Reading the Comics, October 2019: Cube Edition


Comic Strip Master Command hoped to give me an easy week, one that would let me finally get ahead on my A-to-Z essays and avoid the last-minute rush to complete tasks. I showed them, though. I can procrastinate more than they can give me breaks. This essay alone I’m writing about ten minutes after you read it.

Eric the Circle for the 7th, by Shoy, is one of the jokes where Eric’s drawn as something besides a circle. I can work with this, though, because the cube is less far from a circle than you think. It gets to what we mean by “a circle”. If it’s all the points that are exactly a particular distance from a given center? Or maybe all the points up to that particular distance from a given center? This seems too reasonable to argue with, so you know where the trick is.

Drawing of a cube. Caption: 'Eric the Circle is trying to develop his Halloween costume into a more 3-dimensional character.'
Eric the Circle for the 7th of October, 2019. This one by Shoy. Essays with appearances by Eric the Circle should be at this link.

The trick is asking what we mean by distance? The ordinary distance that normal people use has a couple names. The Euclidean distance, often. Or Euclidean metric. Euclidean norm. It has some fancier names that can wait. Give two points. You can find this distance easily if you have their coordinates in a Cartesian system. (There’s infinitely many Cartesian systems you could use. You can pick whatever one you like; the distance will be the same whatever they are.) That’s that thing about finding the distance between corresponding coordinates, squaring those distances, adding that up, and taking the square root. And that’s good.

That’s not our only choice, though. We can make a perfectly good distance using other rules. For example, take the difference between corresponding coordinates, take the absolute value of each, and add all those absolute values up. This distance even has real-world application. It’s how far it is to go from one place to another on a grid of city squares, where it’s considered poor form to walk directly through buildings. There’s another. Instead of adding those absolute values up? Just pick the biggest of the absolute values. This is another distance. In it, circles look like squares. Or, in three dimensions, spheres look like cubes.

T-Rex: 'The world of tomorrow! We detect evidence of alien life: a signal from a distant solar system. It starts out simple, numbers, counting. Then it gets to more advanced mathematics. Then: instructions for decoding images about to be transmitted. In months we go from hoping we're not alone to about to find out what they look like!' Dromiceiomimus: 'What do they look like?' T-Rex: 'WHO KNOWS? We're at the frontier of imagination here!' Utahraptor: 'Maybe they look like puppies?' T-Rex: 'Please! No way aliens look like puppies!' Utahraptor: 'For generations we wondered if we're alone. Today we know: no. Life is out there, and it's puppies. Our universe is teeming with puppies.' T-Rex: 'Fine, I accept this version of events. THE END.'
Ryan North’s Dinosaur Comics for the 9th of October, 2019. This and other essays with Dinosaur Comics discussed should be at this link.

Ryan North’s Dinosaur Comics for the 9th builds on a common science fictional premise, that contact with an alien intelligence is done through mathematics first. It’s a common supposition in science fiction circles, and among many scientists, that mathematics is a truly universal language. It’s hard to imagine a species capable of communication with us that wouldn’t understand two and two adding up to four. Or about the ratio of a circle circumference to its diameter being independent of that diameter. Or about how an alternating knot for which the minimum number of crossing points is odd can’t ever be amphicheiral.

All right, I guess I can imagine a species that never ran across that point. Which is one of the things we suppose in using mathematics as a universal language. Its truths are indisputable, if we allow the rules of logic and axioms and definitions that we use. And I agree I don’t know that it’s possible not to notice basic arithmetic and basic geometry, not if one lives in a sensory world much like humans’. But it does seem to me at least some of mathematics is probably idiosyncratic. In representation at least; certainly in organization. I suspect there may be trouble in using universal and generically true things to express something local and specific. I don’t know how to go from deductive logic to telling someone when my birthday is. Well, I’m sure our friends in the philosophy department have considered that problem and have some good thoughts we can use, if there were only some way to communicate with them.

Mathematician filling a blackboard with: 'ENOUGH = [ long ,multi-level expression ] = ENOUGH'. Caption: 'Theoretical mathematician proving why he's had all he can take.'
Bill Whitehead’s Free Range for the 12th of October, 2019. Essays that have some mention of Free Range should appear at this link.

Bill Whitehead’s Free Range for the 12th is your classic blackboard-full-of-symbols. I like the beauty of the symbols used. I mean, the whole expression doesn’t parse, but many of the symbols do and are used in reasonable ways. Long trailing strings of arrows to extend one line to another are common and reasonable too. In the middle of the second line is \vec{n = z}, which doesn’t make sense, but which doesn’t make sense in a way that seems authentic to working out an idea. It’s something that could be cleaned up if the reasoning needed to be made presentable.


Later this week I’ll run a list of the fair number of comics that mentioned mathematics along the way to a joke, but that don’t do much with that mention. And this week in the A-to-Z sequence should see both M and N given their chances.

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Reading the Comics, September 28, 2019: Modeling Edition


The second half of last week’s mathematically-themed comic strips had an interesting range of topics. Two of them seemed to circle around the making of models. So that’s my name for this installment.

Ryan North’s Dinosaur Comics for the 26th has T-Rex trying to build a model. In this case, it’s to project how often we should expect to see a real-life Batman. T-Rex is building a simple model, which is fine. Simple models, first, are usually easier to calculate with. How they differ from reality can give a guide to how to make a more complex model. Or they can indicate the things that have to be learned in order to make a more complex model. The difference between a model’s representation and the observed reality (or plausibly expected reality) can point out problems in one’s assumptions, too.

T-Rex: 'Start with the number of children born to billionaires each year! Multiply by the chance of someone becoming an Olympic athlete! And multiply that by the unfortunate chance someone will witness their parents become victims of a violent crime as a child!' Dromiceiomimus: 'Good god! You're calculating---' T-Rex: 'YES. The expected real-life Batman generation rate.' Utahraptor: 'What do you get?' T-Rex: 'There's only about 1000 billionaires worldwide.' Utahraptor: 'And there were 2600 athletes last Olympics, so your odds are 1 in 2,307,692 of such peak physicality.' T-Rex: 'And if we estimate a 0.0001 chance of parent murder then ... that's one Batman every 25 million years, assuming every billionaire has a child each year. And is murdered each year. And I didn't even work the odds of becoming friends with Superman. I hate to say it, but reality SUCKS sometimes.'
Ryan North’s Dinosaur Comics for the 26th of September, 2019. Essays featuring discussion of some topic raised by Dinosaur Comics should appear at this link.

