I’m writing this on Thursday, because I’m expecting to be busy Friday and Saturday. It might be a good policy if I planned the deadline for all these Reading the Comics posts to be a couple days before publishing. But it’ll probably ever come to that. I am not yet begun resisting treating this blog like a professional would. Well, what’s been interesting this week so far have been comic strips presenting or about story problems. That’s enough for a theme.

Mac King and Bill King’s Magic in a Minute for the 13th sets out a recreational mathematics puzzle: set twelve things into three bowls so that every bowl has an odd number of things in them. That’s an easy problem to declare impossible, with parity giving us the clue. This comic’s used just that trick before. Possibly the Kings remembered this, and so given answer that plays with what it means to be “in” a bowl.

Olivia Jaimes’s Nancy for the 13th makes the familiar complaint that story problems aren’t “useful”. Perhaps not; she can cut the Gordion knot for most common setups. Well, students aren’t likely to get problems for which there’s no other way to find a solution. I suppose what’s happening is that many mathematical puzzles come from questions like, what’s the least amount of information you need to deduce something? Or what’s an indirect way to find that? Mathematicians are often drawn to questions like this. At least Nancy has found there are problems she’s legitimately interested in, questions about how to do a thing she finds important.

Mort Walker and Dik Browne’s vintage Hi and Lois for the 17th has Chip’s new arithmetic book trying to be more relevant. Chip’s still bored by the problem, which chooses to be about foreign aid. (In the 50s and 60s comics discovered it was very funny that the United States would just give money to other countries and not get anything back out of it except maybe their economies staying stable or their countries not going to war too much.)

One thing to worry about during an A To Z sequence is how busy Comic Strip Master Command will decide I need to be. I’m glad to say that this first week, it wasn’t too overly busy. Even the comic strips that are most on topic are not ones that need too much explanation. They’re also all reruns from their original publication, although I don’t know the dates that any of these first ran. A casual search doesn’t find that I said anything about these in their previous appearances.

Mac King and Bill King’s Magic in a Minute for the 1st is a rerun printed without the editor reading the thing. If they had, they’d have edited the 13 to be a 19. As the explanation at the bottom of the page almost makes clear, the ‘magic number’ produced by this will be the last two digits of the current year. After all, your age (at the end of this year) will be this year minus the year of your birth.

That this can be used for a magic trick relies on two things. One is that while, yes, anyone who thinks about it sees the relationship between their birth year, their age, and the current year, the magic trick is done before they can do that thinking. They’re too busy calculating, and then counting out cards and trying to see where this is going. Calculating without thinking about why this calculation is dangerous for mathematics. But it allows for some recreational fun. the other thing this trick depends on is showmanship: the purpose of the calculation is meant to be surprising enough, and delightful enough, that people won’t care to deconstruct its logic.

Morrie Turner’s Wee Pals rerun for the 1st is your basic joke about the kid subverting a word problem. But it also shows a bit why mathematicians get trained to make as explicit as possible their assumptions. This saves us from dumb mistakes, but at the cost of putting a prologue to anything we do want to ask. But it’s a legitimate part of mathematics to look at the questions someone else has asked and find their unstated assumptions, the things that could be true and would make their claims wrong.

Ryan North’s Dinosaur Comics rerun for the 2nd presents Utahraptor struggling with a mathematics problem. This is in character for him and for the comic. The particular problem is a classic recreational mathematics puzzle. Given a balance that can only give relative weights, and that you can use up to three times, find the one ball out of twelve which is of a different weight. It’s also a classic information theory problem. We know we can solve it, though. Each weighing gives us information about which of the twelve balls might, or might not, be abnormal. There is enough information in these three weighings to pick out which ball is the unusual one.

Granted, though, just knowing three weighings are enough doesn’t tell us what to weigh, or in what order. I haven’t looked at the GoComics comments. But there are likely at least three people who’ve explained some way to do it. It’s worth playing with the problem a while to see if you have any good ideas. You can use coins if you want to play with possibilities.

Ryan North’s Dinosaur Comics rerun for the 6th is, as the last panels suggest, a sequel to a comic rerun in mid-August. The question of whether the word ‘heterological’ is itself heterological is a recasting of one of Eubulides’s paradoxes. It’s the problem of working out whether a self-referential statement can be true. Or false. It shouldn’t surprise us that common language statements can defy being called true or false. But definitions are so close to logical structures that it’s hard to see why these refuse to fit. The problem is silly, but why it’s silly is hard to say.

