And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.

David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.

Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.

I like this scheme where I use the Sunday publication slot to list comics that mention mathematics without inspiring conversation. I may need a better name for that branch of the series, though. But, nevertheless, here are comic strips from last week that don’t need much said about them.

John Deering’s Strange Brew for the 24th features Pythagoras, here being asked about his angles. I’m not aware of anything actually called a Pythagorean Angle, but there’s enough geometric things with Pythagoras’s name attached for the joke to make sense.

Maria Scrivan’s Half Full for the 25th is a Venn Diagram joke for the week. It doesn’t quite make sense as a Venn Diagram, as it’s not clear to me that “invasive questions” is sensibly a part of “food”. But it’s a break from every comic strip doing a week full of jokes about turkeys preferring to not be killed.

Tony Carrillo’s F Minus for the 26th is set in mathematics class. And talks about how the process of teaching mathematics is “an important step on the road to hating math”, which is funny because it’s painfully true.

And so the Reading the Comics posts have returned to Sunday after a month in exile to Tuesdays. I’m curious whether Sunday is actually the best day to post my signature series of essays, since everybody is usually doing stuff on the weekends. Tuesdays more people are at work and looking for other things to think about. But at least for the duration of the A to Z series there’s not a good time to schedule them besides Sundays. So Sundays it is and I’ll possibly think things over again in December, if all goes well.

Ralph Hagen’s The Barn for the 27th poses a question that’s ridiculous when you look at it. Why should being twenty times as old as your newborn (sic) when you’re twenty years old imply you’d be twenty times as old as the newborn when you’re sixty? Age increases linearly. The ratios between ages, though, those decrease, in a ratio asymptotically approaching 1. So as far as that goes, this strip isn’t much of anything.

But I do like how it captures the way a mathematics puzzle can come from nowhere. Often interesting ones seem to generate themselves. You notice a pattern and wonder whether it reaches some interesting point. If you convince yourself it does, you wonder when it does. If it does not, you wonder why it can’t. This is the fun sort of mathematics, and you create it by looking at the two separate tile patterns in the kitchen or, as here, thinking about the ages of parent and child. Anything that catches the imagination of a bored mind. It’s fun being there.

Rory (the sheep) makes a common enough slip. Saying a twenty-year-old with a newborn is twenty times as old as the newborn is, implicitly, saying the newborn is one year old. This kind of error is so common it’s got a folksy name, the “fencepost error”. It has a more respectable name, for its LinkedIn profile, the “off-by-one error”. But you see the problem. Say that your birthday is the 1st of September. How many times were you alive on the 1st of September by the time you’re ten years old? Eleven times, the first one being the one you were born on, with one more counted up each year you’d lived. This was probably more clear before I explained it.

John Rose’s Barney Google and Snuffy Smith for the 27th has Mis Prunelly complimenting Jughaid’s creativity, but not wanting it in arithmetic. There is creativity in mathematics. And there is great value in calculating something in an original way. There’s value in calculating things wrong, too, if it’s an approximate calculation. Knowing whether your answer is nearer 10 or 20 is of some value, and it might be all that you in fact want. That’s being wrong in a productive way, though.

Harry Bliss and Steve Martin’s Bliss for the 27th uses a string of mathematical symbols as emblem of genius. Most of the symbols look just near enough meaningful that I wonder if Bliss and Martin got a mathematician friend of theirs to give them some scraps. Why I say mathematician rather than, say, physicist is because some of the lines look more mathematician than physicist.

The most distinctive one, to me, is right above Dumbo’s pencil and trunk there: . This is the kind of equation you’ll see all the time in group theory. It’s an important field of mathematics, the one studying sets that work like arithmetic does. This starts with groups, which have a set of things and a binary operation between those things. Think of it as either addition or multiplication. You notice that already looks like multiplication. ‘g’ and ‘h’ serve, for group theory, the roles that ‘x’ and ‘y’ do in (high school) algebra. ‘x’ and ‘y’ mean some number, whose value we might or might not care about. Similarly, ‘g’ and ‘h’ are some elements, things in the set for our group. We might or might not care which ones they are. means the identity element, the thing which won’t change the value of the other partner in an operation. The thing that works like zero for addition, or like one for multiplication. And means the inverse of : the thing which, added (or multiplied) to gives us the identity element. So if we were talking addition and were 5, then would be -5. This might not sound like very much, but we can make it complicated.