For example, T-Rex supposes that a Batman needs to have billionaire parents. This makes for a tiny number of available parents. But surely what’s important is that a Batman be wealthy enough he doesn’t have to show up to any appointments he doesn’t want to make. Having a half-billion dollars, or a “mere” hundred million, would allow that. Even a Batman who had “only” ten million dollars would be about as free to be a superhero. Similarly, consider the restriction to Olympic athletes. Astronaut Ed White, who on Gemini IV became the first American to walk in space, was not an Olympic athlete; but he certainly could have been. He missed by a split-second in the 400 meter hurdles race. Surely someone as physically fit as Ed White would be fit enough for a Batman. Not to say that “Olympic athletes or NASA astronauts” is a much bigger population than “Olympic athletes”. (And White was unusually fit even for NASA astronauts.) But it does suggest that merely counting Olympic athletes is too restrictive.

But that’s quibbling over the exact numbers. The process is a good rough model. List all the factors, suppose that all the factors are independent of one another, and multiply how likely it is each step happens by the population it could happen to. It’s hard to imagine a simpler model, but it’s a place to start.

'When Juanita entered the picture, the love triangle between Ken, Debra, and Bill became a love rhombus. But only when they convened at opposite, equal acute angles and opposite, equal obtuse angles. Otherwise, they were just a parallelogram looking for a good time.'
Greg Wallace’s Nothing Is Not Something for the 26th of September, 2019. I don’t seem to have tagged this strip before! Well, this essay and any future ones based on Nothing Is Not Something should appear at this link.

Greg Wallace’s Nothing Is Not Something for the 26th is a bit of a geometry joke. It’s built on the idiom of the love triangle, expanding it into more-sided shapes. Relationships between groups of people like this can be well-represented in graph theory, with each person a vertex, and each pair of involved people an edge. There are even “directed graphs”, where each edge contains a direction. This lets one represent the difference between requited and unrequited interests.

Sophie, dog, to Conspiracy Squirrels who have a drill digging up ground: 'What're you doing?' Left Squirrel: 'Digging all through the Earth.' Right Squirrel: 'To prove it's not flat.' Sophie: 'Ambitious. You know there's easier ways to prove Earth's round?' Right: 'ROUND?' Left: 'The Earth is a smushed rhombus. Everyone knows that!' Right: 'Where'd you go to school, Eddie Bravo University?' Left: 'If Earth is round what keeps it from rolling into the sun?' Sophie: 'OK, then, careful not to fall out the other side. Gravity's a conspiracy created by Canada geese to keep us out of the sky.' Left: 'Really?' Right, dashing off: 'I'll get the magnetic boots!'
Brian Anderson’s Dog Eat Doug for the 27th of September, 2019. The essays exploring some topic raised by Dog Eat Doug should appear at this link.

Brian Anderson’s Dog Eat Doug for the 27th has Sophie the dog encounter some squirrels trying to disprove a flat Earth. They’re not proposing a round Earth either; they’ve gone in for a rhomboid. Sophie’s right to point out that drilling is a really hard way to get through the Earth. That’s a practical matter, though.

Is it possible to tell something about the shape of a whole thing from a small spot? In the terminology, what kind of global knowledge can we get from local information? We can do some things. For example, we can draw a triangle on the surface of the Earth and measure the interior angles to see what they sum to. If this could be done perfectly, finding that the interior angles add up to more than 180 degrees would show the triangle’s on a spherical surface. But that also has practical limitations. Like, if we find that locally the planet is curved then we can rule out it being entirely flat. But it’s imaginable that we’d be on the one dome of an otherwise flat planet. At some point you have to either assume you’re in a typical spot, or work out ways to find what’s atypical. In the Conspiracy Squirrels’ case, that would be the edge between two faces of the rhomboid Earth. Then it becomes something susceptible to reason.

Mathematician at chalkboard full of symbols: 'Thus we arrive at the conclusion that one could go to a pay-by-weight salad bar and earn money by eating cheese, which is clearly impossible.' Caption: 'Disproving the idea of negative mass was remarkably easy.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of September, 2019. It’s not literally true that every Reading the Comics essay includes this strip. The essays with Saturday Morning Breakfast Cereal in them are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th has the mathematician making another model. And this is one of the other uses of a model: to show a thing can’t happen, show that it would have results contrary to reason. But then you have to validate the model, showing that its premises do represent reality so well that its conclusion should be believed. This can be hard. There’s some nice symbol-writing on the chalkboard here, although I don’t see that they parse. Particularly, the bit on the right edge of the panel, where the writing has a rotated-by-180-degrees ‘E’ followed by an ‘x’, a rotated-by-180-degrees ‘A’, and then a ‘z’, is hard to fit inside an equation like this. The string of symbols mean “there exists some x for which, for all z, (something) is true”. This fits at the start of a proof, or before an equation starts. It doesn’t make grammatical sense in the middle of an equation. But, in the heat of writing out an idea, mathematicians will write out ungrammatical things. As with plain-text writing, it’s valuable to get an idea down, and edit it into good form later.

Computer, talking to itself in a MICR-inspired font: 'Many people are amazed at the complex mathematical ability of a computer. Actually though, the concept is quite simple! Inside we're just filled with thousands of toes that we can count on!'
Tom Batiuk’s Funky Winkerbean Vintage for the 28th of September, 2019. This strip originally ran the 15th of November, 1973. Both 1970s-era vintage and such 2010s-era modern Funky Winkerbean strips which inspire discussion should be at this link.

Tom Batiuk’s Funky Winkerbean Vintage for the 28th sees the school’s Computer explaining the nature of its existence, and how it works. Here the Computer claims to just be filled with thousands of toes to count on. It’s silly, but it is the case that there’s no operation a computer does that isn’t something a human can do, manually. If you had the paper and the time you could do all the steps of a Facebook group chat, a game of SimCity, or a rocket guidance computer’s calculations. The results might just be impractically slow.