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

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

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

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

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

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

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

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

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

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

So this past week has been a curious blend for the mathematically-themed comics. There were many comics mentioning some mathematical topic. But that’s because Grand Advenue and Luann Againn — reprints of early 90s Luann comics — have been doing a lot of schoolwork. There’s a certain repetitiveness to saying, “and here we get a silly answer to a story problem” four times over. But we’ll see what I do with the work.

Mark Anderson’s Andertoons for the 7th is Mark Anderson’s Andertoons for the week. Very comforting to see. It’s a geometry-vocabulary joke, with Wavehead noticing the similar ends of some terms. I’m disappointed that I can’t offer much etymological insight. “Vertex”, for example, derives from the Latin for “highest point”, and traces back to the Proto-Indo-European root “wer-”, meaning “to turn, to bend”. “Apex” derives from the Latin for “summit” or “extreme”. And that traces back to the Proto-Indo-European “ap”, meaning “to take, to reach”. Which is all fine, but doesn’t offer much about how both words ended up ending in “ex”. This is where my failure to master Latin by reading a teach-yourself book on the bus during my morning commute for three months back in 2002 comes back to haunt me. There’s probably something that might have helped me in there.

Mac King and Bill King’s Magic in a Minute for the 7th is an activity puzzle this time. It’s also a legitimate problem of graph theory. Not a complicated one, but still, one. Graph theory is about sets of points, called vertices, and connections between points, called edges. It gives interesting results for anything that’s networked. That shows up in computers, in roadways, in blood vessels, in the spreads of disease, in maps, in shapes.

One common problem, found early in studying graph theory, is about whether a graph is planar. That is, can you draw the whole graph, all its vertices and edges, without any lines cross each other? This graph, with six vertices and three edges, is planar. There are graphs that are not. If the challenge were to connect each number to a 1, a 2, and a 3, then it would be nonplanar. That’s a famous non-planar graph, given the obvious name K_{3, 3}. A fun part of learning graph theory — at least fun for me — is looking through pictures of graphs. The goal is finding K_{3, 3} or another one called K_{5}, inside a big messy graph.

Mike Thompson’s Grand Avenue for the 8th has had a week of story problems featuring both of the kid characters. Here’s the start of them. Making an addition or subtraction problem about counting things is probably a good way of making the problem less abstract. I don’t have children, so I don’t know whether they play marbles or care about them. The most recent time I saw any of my niblings I told them about the subtleties of industrial design in the old-fashioned Western Electric Model 2500 touch-tone telephone. They love me. Also I’m not sure that this question actually tests subtraction more than it tests reading comprehension. But there are teachers who like to throw in the occasional surprisingly easy one. Keeps students on their toes.

Greg Evans’s Luann Againn for the 10th is part of a sequence showing Gunther helping Luann with her mathematics homework. The story started the day before, but this was the first time a specific mathematical topic was named. The point-slope form is a conventional way of writing an equation which corresponds to a particular line. There are many ways to write equations for lines. This is one that’s convenient to use if you know coordinates for one point on the line and the slope of the line. Any coordinates which make the equation true are then the coordinates for some point on the line.

Doug Savage’s Savage Chickens for the 10th tosses in a line about logical paradoxes. In this case, using a classic problem, the self-referential statement. Working out whether a statement is true or false — its “truth value” — is one of those things we expect logic to be able to do. Some self-referential statements, logical claims about themselves, are troublesome. “This statement is false” was a good one for baffling kids and would-be world-dominating computers in science fiction television up to about 1978. Some self-referential statements seem harmless, though. Nobody expects even the most timid world-dominating computer to be bothered by “this statement is true”. It takes more than just a statement being about itself to create a paradox.

And a last note. The blog hardly needs my push to help it out, but, sometimes people will miss a good thing. Ben Orlin’s Math With Bad Drawings just ran an essay about some of the many mathematics-themed comics that Hilary Price and Rina Piccolo’s Rhymes With Orange has run. The comic is one of my favorites too. Orlin looks through some of the comic’s twenty-plus year history and discusses the different types of mathematical jokes Price (with, in recent years, Piccolo) makes.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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.

It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.

Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.

Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.

Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.

Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.

There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.

Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.

Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)

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 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.

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.