Also distinctive to me: that first line. I’m not perfectly sure I’m transcribing this right. But it looks a good deal to me like the binomial distribution. This is the probability of seeing something happen k times, if you give it n chances to happen, and every chance has the same probability p of it happening. The formula isn’t quite right. It’s missing a power on the (1 – p) term at the end. But it’s wrong in ways that make sense for the need to draw something legible.

Just under Dumbo’s pencil, too, is a line that I had to look up how to render in WordPress’s LaTeX. It’s the one about . The union symbol, the U there, speaks of set theory. It means to form a new set, one that has all the elements in the set called X or the set called Y or both. The straight vertical lines flanking these set names or descriptions are how we describe taking the norm, finding the size, of a set. This is ordinarily how many things are inside the set. If the sets X and Y have no elements in common, then the size of the union of X and Y will be the size of the set X plus the size of the set Y.

There’s other lines that come near making sense. The line about has the form of the “mapping” way to define a function. I just don’t understand what the rule here means. The final line, , first … well, this sort of e-raised-to-the-minus-something-squared form turns up all the time. But second, to end a bit of work with an exclamation point really captures the surprise and joy of having reached a goal. Mathematicians take delight in their work, like you’d expect.

Maria Scrivan’s Half Full for the 29th is a Rubik’s Cube joke. A variation of it ran back in June 2018. I hate that this time I noticed that on the right, the cubelet — with white on top, red on the lower left, and green on the lower right — is inconsistent with the ordered cube. The corresponding cubelet there has blue on top, red on the lower left, and green on the lower right. Well, maybe the cube on the right had its color stickers applied differently. This is a little thing. But it’s close to a problem that turns up all the time in representing geometry. It’s easy to say you have, say, axes going in the x, y, and z directions. But which direction is x? Which is y? Which is z? You can lay all three out so every pair makes a right angle. Whatever way you lay them out will turn out to be, up to a rotation, one of two patterns. Let’s say the x axis points east, and the y axis points north. Then the z axis can point up. Or it can point down. You can pick which one makes sense for your problem. The two choices are mirror images of the other. You get primed to notice this when you do mathematical physics. The Rubik’s Cube on the left is just this kind of representation, with (let’s say) the red face pointing in the x direction, the green face pointing in the y direction, and the blue pointing in the z direction. Which is a lot of thought to put into what was an arbitrary choice, as I’m sure the cartoonist (or whoever did the coloring) just wanted a cube that looked attractive.

Bill Holbrook’s On The Fastrack for the 18th is an anthropomorphic numerals joke. It’s part of Holbrook’s style to draw metaphors as literal happenings. It’s also a variation on a joke Holbrook used just last month, depicting then the phrase “accepting his numbers”. What I said about “accepting numbers” transfers over naturally to “trusting numbers”. It’s not that a number itself means anything. It’s that numbers are used to represent some narrative. If we can’t believe the narrative, we don’t believe the numbers. And the numbers used to represent something can give us reasons to trust, or reject, a narrative.

Eric the Circle for the 18th I can dub an anthropomorphic geometry joke for the week. At least it brings up one of the handful of geometry facts that people remember outside school. The relationship between the circumference and the diameter (or radius, if you rather) of a circle has been known just forever. It has the advantage of going through π, supporting and being supported by that celebrity number. … I’m not quite sure about the logic of this joke, though. My experience is that guys at least are fairly good about knowing their waist size (if you don’t know, it’s 38, although a 40 can feel so comfortable, and they’re sure they can wear a 36). Radius is a harder thing to keep in mind. But maybe it’s different for circles.

Russell Myers’s Broom Hilda for the 19th is a student-and-teacher problem. One thing is that Nerwin’s not wrong. It’s just that simply saying something true isn’t enough. We want to say things that are true and interesting.

But “you add two numbers and get a number” can be interesting. It depends on context. For example, in group theory, we will start by describing groups as a collection of things and an operation which works like addition. What does it mean to work like addition? Here, it means if you add two things from the collection, you get something from the collection. The collection of things is “closed” under your operation. And mathematical operations defined this abstractly — or defined this vaguely, if you don’t like the way it goes — can be great. We’re introduced to vectors, for example, as “ordered sets of numbers”. And that definition works all right. But when you start thinking of them instead as “things you can add to vectors and get other vectors out” you gain new power. You can use the mechanism developed for ordered sets of numbers to describe many things, including matrices and functions and shapes. But when we do that we’re saying things about how addition works, rather than what this particular addition is.