And that’s finished the comic strips of last week! Sunday I should have a new Reading the Comics post. And then tomorrow I hope to resume the Fall 2019 A to Z series with ‘J’. Thanks for reading.

Reading the Comics, September 7, 2019: Dinosaur Follow-Up Edition


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.

Here's a simple card trick to use any elephants you have. Ask a friend to write the last two digits of the year he was born. Below that, write the age he'll be at the end of this year. Have your friend add his age to the year he was born; this will be his lucky number. Place a deck of cards on a table and tell your friend to deal out a number of cards equal to his lucky number. If his number has three digits, just use the last two. Instruct your friend to look at the last card he deals out, to not show it to you, but to close his eyes and concentrate on it. While your friend's eyes are closed, bring in your elephant in a blanket displaying the card he chose! He'll be impressed! (The secret is this will always be the same number: the last two digits of the current year, so, before you start place the card you want in the right spot and 'reveal' that.)
Mac King and Bill King’s Magic in a Minute for the 1st of September, 2019. Cute little arithmetic, logic, and related mathematical puzzles from Other appearances by Magic In A Minute should be behind this link.

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.

Oliver: 'Here's a math quiz for you, Mikki. If I gave you three jelly beans and George gave you five jelly beans, how many jelly beans would you have?' Mikke: 'Eleven jelly beans, Oliver.' Oliver: 'Wrong! The answer is eight!' Mikki: 'No, it's eleven. I already have three jelly beans.'
Morrie Turner’s Wee Pals rerun for the 1st of September, 2019. Essays inspired by a refusal of any of the Wee Pals to do word problems should appear at this link.

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.

Trouble 4 Utahraptor comics! T-Rex: 'Oh no! Is Utahraptor in trouble again? Has trouble finally come ... for Utahraptor? What can we do, Dromiceiomius?' Dromiceiomimus: 'I imagine the first step is to ascertain what kind of trouble he's in!' Utahraptor: 'You called, T-Rex?' 'I heard you're in trouble!' 'I am! It seems I get into trouble five times a week, but luckily I've got great pals to help me out!' T-Rex: 'That's me!' Utahraptor: 'Yes, so here's my trouble. I have 12 identical balls, but one is EITHER heavier or lighter than the rest, and I've got a balance scale that measures relative weight.' 'This sounds complicated!' 'And I can only use the scale 3 times to find out which ball is different!' 'T-Rex: I'm going home okay.'
Ryan North’s Dinosaur Comics rerun for the 2nd of September, 2019. There are a good number of Dinosaur Comics discussed in essays here.

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.

T-Rex: 'The best words MIGHT be autological words, you guys! These are words that describe something that also describes the word itself! For example, the word 'short' is itself short, and the word 'multisyllabic' has more than one syllable in it!' Dromiceiomius: 'And the word 'understandable' is itself understandable, so it's autological too!' T-Rex: 'Just like '2oig3nt2as2y' which is a word I just made up that means 'annoying to say'!' Utahraptor: 'There's the opposite too, heterological words that don't describe themselves, like 'long'!' T-Rex: 'Neat!' Utahraptor: 'So is 'heterological' heterological?' T-Rex: 'Well, if it IS then it's self-describing, which means autological. And if it ISN'T, then it's autological again too. Huh. If this paradox is supposed to make me trip balls, you should know I've taken the precaution of having them TIGHTLY SECURED.' Later. Utahraptor: 'Hey, who put all my balls in the closet? And then tied the closet handle shut and attached a note 'NEVER AGAIN'?' T-Rex: 'Sir, calm down! You can thank me whenever!!'
Ryan North’s Dinosaur Comics rerun for the 6th of September, 2019. There are an equal number of Dinosaur Comics discussed in essays here.

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.


There were also comics so casual in their mention of mathematics that I don’t have essays to write about them. I’ll list those soon, all going well, at this link. And then Tuesday, I hope, resume the Fall 2019 A-to-Z Sequence. Thanks for reading.

Reading the Comics, August 23, 2019: Basics of Logic Edition


While there were a good number of comic strips to mention mathematics this past week, there were only a few that seemed substantial to me. This works well enough. This probably is going to be the last time I keep the Reading the Comics post until after Sunday, at least until the Fall 2019 A To Z is finished.

And I’m still open to topics for the first third of the alphabet. If you’d like to see my try to understand a thing of your choice please nominate one or more concepts over at this page. You might be the one to name a topic I can’t possibly summarize!

Gordon Bess’s Redeye rerun for the 18th is a joke building on animals’ number sense. And, yeah, about dumb parents too. Horses doing arithmetic have a noteworthy history. But more in the field of understanding how animals learn, than in how they do arithmetic. In particular in how animals learn to respond to human cues, and how slight a cue has to be to be recognized and acted on. I imagine this reflects horses being unwieldy experimental animals. Birds — pigeons and ravens, particularly — make better test animals.

Kid: 'I've taught Loco [the horse] how to add!' Dad: 'You couldn't teach that stupid horse to come in out of the rain, let alone add.' Kid: 'He can too add! Just watch! OK, Loco, how much is two plus two?' Loco taps his foot four times. 'Four taps!' Dad: 'See! I told you he was a stupid horse!'
Gordon Bess’s Redeye rerun for the 18th of August, 2019. It originally ran the 1st of April, 1973. Essays with mention of Redeye are at this link. This seems to be the first time in over a year the strip has included an actual image and not just a casual “oh, this also mentioned mathematics” line.

Art Sansom and Chip Sansom’s The Born Loser for the 18th gives a mental arithmetic problem. It’s a trick question, yes. But Brutus gives up too soon on what the problem is supposed to be. Now there’s no calculating, in your head, exactly how many seconds are in a year; that’s just too much work. But an estimate? That’s easy.

At least it’s easy if you remember one thing: a million seconds is about eleven and a half days. I find this easy to remember because it’s one of the ideas used all the time to express how big a million, a billion, and a trillion are. A million seconds are about eleven and a half days. A billion seconds are a little under 32 years. A trillion seconds are about 32,000 years, which is about how long it’s been since the oldest known domesticated dog skulls were fossilized. I’m sure that gives everyone a clear idea of how big a trillion is. The important thing, though, is that a million seconds is about eleven and a half days.