It was an ordinary enough week when I realized I wasn’t sure about the name of the schoolmarm in Barney Google and Snuffy Smith. So I looked it up on Comics Kingdom’s official cast page for John Rose’s comic strip. And then I realized something about the Smiths’ next-door neighbor Elviney and Jughaid’s teacher Miss Prunelly:

Are … are they the same character, just wearing different glasses? I’ve been reading this comic strip for like forty years and I’ve never noticed this before. I’ve also never heard any of you all joking about this, by the way, so I stand by my argument that if they’re prominent enough then, yes, glasses could be an adequate disguise for Superman. Anyway, I’m startled. (Are they sisters? Cousins? But wouldn’t that make mention on the cast page? There are missing pieces here.)

Mac King and Bill King’s Magic In A Minute feature for the 10th sneaks in here yet again with a magic trick based in arithmetic. Here, they use what’s got to be some Magic Square-based technology for a card trick. This probably could be put to use with other arrangements of numbers, but cards have the advantage of being stuff a magician is likely to have around and that are expected to do something weird.

Susan Camilleri Konair’s Six Chix for the 13th name-drops mathematics as the homework likely to be impossible doing. I think this is the first time Konair’s turned up in a Reading The Comics survey.

Thom Bluemel’s Birdbrains for the 13th is an Albert Einstein Needing Help panel. It’s got your blackboard full of symbols, not one of which is the famous E = mc^{2} equation. But given the setup it couldn’t feature that equation, not and be a correct joke.

John Rose’s Barney Google for the 14th does a little more work than necessary for its subtraction-explained-with-candy joke. I non-sarcastically appreciate Rose’s dodging the obvious joke in favor of a guy-is-stupid joke.

Niklas Eriksson’s Carpe Diem for the 14th is a kind of lying-with-statistics joke. That’s as much as it needs to be. Still, thought always should go into exactly how one presents data, especially visually. There are connotations to things. Just inverting an axis is dangerous stuff, though. The convention of matching an increase in number to moving up on the graph is so ingrained that it should be avoided only for enormous cause.

This joke also seems conceptually close, to me, to the jokes about the strangeness of how a “negative” medical test is so often the good news.

Olivia Walch’s Imogen Quest for the 15th is not about solitaire. But “solving” a game by simulating many gameplays and drawing strategic advice from that is a classic numerical mathematics trick. Whether a game is fun once it’s been solved so is up to you. And often in actual play, for a game with many options at each step, it’s impossible without a computer to know the best possible move. You could use simulations like this to develop general guidelines, and a couple rules that often pan out.

It was again a week just busy enough that I’m comfortable splitting the Reading The Comments thread into two pieces. It’s also a week that made me think about cake. So, I’m happy with the way last week shaped up, as far as comic strips go. Other stuff could have used a lot of work Let’s read.

Stephen Bentley’s Herb and Jamaal rerun for the 13th depicts “teaching the kids math” by having them divide up a cake fairly. I accept this as a viable way to make kids interested in the problem. Cake-slicing problems are a corner of game theory as it addresses questions we always find interesting. How can a resource be fairly divided? How can it be divided if there is not a trusted authority? How can it be divided if the parties do not trust one another? Why do we not have more cake? The kids seem to be trying to divide the cake by volume, which could be fair. If the cake slice is a small enough wedge they can likely get near enough a perfect split by ordinary measures. If it’s a bigger wedge they’d need calculus to get the answer perfect. It’ll be well-approximated by solids of revolution. But they likely don’t need perfection.

This is assuming the value of the icing side is not held in greater esteem than the bare-cake sides. This is not how I would value the parts of the cake. They’ll need to work something out about that, too.

Mac King and Bill King’s Magic in a Minute for the 13th features a bit of numerical wizardry. That the dates in a three-by-three block in a calendar will add up to nine times the centered date. Why this works is good for a bit of practice in simplifying algebraic expressions. The stunt will be more impressive if you can multiply by nine in your head. I’d do that by taking ten times the given date and then subtracting the original date. I won’t say I’m fond of the idea of subtracting 23 from 230, or 17 from 170. But a skilled performer could do something interesting while trying to do this subtraction. (And if you practice the trick you can get the hang of the … fifteen? … different possible answers.)

Bill Amend’s FoxTrot rerun for the 14th mentions mathematics. Young nerd Jason’s trying to get back into hand-raising form. Arithmetic has considerable advantages as a thing to practice answering teachers. The questions have clear, definitely right answers, that can be worked out or memorized ahead of time, and can be asked in under half a panel’s word balloon space. I deduce the strip first ran the 21st of August, 2006, although that image seems to be broken.