You know, on reflection, I’m not sure that Eric the Circle was more worthy of discussion than that Barney Google was. Hm.

To close out last week’s comics I got a bunch of strips that were repeats, or that touch on topics I’ve discussed quite a bit around these parts already. I’m pretty sure all the words I have here are new in their specific organization. The words themselves are pretty old.

Maria Scrivan’s Half Full for the 4th is the Rubik’s Cube joke for the week. I ought to write up a proper description of the algebra of Rubik’s Cubes. The real stuff is several books’ worth of material, yes. But a couple hundred words about what’s interesting should be doable. … Or I could just ask folks if they’ve read good descriptions of the group theory that cubes show off. I’m always open to learning other people have said stuff better than me. This is part of why I’ve never published an essay about Cantor’s Diagonal Proof; many people have written such essays and I couldn’t add anything useful to that heap of words.

Ryan North’s Dinosaur Comics for the 5th is about the heap paradox. Or the sorites paradox, depending on what book you’ve been reading from. The problem is straightforward enough. As God, in the strip says, a big pile of sand is clearly a heap. One or two grains of sand is clearly not. If you remove grains from the heap, eventually, you lose the heap-ness. T-Rex suggests solving the question of when that happens by statistical survey, finding what people on average find to be the range where things shift over.

As with many attempts to apply statistical, or experimental, methods to philosophical questions it misses the point. There are properties that things seem to have only as aggregations. Where do they come from? How can there be something true about a collection of things that isn’t true about any part of the thing? This is not just about messy real-world properties either; we can say stuff about groups of mathematical objects that aren’t true about individual objects within the set. For example, suppose we want to draw a real number at random, uniformly, from the continuous interval 0 to 10. There’s a 50% chance we’ll draw a number greater than 5. The chance of drawing any specific number greater than 5, though, is zero. But we can always draw one. Something weird is happening here, as often happens with questions we’ve been trying to answer for thousands of years.

Norm Feuti’s Retail for the 6th is a new strip, although the joke’s appeared before. There’s some arithmetic calculations that are easy to do, or that become easy because you do them a lot. Or because you see them done a lot and learn what the patterns are. A handful of basic tricks — like that 80 percent off is 20 percent of something, or that 20 percent of a thing is one-fifth the original thing — can be stunning. Stage magicians find the same effect.

Bob Shannon’s Tough Town for the 7th is the anthropomorphized abacus joke of the week. Been a while since we had one of those. I suppose an adding machine would be at least as good a representative of the abstract concept of doing arithmetic, but it’s likely harder to draw too. This is just tiring to draw.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th presents the old complaint about mathematics’s utility, here in an ancient setting. I’m intereste that the caveman presents counting in terms of matching up other things to his fingers. We use this matching of one set of things to another even today. It gets us to ordinal and cardinal numbers, and the to what we feel pretty sure about with infinitely large sets. An idea can be ancient and basic and still be vital.

Steve Sicula’s Home and Away for the 9th is about the hatred people profess for mathematics. Some of that is more hatred of how it’s taught, which is too often as a complicated and apparently pointless activity. Some of that is hatred of how it’s used, since it turns up in a lot of jobs. And for some reason we’ve designed society so that we do jobs we don’t like. I don’t know why we think that’s a good idea. We should work on that.

I should have got to this yesterday; I don’t know. Something happened. Should be back to normal Sunday.

Bill Rechin’s Crock rerun for the 26th of April does a joke about picking-the-number-in-my-head. There’s more clearly psychological than mathematical content in the strip. It shows off something about what people understand numbers to be, though. It’s easy to imagine someone asked to pick a number choosing “9”. It’s hard to imagine them picking “4,796,034,621,322”, even though that’s just as legitimate a number. It’s possible someone might pick π, or e, but only if that person’s a particular streak of nerd. They’re not going to pick the square root of eleven, or negative eight, or so. There’s thing that are numbers that a person just, offhand, doesn’t think of as numbers.

Mark Anderson’s Andertoons for the 26th sees Wavehead ask about “borrowing” in subtraction. It’s a riff on some of the terminology. Wavehead’s reading too much into the term, naturally. But there are things someone can reasonably be confused about. To say that we are “borrowing” ten does suggest we plan to return it, for example, and we never do that. I’m not sure there is a better term for this turning a digit in one column to adding ten to the column next to it, though. But I admit I’m far out of touch with current thinking in teaching subtraction.