Hattie: 'Betcha a buck I can ask a question you can't answer!' Brutus: 'You're on!' Hattie: 'How many seconds are in a year?' Brutus: 'Without a calculator I have no idea.' Hattie: 'It's easy! There are 12 seconds in a year.' Brutus: 'No way that's correct!' Hattie: 'Sure, there's 12 months in a year, so there's January 2nd, February 2nd, and so on through December 2nd. Twelve seconds, pay me!'
Art Sansom and Chip Sansom’s The Born Loser for the 18th of August, 2019. Appearances by The Born Loser should be at this link.

So. Think of the year. There are — as the punch line to Hattie’s riddle puts it — twelve 2nd’s in the year. So there are something like a million seconds spent each year on days that are the 2nd of the month. There about a million seconds spent each year on days that are the 1st of the month, too. There are about a million seconds spent each year on days that are the 3rd of the month. And so on. So, there’s something like 31 million seconds in the year.

You protest. There aren’t a million seconds in twelve days; there’s a million seconds in eleven and a half days. True. Also there aren’t 31 days in every month; there’s 31 days in seven months of the year. There’s 30 days in four months, and 28 or 29 in the remainder. That’s fine. This is mental arithmetic. I’m undercounting the number of seconds by supposing that a million seconds makes twelve days. I’m overcounting the number of seconds by supposing that there are twelve months of 31 days each. I’m willing to bet this undercount and this overcount roughly balance out. How close do I get?

There are 31,536,000 seconds in a common year. That is, a non-leap-year. So “31 million” is a bit low. But it’s not bad for working without a calculator.

T-Rex: 'Everyone! Check this out and I hope you haven't left your balls on the floor because you'll trip on them when you hear this. Ready? 'This sentence is a lie!' Get it? Welcome to Paradox Towne, population: you!' Utahraptor: 'This paradox is ancient.' T-Rex 'WHAT'S THAT? YOU CAN'T HEAR YOUR OWN CRITICISM AS YOU QUICK-TRIP BALLS.' Dromiceiomimus: 'It's old. We've all heard this and dealt with it. Personally, I said 'Oh, I get it'.' T-Rex: 'You say 'Oh, I get it' but here in Paradox Towne that actually means, 'Oh balls, I am here to trip you!' Hours later Paradox Towne is stil infested by balls. You strap a shotgun to your back and set off alone downtown ... oh wow, *this* must be how Shakespeare felt!'
Ryan North’s Dinosaur Comics for the 19th of August, 2019. This and other essays with Dinosaur Comics under discussion should be at this link.

Ryan North’s Dinosaur Comics for the 19th lays on us the Eubulides Paradox. It’s traced back to the fourth century BCE. Eubulides was a Greek philosopher, student of “Not That” Euclid of Megara. We know Eubulides for a set of paradoxes, including the Sorites paradox. As T-Rex’s friends point out, we’ve all heard this paradox. We’ve all gone on with our lives, knowing that the person who said it wanted us to say they were very clever. Fine.

But if we take this seriously we find … this keeps not being simple. We can avoid the problem by declaring self-referential statements exist outside of truth or falsity. This forces us to declare the sentence “this sentence is true” can’t be true. This seems goofy. We can avoid the problem by supposing there are things that are neither true nor false. That solves our problem here at the mere cost of ruining our ability to prove stuff by contradiction. There’s a lot of stuff we prove by contradiction. It’s hard to give that all up for this (Although, so far as I’m aware, anything that can be proved by contradiction can also be proven by a direct line of reasoning. The direct line may just be tedious.) We can solve this problem by saying that our words are fuzzy imprecise things. This is true enough, as see any time my love and I debate how many things are in “a couple of things”. But declaring that we just can’t express the problem well enough to answer it seems like running away from the question. We can resolve things by accepting there are limits to what can be proved by logic. Gödel’s Incompleteness Theorem shows that any interesting enough logic system has statements that are true but unprovable. A version of this paradox helps us get to this interesting conclusion.

So this is one of those things it should be easy to laugh off, but why it should be easy is hard.

Alan Turing, holding a club: 'The Halting Problem is easy to solve. If the program runs too long, I take this stick and beat the computer until it stops.' Caption: 'What if Alan Turing had been an engineer?'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of August, 2019. In case I ever mention Saturday Morning Breakfast Cereal in an essay you’ll see it at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st is about the other great logic problem of the 20th century. The Halting Problem here refers to Turing Machines. This is the algorithmic model for computing devices. It’s rather abstract, so the model won’t help you with your C++ homework, but nothing will. But it turns out we can represent a computer running a program as a string of cells. Each cell holds one of a couple possible values. The program is a series of steps. Each step starts at one cell. The program resets the value of that cell to something dictated by the algorithm. Then, the program moves focus to another cell, again as the algorithm dictates. Do enough of this and you get SimCity 2000. I don’t know all the steps in-between.

So. The Halting Program is this: take a program. Run it. What happens in the long run? Well, it does something or other, yes. But there’s three kinds of things it can do. It can run for a while and then finish, that is, ‘halt’. It can run for a while and then get into a repeating loop, after which it repeats things forever. It can run forever without repeating itself. (Yeah, I see the structural resemblance to terminating decimals, repeating decimals, and irrational numbers too, but I don’t know of any link there.) The Halting Problem asks, if all we know is the algorithm, can we know what happens? Can we say for sure the program will always end, regardless of what the data it works on are? Can we say for sure the program won’t end if we feed it the right data to start?

If the program is simple enough — and it has to be extremely simple — we can say. But, basically, if the program is complicated enough to be even the least bit interesting, it’s impossible to say. Even just running the program isn’t enough: how do you know the difference between a program that takes a trillion seconds to finish and one that never finishes?

For human needs, yes, a program that needs a trillion seconds might as well be one that never finishes. Which is not precisely the joke Weinersmith makes here, but is circling around similar territory.