Ed Allison’s Unstrange Phenomena for the 14th suggests changes in the definition of the mile and the gallon to effortlessly improve the fuel economy of cars. As befits Allison’s Dadaist inclinations the numbers don’t work out. As it is, if you defined a New Mile of 7,290 feet (and didn’t change what a foot was) and a New Gallon of 192 fluid ounces (and didn’t change what an old fluid ounce was) then a 20 old-miles-per-old-gallon car would come out to about 21.7 new-miles-per-new-gallon. Commenter Del_Grande points out that if the New Mile were 3,960 feet then the calculation would work out. This inspires in me curiosity. Did Allison figure out the numbers that would work and then make a mistake in the final art? Or did he pick funny-looking numbers and not worry about whether they made sense? No way to tell from here, I suppose. (Allison doesn’t mention ways to get in touch on the comic’s About page and I’ve only got the weakest links into the professional cartoon community.)

Patrick Roberts’s Todd the Dinosaur for the 15th mentions long division as the stuff of nightmares. So it is. I guess MathWorld and Wikipedia endorse calling 128 divided by 4 long division, although I’m not sure I’m comfortable with that. This may be idiosyncratic; I’d thought of long division as where the divisor is two or more digits. A three-digit number divided by a one-digit one doesn’t seem long to me. I’d just think that was division. I’m curious what readers’ experiences have been.

So this past week saw a lot of comic strips with some mathematical connection put forth. There were enough just for the 26th that I probably could have done an essay with exclusively those comics. So it’s another split-week edition, which suits me fine as I need to balance some of my writing loads the next couple weeks for convenience (mine).

Tony Cochrane’s Agnes for the 25th of June is fun as the comic strip almost always is. And it’s even about estimation, one of the things mathematicians do way more than non-mathematicians expect. Mathematics has a reputation for precision, when in my experience it’s much more about understanding and controlling error methods. Even in analysis, the study of why calculus works, the typical proof amounts to showing that the difference between what you want to prove and what you can prove is smaller than your tolerance for an error. So: how do we go about estimating something difficult, like, the number of stars? If it’s true that nobody really knows, how do we know there are some wrong answers? And the underlying answer is that we always know some things, and those let us rule out answers that are obviously low or obviously high. We can make progress.

Russell Myers’s Broom Hilda for the 25th is about one explanation given for why time keeps seeming to pass faster as one age. This is a mathematical explanation, built on the idea that the same linear unit of time is a greater proportion of a young person’s lifestyle so of course it seems to take longer. This is probably partly true. Most of our senses work by a sense of proportion: it’s easy to tell a one-kilogram from a two-kilogram weight by holding them, and easy to tell a five-kilogram from a ten-kilogram weight, but harder to tell a five from a six-kilogram weight.

As ever, though, I’m skeptical that anything really is that simple. My biggest doubt is that it seems to me time flies when we haven’t got stories to tell about our days, when they’re all more or less the same. When we’re doing new or exciting or unusual things we remember more of the days and more about the days. A kid has an easy time finding new things, and exciting or unusual things. Broom Hilda, at something like 1500-plus years old and really a dour, unsociable person, doesn’t do so much that isn’t just like she’s done before. Wouldn’t that be an influence? And I doubt that’s a complete explanation either. Real things are more complicated than that yet.

Mac and Bill King’s Magic In A Minute for the 25th features a form-a-square puzzle using some triangles. Mathematics? Well, logic anyway. Also a good reminder about open-mindedness when you’re attempting to construct something.

Norm Feuti’s Retail for the 26th is about how you get good at arithmetic. I suspect there’s two natural paths; you either find it really interesting in your own right, or you do it often enough you want to find ways to do it quicker. Marla shows the signs of learning to do arithmetic quickly because she does it a lot: turning “30 percent off” into “subtract ten percent three times over” is definitely the easy way to go. The alternative is multiplying by seven and dividing by ten and you don’t want to multiply by seven unless the problem gives a good reason why you should. And I certainly don’t fault the customer not knowing offhand what 30 percent off $25 would be. Why would she be in practice doing this sort of problem?

Johnny Hart’s Back To B.C. for the 26th reruns the comic from the 30th of December, 1959. In it … uh … one of the cavemen guys has found his calendar for the next year has too many days. (Think about what 1960 was.) It’s a common problem. Every calendar people have developed has too few or too many days, as the Earth’s daily rotations on its axis and annual revolution around the sun aren’t perfectly synchronized. We handle this in many different ways. Some calendars worry little about tracking solar time and just follow the moon. Some calendars would run deliberately short and leave a little stretch of un-named time before the new year started; the ancient Roman calendar, before the addition of February and January, is famous in calendar-enthusiast circles for this. We’ve now settled on a calendar which will let the nominal seasons and the actual seasons drift out of synch slowly enough that periodic changes in the Earth’s orbit will dominate the problem before the error between actual-year and calendar-year length will matter. That’s a pretty good sort of error control.