Greg Cravens’s The Buckets for the 26th is kind of a practical probability question. And psychology also, since most of the time we don’t put shirts on wrong. Granted there might be four ways to put a shirt on. You can put it on forwards or backwards, you can put it on right-side-out or inside-out. But there are shirts that are harder to mistake. Collars or a cut around the neck that aren’t symmetric front-to-back make it harder to mistake. Care tags make the inside-out mistake harder to make. We still manage it, but the chance of putting a shirt on wrong is a lot lower than the 75% chance we might naively expect. (New comic tag, by the way.)

Charles Schulz’s Peanuts rerun for the 27th is surely set in mathematics class. The publication date interests me. I’m curious if this is the first time a Peanuts kid has flailed around and guessed “the answer is twelve!” Guessing the answer is twelve would be a Peppermint Patty specialty. But it has to start somewhere.

Knowing nothing about the problem, if I did get the information that my first guess of 12 was wrong, yeah, I’d go looking for 6 or 4 as next guesses, and 12 or 48 after that. When I make an arithmetic mistake, it’s often multiplying or dividing by the wrong number. And 12 has so many factors that they’re good places to look. Subtracting a number instead of adding, or vice-versa, is also common. But there’s nothing in 12 by itself to suggest another place to look, if the addition or subtraction went wrong. It would be in the question which, of course, doesn’t exist.

Maria Scrivan’s Half-Full for the 28th is the Venn Diagram joke for this week. It could include an extra circle for bloggers looking for content they don’t need to feel inspired to write. This one isn’t a new comics tag, which surprises me.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th uses the M&oum;bius Strip. It’s an example of a surface that you could just go along forever. There’s nothing topologically special about the M&oum;bius Strip in this regard, though. The mathematician would have as infinitely “long” a résumé if she tied it into a simple cylindrical loop. But the M&oum;bius Strip sounds more exotic, not to mention funnier. Can’t blame anyone going for that instead.

So Mark Anderson’s Andertoons has been missing from the list of mathematically-themed the last couple weeks. Don’t think I haven’t been worried about that. But it’s finally given another on-topic-enough strip and I’m not going to include it here. I’ve had a terrible week and I’m going to use the comics we got in last week slowly.

Greg Evans’s Luann Againn for the 10th reprints the strip of the 10th of December, 1989. And as often happens, mathematics is put up as the stuff that’s too hard to really do. The expressions put up don’t quite parse; there’s nothing to solve. But that’s fair enough for a panicked brain. To not recognize what the problem even is makes it rather hard to solve.

Ruben Bolling’s Super-Fun-Pak Comix for the 10th is an installation of Quantum Mechanic, playing on the most fun example of non-commutative processes I know. That’s the uncertainty principle, which expresses itself as pairs of quantities that can’t be precisely measured simultaneously. There are less esoteric kinds of non-commutative processes. Like, rotating something 90 degrees along a horizontal and then along a vertical axis will turn stuff different from 90 degrees vertical and then horizontal. But that’s too easy to understand to capture the imagination, at least until you’re as smart as an adult and as thoughtful as a child.

Jeff Stahler’s Moderately Confused for the 11th features the classic blackboard full of equations, this time to explain why Christmas lights wouldn’t work. There is proper mathematics in lights not working. It’s that electrical-engineering work about the flow of electricity. The problem is, typically, a broken or loose bulb. Maybe a burnt-out fuse, although I have never fixed a Christmas lights problem by replacing the fuse. It’s just something to do so you can feel like you’ve taken action before screaming in rage and throwing the lights out onto the front porch. More interesting to me is the mathematics of strands getting tangled. The idea — a foldable thread, marked at regular intervals by points that can hook together — seems trivially simple. But it can give insight into how long molecules, particularly proteins, will fold together. It may help someone frustrated to ponder that their light strands are knotted for the same reasons life can exist. But I’m not sure it ever does.

Stephan Pastis’s Pearls Before Swine for the 13th is one of those equations-of-everything jokes. Naturally it features a panel full of symbols that, to my eye, don’t parse. There are what look like syntax errors, for example, with the one that anyone could see the { mark that isn’t balanced by a }. But when someone works rough they will, often, write stuff that doesn’t quite parse. Think of it as an artist’s rough sketch of a complicated scene: the lines and anatomy may be gibberish, but if the major lines of the composition are right then all is well.