Wavehead, having divided 19 by 4 on the chalkboard and gotten 4 r 3: 'So there's a little leftover? Great! I can use that in some other math later in the week!'
Mark Anderson’s Andertoons for the 23rd of August, 2019. Tune in to all the times Wavehead says something to a teacher with the Andertoons-based essays at this link.

Mark Anderson’s Andertoons for the 23rd is the Mark Anderson’s Andertoons for the week. And it teases my planned post for Thursday, available soon at this link. Thanks for reading.

Reading the Comics, February 9, 2019: Garfield Outwits Me Edition


Comic Strip Master Command decreed that this should be a slow week. The greatest bit of mathematical meat came at the start, with a Garfield that included a throwaway mathematical puzzle. It didn’t turn out the way I figured when I read the strip but didn’t actually try the puzzle.

Jim Davis’s Garfield for the 3rd is a mathematics cameo. Working out a problem is one more petty obstacle in Jon’s day. Working out a square root by hand is a pretty good tedious little problem to do. You can make an estimate of this that would be not too bad. 324 is between 100 and 400. This is worth observing because the square root of 100 is 10, and the square root of 400 is 20. The square of 16 is 256, which is easy for me to remember because this turns up in computer stuff a lot. But anyway, numbers from 300 to 400 have square roots that are pretty close to but a little less than 20. So expect a number between 17 and 20.

Jon swipes his card at a supermarket checkout. The reader asks: 'Would you like to donate a dollar to charity today?' (Boop.) 'Enter PIN.' (boop boop boop boop.) 'Your total is $3.24. Is this correct?' (Boop.) 'What is the square root of 324? Please show your work.' Jon: 'ALL I WANT IS A BAG OF CHEESE DOODLES!' Garfield: 'DON'T WE ALL?!!'
Jim Davis’s Garfield for the 3rd of February, 2019. Other essays featuring Garfield would be at this link. But somehow this is the first time I’ve had something to write about based in Garfield. Huh.

But after that? … Well, it depends whether 324 is a perfect square. If it is a perfect square, then it has to be the square of a two-digit number. The first digit has to be 1. And the last digit has to be an 8, because the square of the last digit is 4. But that’s if 324 is a perfect square, which it almost certainly is … wait, what? … Uh .. huh. Well, that foils where I was going with this, which was to look at a couple ways to do square roots.

One is to start looking at factors. If a number is equal to the product of two numbers, then its square root is the product of the square roots of those numbers. So dividing your suspect number 324 by, say, 4 is a great idea. The square root of 324 would be 2 times the square root of whatever 324 ÷ 4 is. Turns out that’s 81, and the square root of 81 is 9 and there we go, 18 by a completely different route.

So that works well too. If it had turned out the square root was something like 2\sqrt{82} then we get into tricky stuff. One response is to leave the answer like that: 2\sqrt{82} is exactly the square root of 328. But I can understand someone who feels like they could use a numerical approximation, so that they know whether this is bigger than 19 or not. There are a bunch of ways to numerically approximate square roots. Last year I worked out a way myself, one that needs only a table of trigonometric functions to work out. Tables of logarithms are also usable. And there are many methods, often using iterative techniques, in which you make ever-better approximations until you have one as good as your situation demands.

Anyway, I’m startled that the cheese doodles price turned out to be a perfect square (in cents). Of course, the comic strip can be written to have any price filled in there. The joke doesn’t depend on whether it’s easy or hard to take the square root of 324. But that does mean it was written so that the problem was surprisingly doable and I’m amused by that.

T-Rex: 'Say the average person can expect to live for 81 years. That's a little over 2.5 billion seconds. 2.5 billion is not that much! I thought I'd compare the seconds in a life to the molecules in a glass of water, but even a gram of water has over ten sextillion molecules in it. Even if I measure my life in NANOSECONDS I'm still not on par with a gram of boring ol' WATER.' Dromiceiomimus: 'Molecules are super tiny, T-Rex! You should measure yourself in bigger units.' T-Rex: 'like ... cubic millimeters?' Utahraptor: 'That'd give you 2500 litres, that's a lot!' T-Rex: 'Dude, that's just a GIANT BATHTUB! I want to visualize my lifespan as something impressive!' Utahraptor: 'OK. 2.5 billion kilometers is enough to make a one-way trip to Saturn and get most of the way back before dying, OR to travel part of the way to Uranus, but again, dying well before you arrive.' LATER: T-Rex: 'Dear audio diary! Today I learned why we measure lifetimes in years and not in 'failed trips to Uranus where only corpses show up at the end'. It's, um, for the reasons you'd expect, basically.'
Ryan North’s Dinosaur Comics for the 4th of February, 2019. Some of the many essays inspired by Dinosaur Comics appear at this link.

Ryan North’s Dinosaur Comics for the 4th goes in some odd directions. But it’s built on the wonder of big numbers. We don’t have much of a sense for how big truly large numbers. We can approach pieces of that, such as by noticing that a billion seconds is a bit more than thirty years. But there are a lot of truly staggeringly large numbers out there. Our basic units for things like distance and mass and quantity are designed for everyday, tabletop measurements. The numbers don’t get outrageously large. Had they threatened to, we’d have set the length of a meter to be something different. We need to look at the cosmos or at the quantum to see things that need numbers like a sextillion. Or we need to look at combinations and permutations of things, but that’s extremely hard to do.

Tube Sock: a white cylinder with two blue stripes near the top. Inner Tube Sock: a white torus with two blue stripes around the narrow radius.
Tom Horacek’s Foolish Mortals for the 4th of February, 2019. This is a new tag. When I am next moved to write about Foolish Mortals the results should be this link. This might be a while. I can find some examples of writing about this strip in 2014, before I tagged the comic strips by name, but not since then.

Tom Horacek’s Foolish Mortals for the 4th is a marginal inclusion for this week’s strips, but it’s a low-volume week. The intended joke is just showing off a “tube sock” and an “inner tube sock”. But it happens to depict these as a cylinder and a torus and those are some fun shapes to play with. Particularly, consider this: it’s easy to go from a flat surface to a cylinder. You know this because you can roll a piece of paper up and get a good tube. And it’s not hard to imagine going from a cylinder to a torus. You need the cylinder to have a good bit of give, but it’s easy to imagine stretching it around and taping one end to the other. But now you’ve got a shape that is very different from a sheet of paper. The four-color map theorem, for example, no longer holds. You can divide the surface of the torus so it needs at least seven colors.