8,978,432 is not anywhere near the number of days that would be taken between 4,000 BC and the present day. It’s not a joke about Bishop Ussher’s famous research into the time it would take to fit all the Biblically recorded events into history. The time is something like 24,600 years ago, a choice which intrigues me. It would make fair sense to declare, what the heck, they lived 25,000 years ago and use that as the nominal date for the comic strip. 24,600 is a weird number of years. Since it doesn’t seem to be meaningful I suppose Hart went, simply enough, with a number that was funny just for being riotously large.

Mark Tatulli’s Heart of the City for the 26th places itself on my Grand Avenue warning board. There’s plenty of time for things to go a different way but right now it’s set up for a toxic little presentation of mathematics. Heart, after being grounded, was caught sneaking out to a slumber party and now her mother is sending her to two weeks of Math Camp. I’m supposing, from Tatulli’s general attitude about how stuff happens in Heart and in Lio that Math Camp will not be a horrible, penal experience. But it’s still ominous talk and I’m watching.

Brian Fies’s Mom’s Cancer story for the 26th is part of the strip’s rerun on GoComics. (Many comic strips that have ended their run go into eternal loops on GoComics.) This is one of the strips with mathematical content. The spatial dimension of a thing implies relationships between the volume (area, hypervolume, whatever) of a thing and its characteristic linear measure, its diameter or radius or side length. It can be disappointing.

Nicholas Gurewitch’s Perry Bible Fellowship for the 26th is a repeat of one I get on my mathematics Twitter friends now and then. Should warn, it’s kind of racy content, at least as far as my usual recommendations here go. It’s also a little baffling because while the reveal of the unclad woman is funny … what, exactly, does it mean? The symbols don’t mean anything; they’re just what fits graphically. I think the strip is getting at Dr Loring not being able to see even a woman presenting herself for sex as anything but mathematics. I guess that’s funny, but it seems like the idea isn’t quite fully developed.

Zach Weinersmith’s Saturday Morning Breakfast Cereal Again for the 26th has a mathematician snort about plotting a giraffe logarithmically. This is all about representations of figures. When we plot something we usually start with a linear graph: a couple of axes perpendicular to one another. A unit of movement in the direction of any of those axes represents a constant difference in whatever that axis measures. Something growing ten units larger, say. That’s fine for many purposes. But we may want to measure something that changes by a power law, or that grows (or shrinks) exponentially. Or something that has some region where it’s small and some region where it’s huge. Then we might switch to a logarithmic plot. Here the same difference in space along the axis represents a change that’s constant in proportion: something growing ten times as large, say. The effective result is to squash a shape down, making the higher points more nearly flat.

And to completely smother Weinersmith’s fine enough joke: I would call that plot semilogarithmically. I’d use a linear scale for the horizontal axis, the gazelle or giraffe head-to-tail. But I’d use a logarithmic scale for the vertical axis, ears-to-hooves. So, linear in one direction, logarithmic in the other. I’d be more inclined to use “logarithmic” plots to mean logarithms in both the horizontal and the vertical axes. Those are useful plots for turning up power laws, like the relationship between a planet’s orbital radius and the length of its year. Relationships like that turn into straight lines when both axes are logarithmically spaced. But I might also describe that as a “log-log plot” in the hopes of avoiding confusion.

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.

I’m writing this a little bit early because I’m not able to include the Saturday strips in the roundup. There won’t be enough to make a split week edition; I’ll just add the Saturday strips to next week’s report. In the meanwhile:

Mac King and Bill King’s Magic in a Minute for the 2nd is a magic trick, as the name suggests. It figures out a card by way of shuffling a (partial) deck and getting three (honest) answers from the other participant. If I’m not counting wrongly, you could do this trick with up to 27 cards and still get the right card after three answers. I feel like there should be a way to explain this that’s grounded in information theory, but I’m not able to put that together. I leave the suggestion here for people who see the obvious before I get to it.

Bil Keane and Jeff Keane’s Family Circus (probable) rerun for the 6th reassured me that this was not going to be a single-strip week. And a dubiously included single strip at that. I’m not sure that lotteries are the best use of the knowledge of numbers, but they’re a practical use anyway.