Most attempts to write an equation for everything are really about writing a description of the fundamental forces of nature. We trust that it’s possible to go from a description of how gravity and electromagnetism and the nuclear forces go to, ultimately, a description of why chemistry should work and why ecologies should form and there should be societies. There are, as you might imagine, a number of assumed steps along the way. I would accept the idea that we’ll have a unification of the fundamental forces of physics this century. I’m not sure I would believe having all the steps between the fundamental forces and, say, how nerve cells develop worked out in that time.

Mark Anderson’s Andertoons makes it overdue appearance for the week on the 14th, with a chalkboard word-problem joke. Amusing enough. And estimating an answer, getting it wrong, and refining it is good mathematics. It’s not just numerical mathematics that will look for an approximate solution and then refine it. As a first approximation, 15 minus 7 isn’t far off 10. And for mental arithmetic approximating 15 minus 7 as 10 is quite justifiable. It could be made more precise if a more exact answer were needed.

Maria Scrivan’s Half Full for the 14th I’m going to call the anthropomorphic geometry joke for the week. If it’s not then it’s just wordplay and I’d have no business including it here.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 14th tosses in the formula describing how strong the force of gravity between two objects is. In Newtonian gravity, which is why it’s the Newton Police. It’s close enough for most purposes. I’m not sure how this supports the cause of world peace.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th names Riemann’s Quaternary Conjecture. I was taken in by the panel, trying to work out what the proposed conjecture could even mean. The reason it works is that Bernhard Riemann wrote like 150,000 major works in every field of mathematics, and about 149,000 of them are big, important foundational works. The most important Riemann conjecture would be the one about zeroes of the Riemann Zeta function. This is typically called the Riemann Hypothesis. But someone could probably write a book just listing the stuff named for Riemann, and that’s got to include a bunch of very specific conjectures.

The rest of last week had more mathematically-themed comic strips than Sunday alone did. As sometimes happens, I noticed an objectively unimportant detail in one of the comics and got to thinking about it. Whether I could solve the equation as posted, or whether at least part of it made sense as a mathematics problem. Well, you’ll see.

Patrick McDonnell’s Mutts for the 25th of September I include because it’s cute and I like when I can feature some comic in these roundups. Maybe there’s some discussion that could be had about what “equals” means in ordinary English versus what it means in mathematics. But I admit that’s a stretch.

Olivia Walch’s Imogen Quest for the 25th uses, and describes, the mathematics of a famous probability problem. This is the surprising result of how few people you need to have a 50 percent chance that some pair of people have a birthday in common. It then goes over to some other probability problems. The examples are silly. But the reasoning is sound. And the approach is useful. To find the chance of something happens it’s often easiest to work out the chance it doesn’t. Which is as good as knowing the chance it does, since a thing can either happen or not happen. At least in probability problems, which define “thing” and “happen” so there’s not ambiguity about whether it happened or not.

Piers Baker’s Ollie and Quentin rerun for the 26th I’m pretty sure I’ve written about before, although back before I included pictures of the Comics Kingdom strips. (The strip moved from Comics Kingdom over to GoComics, which I haven’t caught removing old comics from their pages.) Anyway, it plays on a core piece of probability. It sets out the world as things, “events”, that can have one of multiple outcomes, and which must have one of those outcomes. Coin tossing is taken to mean, by default, an event that has exactly two possible outcomes, each equally likely. And that is near enough true for real-world coin tossing. But there is a little gap between “near enough” and “true”.

Rick Stromoski’s Soup To Nutz for the 27th is your standard sort of Dumb Royboy joke, in this case about him not knowing what percentages are. You could do the same joke about fractions, including with the same breakdown of what part of the mathematics geek population ruins it for the remainder.

Nate Fakes’s Break of Day for the 28th is not quite the anthropomorphic-numerals joke for the week. Anthropomorphic mathematics problems, anyway. The intriguing thing to me is that the difficult, calculus, problem looks almost legitimate to me. On the right-hand-side of the first two lines, for example, the calculation goes from

to

This is a little sloppy. The first line ought to end in a ‘dt’, and the second ought to have a constant of integration. If you don’t know what these calculus things are let me explain: they’re calculus things. You need to include them to express the work correctly. But if you’re just doing a quick check of something, the mathematical equivalent of a very rough preliminary sketch, it’s common enough to leave that out.

It doesn’t quite parse or mean anything precisely as it is. But it looks like the sort of thing that some context would make meaningful. That there’s repeated appearances of , or , particularly makes me wonder if Frakes used a problem he (or a friend) was doing for some reason.

Mark Anderson’s Andertoons for the 29th is a welcome reassurance that something like normality still exists. Something something student blackboard story problem something.