Wiley's Dictionary, as read by Peter: 'Logarithm. A downed tree with dance moves.'
Mastroianni and Hart’s B.C. for the 5th of February, 2019. Essays describing some aspect of B.C., whether the current run or the vintage 1960s reruns, appear at this link.

Mastroianni and Hart’s B.C. for the 5th is a bit of wordplay. As I said, this was a low-volume week around here. The word “logarithm” derives, I’m told, from the modern-Latin ‘logarithmus’. John Napier, who advanced most of the idea of logarithms, coined the term. It derives from ‘logos’, here meaning ‘ratio’, and ‘re-arithmos’, meaning ‘counting number’. The connection between ratios and logarithms might not seem obvious. But suppose you have a couple of numbers, and we’ll reach deep into the set of possible names and call them a, b, and c. Suppose a ÷ b equals b ÷ c. Then the difference between the logarithm of a and the logarithm of b is the same as the difference between the logarithm of b and the logarithm of c. This lets us change calculations on numbers to calculations on the ratios between numbers and this turns out to often be easier work. Once you’ve found the logarithms. That can be tricky, but there are always ways to do it.

Mother: 'Maggot, help Otis with his math homework. Explain fractions to him.' Maggot, to Otis: 'Well, it's like when you drop a beer bottle and it breaks into a lot of pieces.'
Bill Rechin’s Crock rerun for the 8th of February, 2019. I have no information about when this strip previously appeared. Essays based on things mentioned in Crock appear at this link. Somehow this isn’t the first time I’ve tagged this comic.

Bill Rechin’s Crock for the 8th is not quite a bit of wordplay. But it mentions fractions, which seem to reliably confuse people. Otis’s father is helpless to present a concrete, specific example of what fractions mean. I’d probably go with change, or with slices of pizza or cake. Something common enough in a child’s life.

And I grant there have been several comic strips here of marginal mathematics value. There was still one of such marginal value. Mark Parisi’s Off The Mark for the 7th has anthropomorphized numerals, in service of a temperature joke.


These are all the mathematically-themed comic strips for the past week. Next Sunday, I hope, I’ll have more. Meanwhile please come around here this week to see what, if anything, I think to write about.

Reading the Comics, July 21, 2018: Infinite Hotels Edition


Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.

The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.

T-Rex: 'David Hilbert was a mathematician and hotelier who was born in 1892. He built an infinite hotel, you guys! THE INFINITE HOTEL: A TRUE STORY. So Hilbert built this infinite hotel that was infinitely big and had infinitely many rooms; I believe this was a matter of some investment. But build it he did, and soon after a bus with infinity people in it showed up, with each of them wanting a room! Lucky for Hilbert he had his infinite hotel, so each guest got a room, and the hotel was filled up to capacity. Nice! But just then another friggin' bus showed up, and it ALSO had infinity people in it!' Utahraptor: 'Nobody builds for TWO infinite buses showing up right after the other!' T-Rex: 'Turns out they do! He just told every guest already there to move into the room that was double their current room number. So the guest in room 3 moved into room 6, and so on! Thus, only the even-numbered rooms were occupied, and everyone on the new bus could have an odd-numbered room!' Utahraptor: 'Amazing!' T-Rex: 'Yep! Anyway! It's my understanding he died an infinitely rich man infinity years later.'
Ryan North’s Dinosaur Comics for the 18th of July, 2018. The strip likely ran sometime before on North’s own web site; I don’t know when.

Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.

And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.

David Hilbert was born in 1862; T-Rex misspoke.

Teacher: 'Sluggo --- describe an octagon.' Sluggo: 'A figure with eight sides and eight angles.' Teacher: 'Correct. Now, Nancy --- describe a sphere'. (She blows a bubble-gum bubble.)
Ernie Bushmiller’s Nancy Classics for the 20th of July, 2018. Originally run, it looks to me, like the 18th of October, 1953.

Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.

I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.

Literal Figures: a Venn diagram of two circles, their disjoint segments labelled 'Different' and their common area labelled 'Same'. A graph, 'Height of Rectangles', a bar chart with several rectangles. A graph, 'Line Usage': a dashed line labelled Dashed; a jagged line labelled Jagged; a curvy line labelled Curvy. A map: 'Global Dot Concentration', with dots put on a map of the world.
John Atkinson’s Wrong Hands for the 20th of July, 2018. So this spoils a couple good ideas for my humor blog’s Statistics Saturdays now that you know I’ve seen this somewhere.

John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.

Ziggy: 'I wish I'd paid more attention in math class! I can't even count the number of times I've had trouble with math!'
Tom II Wilson’s Ziggy for the 20th of July, 2018. Tom Wilson’s still credited with the comic strip, though he died in 2011. I don’t know whether this indicates the comic is in reruns or what.

Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.

Cop: 'You were travelling at 70 miles per hour. How much later would you have arrived if you were only going 60?' Eno: 'No fair --- I hate word problems!'
Glenn McCoy and Gary McCoy’s The Duplex for the 21st of July, 2018. So the strip is named The Duplex because it’s supposed to be about two families in the same, uh, duplex: this guy with his dog, and a woman with her cat. I was reading the strip for years before I understood that. (The woman doesn’t show up nearly so often, or at least it feels like that.)

Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.

There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:

t = \frac{d}{v - s} = \frac{d}{v}\left(1 + \left(\frac{s}{v}\right) + \left(\frac{s}{v}\right)^2 + \left(\frac{s}{v}\right)^3 + \left(\frac{s}{v}\right)^4 + \left(\frac{s}{v}\right)^5 + \cdots \right)

‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, \left(\frac{s}{v}\right) — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So \left(\frac{s}{v}\right)^2 is tiny. And \left(\frac{s}{v}\right)^3 is impossibly tiny. And \left(\frac{s}{v}\right)^4 is ridiculously tiny. You make an error in dropping these \left(\frac{s}{v}\right) squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:

\frac{d}{v} \left(1 + \frac{s}{v}\right)

Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.