Bill Bettwy’s Take It From The Tinkersons for the 6th is part of the universe of students resisting class. I can understand the motivation problem in caring about numbers of apples that satisfy some condition. In the role of distinct objects whose number can be counted or deduced cards are as good as apples. In the role of things to gamble on, cards open up a lot of probability questions. Counting cards is even about how the probability of future events changes as information about the system changes. There’s a lot worth learning there. I wouldn’t try teaching it to elementary school students.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 6th uses mathematics as the stuff know-it-alls know. At least I suppose it is; Doctor Know It All speaks of “the pathagorean principle”. I’m assuming that’s meant to be the Pythagorean theorem, although the talk about “in any right triangle the area … ” skews things. You can get to stuf about areas of triangles from the Pythagorean theorem. One of the shorter proofs of it depends on the areas of the squares of the three sides of a right triangle. But it’s not what people typically think of right away. But he wouldn’t be the first know-it-all to start blathering on the assumption that people aren’t really listening. It’s common enough to suppose someone who speaks confidently and at length must know something.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th builds on the form of a classic puzzle, about a sequence indexed to the squares of a chessboard. The story being riffed on is a bit of mathematical legend. The King offered the inventor of chess any reward. The inventor asked for one grain of wheat for the first square, two grains for the second square, four grains for the third square, eight grains for the fourth square, and so on, through all 64 squares. An extravagant reward, but surely one within the king’s power to grant, right? And of course not: by the 64th doubling the amount of wheat involved is so enormous it’s impossibly great wealth.

The father’s offer is meant to evoke that. But he phrases it in a deceptive way, “one penny for the first square, two for the second, and so on”. That “and so on” is the key. Listing a sequence and ending “and so on” is incomplete. The sequence can go in absolutely any direction after the given examples and not be inconsistent. There is no way to pick a single extrapolation as the only logical choice.

We do it anyway, though. Even mathematicians say “and so on”. This is because we usually stick to a couple popular extrapolations. We suppose things follow a couple common patterns. They’re polynomials. Or they’re exponentials. Or they’re sine waves. If they’re polynomials, they’re lower-order polynomials. Things like that. Most of the time we’re not trying to trick our fellow mathematicians. Or we know we’re modeling things with some physical base and we have reason to expect some particular type of function.

In this case, the $1.27 total is consistent with getting two cents for every chess square after the first. There are infinitely many other patterns that would work, and the kid would have been wise to ask for what precisely “and so on” meant before choosing.

Berkeley Breathed’s Bloom County 2017 for the 7th is the climax of a little story in which Oliver Wendell Holmes has been annoying people by shoving scientific explanations of things into their otherwise pleasant days. It’s a habit some scientifically-minded folks have, and it’s an annoying one. Many of us outgrow it. Anyway, this strip is about the curious evidence suggesting that the universe is not just expanding, but accelerating its expansion. There are mathematical models which allow this to happen. When developing General Relativity, Albert Einstein included a Cosmological Constant for little reason besides that without it, his model would suggest the universe was of a finite age and had expanded from an infinitesimally small origin. He had grown up without anyone knowing of any evidence that the size of the universe was a thing that could change.

Anyway, the Cosmological Constant is a puzzle. We can find values that seem to match what we observe, but we don’t know of a good reason it should be there. We sciencey types like to have models that match data, but we appreciate more knowing why the models look like that and not anything else. So it’s a good problem some of the cosmologists have been working on. But we’ve been here before. A great deal of physics, especially in the 20th Century, has been driven by looking for reasons behind what look like arbitrary points in a successful model. If Oliver were better-versed in the history of science — something scientifically minded people are often weak on, myself included — he’d be less easily taunted by Opus.

I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.

Since I suspect that the comics roundup posts are the most popular ones I post, I’m very glad to see there was a bumper crop of strips among the ones I read regularly (from King Features Syndicate and from gocomics.com) this past week. Some of those were from cancelled strips in perpetual reruns, but that’s fine, I think: there aren’t any particular limits on how big an electronic comics page one can have, after all, and while it’s possible to read a short-lived strip long enough that you see all its entries, it takes a couple go-rounds to actually have them all memorized.

The first entry, and one from one of these cancelled strips, comes from Mark O’Hare’s Citizen Dog, a charmer of a comic set in a world-plus-talking-animals strip. In this case Fergus has taken the place of Maggie, a girl who’s not quite ready to come back from summer vacation. It’s also the sort of series of questions that it feels like come at the start of any class where a homework assignment’s due.