff like basic arithmetic. I wouldn’t want to try taking the cosine of an apple, though. Separating the quantity of a thing from the kind of thing measured is one of those subtle breakthroughs. It’s one of the ways that, for example, modern calculations differ from those of the Ancient Greeks. But it does mean thinking of numbers in, we’d say, a more abstract way than they did, and in a way that seems to tax us more.

Wallace the Brave recently had a book collection published, by the way. I mention because this is one of a handful of comics with a character who likes pinball, and more, who really really loves the Williams game FunHouse. This is an utterly correct choice for favorite pinball game. It’s one of the games that made me a pinball enthusiast.

Ryan North’s Dinosaur Comics rerun for the 19th I mention on loose grounds. In it T-Rex suggests trying out an alternate model for how gravity works. The idea, of what seems to be gravity “really” being the shade cast by massive objects in a particle storm, was explored in the late 17th and early 18th century. It avoids the problem of not being able to quite say what propagates gravitational attraction. But it also doesn’t work, analytically. We would see the planets orbit differently if this were how gravity worked. And there’s the problem about mass and energy absorption, as pointed out in the comic. But it can often be interesting or productive to play with models that don’t work. You might learn something about models that do, or that could.