Wellington: 'First our teacher says 25 plus 25 equals 50. Then she says 30 and 20 equals 50. Then she says 10 and 40 equals 50. Finally she says 15 and 35 equals 50. Shouldn't we have a teacher who can make up her mind?'
Morrie Turner’s Wee Pals rerun for the 21st of July, 2018. Originally ran the 22nd of July, 2013.

Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.

And this is marginal but the “Soul Circle” personal profile for the 20th of July — rerun from the 20th of July, 2013 — was about Dr Cecil T Draper, a mathematics professor.


You can get to this and more Reading the Comics posts at this link. Other essays mentioning Dinosaur Comics are at this link. Essays that describe Nancy, vintage and modern, are at this link. Wrong Hands gets discussed in essays on this link. Other Ziggy-based essays are at this link. The Duplex will get mentioned in essays at this link if any other examples of the strip get tagged here. And other Wee Pals strips get reviewed at this link.

Reading the Comics, March 9, 2018: Some Old Lines Edition


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.

Partly scrambled Rubik's Cube to a solved one: 'Rough week.'
Maria Scrivan’s Half Full for the 4th of June, 2018. Yeah, uh, it me.

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.

God: 'T-Rex let's say you have a giant heap of sand and I remove one grain of it at a time.' T-Rex: 'Ooh, let's!' God: 'Clearly when there's only one grain of sand left it's not a heap anymore!' T-Rex: 'Clearly!' God: 'Aha my friend but when precisely did it switch from heap to non-heap?' T-Rex: 'I dunno! At some fuzzy point if would switch for most observers from 'heap' to, say, 'small pine', and there we can draw the line. Language isn't that precise.' God: 'Listen this is a classic paradox of Eubulides of Miletus came up with over 2000 years ago. You need to have your mind blown now okay.' T-Rex: 'Sounds kinda dumb to me!' Utahraptor: 'What does?' T-Rex: 'The point at which a shrinking heap of sand becomes a non-heap. Clearly I'm supposed to struggle with an arbitrary threshold, because piles on either side of it look much the same. But it's just language! Look at statistical usage of the word 'heap', decide using that average, end of story. Oh, snap, philosophers! Did T-Rex just totally school you with his statistically-based descriptivist approach to semantics? IT APPEARS THAT HE TOTALLY DID! It also appears he's speaking in the third person because he's so impressed with his awesome self!'
Ryan North’s Dinosaur Comics for the 5th of June, 2018. I get that part of the setup of these comics is that T-Rex is nerdy-smart, but I can also imagine the philosophers rolling their eyes at how he’s missed the point. Maybe if he were asked about the density of a single molecule of water he’d understand better why the question can’t be obvious. (And T-Rex does sometimes revisit issues with deeper understanding of the issues. This might have happened between when this strip first appeared on qwantz.com and when it appeared on GoComics.com.

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.

Customer: 'How much will this be at 80% off?' Clerk: 'Ten bucks.' Customer: 'How did you do that in your head so fast?' Clerk: '20% of fifty is ten.' Customer: 'Wow! So you're some kind of super math genius?' Customer: 'Sure.'
Norm Feuti’s Retail for the 6th of June, 2018. This joke, though not this strip, was also run the 26th of June, 2017. There I share my one great retail-mathematics anecdote.

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.

Rita: 'Tell your group I expect them to give me 110%! Keep in mind, reviews are coming!' Jay: 'Rita --- you should realize that it's impossible to give more than 100%!' Rita: 'No --- not with that kind of attitude!'
John Zakour and Scott Roberts’s Working Daze for the 6th of June, 2018. It ran the 22nd of October, 2014, although that was as part of a “Best Of” week. No idea when it originally ran.

John Zakour and Scott Roberts’s Working Daze for the 6th is another chance for me to talk about the supposed folly of giving 110 percent. Or point you to where I did already. I’m forgiving of the use of the phrase.

Abacus at the bar: 'If you ever find yourself working for Weinstein as a bookkeeper, let me offer you sum advice ... never use the phrase, 'Harvey, you can count on me'.' Hostess: 'Thanks for the tip.'
Bob Shannon’s Tough Town for the 7th of June, 2018. The strip is one about all sorts of odd creatures hanging out in the bar, so, you’re not misunderstanding this.

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.

Cave-person Father: 'Me have method for knowing how many rocks you have. Called 'counting'. Put up fingers, then say --- ' Cave-person Kid: 'We ever use this in REAL LIFE?' Caption: The First Math Class.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th of June, 2018. Admit I do wonder how often cave people needed to track the number of rocks they had. I mean, how often do we need to count our rocks? Aren’t the rocks themselves an adequate representation of the number of rocks around?

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.

Karen: 'Uuuhhhhggghh!!! I hate math!!!' Dad: 'First of all, don't say 'hate'. It's a very strong word. Secondly, you will always need math. Even if you're in sales like me. In fact, I'm using math right now. I'm figuring out where I stand against my quota for this quarter. Observe ... I take this number, add it to that one. Take a percentage of this value and subtract it here. See, that's my number ... ... ... I hate math.'
Steve Sicula’s Home and Away rerun for the 9th of June, 2018. The strip originally ran the 6th of March, 2011. … How does Karen there say “Uuuhhhggghh”?

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.

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


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

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

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

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

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

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

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

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

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

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

Reading the Comics, September 19, 2017: Visualization Edition


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

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

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

'How are you and David doing?' 'Better, with counseling.' (As Ben takes his drink bottle.) 'But sometimes he still clings to hope that Ben's autism is 'curable'. Admittedly, I've wondered that myself. Then Ben strips naked and solves a trigonometry problem.' 'Whoa.' (Ben throws his drink bottle in the air and says) 'A = (1/2)(4)(2) sin 45 deg.'
Terri Libenson’s Pajama Diaries for the 18th of September, 2017. When I first read this I assumed that of course the base of the triangle had length 4 and the second leg, at a 45-degree angle to that, had length 2, and I wondered if those numbers could be consistent for a triangle to exist. Of course they could, though. There is a bit of fun to be had working out whether a particular triangle could exist from knowing its side lengths, though.

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

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

Guest Woman: 'Did you know the King was having trouble controlling the young hotheads in his own tribe?' Phantom: 'Yes. He's an old friend of mine. He probably looks like an ignorant savage to you. Actually, he speaks seven languages, is an expert mathematician, and plays a fine hand of poker.' Guest Woman: 'What?' Cut to the King, in his hut, reading The Theory Of Relativity. 'Thank goodness that's over ... Now where was I?'
Lee Falk and Wilson McCoy’s The Phantom for the 10th of August, 1950 and rerun the 18th of September, 2017. For my money, just reading a mathematics book doesn’t take. I need to take notes, as if it were in class. I don’t quite copy the book, but it comes close.

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

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

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

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

Reading the Comics, March 11, 2017: Accountants Edition


And now I can wrap up last week’s delivery from Comic Strip Master Command. It’s only five strips. One certainly stars an accountant. one stars a kid that I believe is being coded to read as an accountant. The rest, I don’t know. I pick Edition titles for flimsy reasons anyway. This’ll do.

Ryan North’s Dinosaur Comics for the 6th is about things that could go wrong. And every molecule of air zipping away from you at once is something which might possibly happen but which is indeed astronomically unlikely. This has been the stuff of nightmares since the late 19th century made probability an important part of physics. The chance all the air near you would zip away at once is impossibly unlikely. But such unlikely events challenge our intuitions about probability. An event that has zero chance of happening might still happen, given enough time and enough opportunities. But we’re not using our time well to worry about that. If nothing else, even if all the air around you did rush away at once, it would almost certainly rush back right away.

'The new SAT multiple-choice questions have 4 answers instead of 5, with no penalty for guessing.' 'Let's see ... so if I took it now ... that would be one chance in four, which would be ... 25%?' 'Yes.' 'But back when I took it, my chances were ... let's see ... um ...' 'Remember, there's no penalty for guessing.'
Steve Kelley and Jeff Parker’s Dustin for the 7th of March, 2017. It’s the title character doing the guessing there. Also, Kelley and Parker hate their title character with a thoroughness you rarely see outside Tom Batiuk and Funky Winkerbean. This is a mild case of it but, there we are.

Steve Kelley and Jeff Parker’s Dustin for the 7th of March talks about the SATs and the chance of picking right answers on a multiple-choice test. I haven’t heard about changes to the SAT but I’ll accept what the comic strip says about them for the purpose of discussion here. At least back when I took it the SAT awarded one point to the raw score for a correct answer, and subtracted one-quarter point for a wrong answer. (The raw scores were then converted into a 200-to-800 range.) I liked this. If you had no idea and guessed on answers you should expect to get one in five right and four in five wrong. On average then you would expect no net change to your raw score. If one or two wrong answers can be definitely ruled out then guessing from the remainder brings you a net positive. I suppose the change, if it is being done, is meant to be confident only right answers are rewarded. I’m not sure this is right; it seems to me there’s value in being able to identify certainly wrong answers even if the right one isn’t obvious. But it’s not my test and I don’t expect to need to take it again either. I can expression opinions without penalty.

Mark Anderson’s Andertoons for the 7th is the Mark Anderson’s Andertoons for last week. It’s another kid-at-the-chalkboard panel. What gets me is that if the kid did keep one for himself then shouldn’t he have written 38?

Brian Basset’s Red and Rover for the 8th mentions fractions. It’s just there as the sort of thing a kid doesn’t find all that naturally compelling. That’s all right I like the bug-eyed squirrel in the first panel.

'The happy couple is about to cut the cake!' 'What kind is it?' 'A math cake.' (It has a square root of 4 sign atop it.)
Bill Holbrook’s On The Fastrack for the 9th of March, 2017. I confess I’m surprised Holbrook didn’t think to set the climax a couple of days later and tie it in to Pi Day.

Bill Holbrook’s On The Fastrack for the 9th concludes the wedding of accountant Fi. It uses the square root symbol so as to make the cake topper clearly mathematical as opposed to just an age.

Reading the Comics, January 14, 2017: Redeye and Reruns Edition


So for all I worried about the Gocomics.com redesign it’s not bad. The biggest change is it’s removed a side panel and given the space over to the comics. And while it does show comics you haven’t been reading, it only shows one per day. One week in it apparently sticks with the same comic unless you choose to dismiss that. So I’ve had it showing me The Comic Strip That Has A Finale Every Day as a strip I’m not “reading”. I’m delighted how thisbreaks the logic about what it means to “not read” an “ongoing comic strip”. (That strip was a Super-Fun-Pak Comix offering, as part of Ruben Bolling’s Tom the Dancing Bug. It was turned into a regular Gocomics.com feature by someone who got the joke.)

Comic Strip Master Command responded to the change by sending out a lot of comic strips. I’m going to have to divide this week’s entry into two pieces. There’s not deep things to say about most of these comics, but I’ll make do, surely.

Julie Larson’s Dinette Set rerun for the 8th is about one of the great uses of combinatorics. That use is working out how the number of possible things compares to the number of things there are. What’s always staggering is that the number of possible things grows so very very fast. Here one of Larson’s characters claims a science-type show made an assertion about the number of possible ideas a brain could hold. I don’t know if that’s inspired by some actual bit of pop science. I can imagine someone trying to estimate the number of possible states a brain might have.

And that has to be larger than the number of atoms in the universe. Consider: there’s something less than a googol of atoms in the universe. But a person can certainly have the idea of the number 1, or the idea of the number 2, or the idea of the number 3, or so on. I admit a certain sameness seems to exist between the ideas of the numbers 2,038,412,562,593,604 and 2,038,412,582,593,604. But there is a difference. We can out-number the atoms in the universe even before we consider ideas like rabbits or liberal democracy or jellybeans or board games. The universe never had a chance.

Or did it? Is it possible for a number to be too big for the human brain to ponder? If there are more digits in the number than there are atoms in the universe we can’t form any discrete representation of it, after all. … Except that we kind of can. For example, “the largest prime number less than one googolplex” is perfectly understandable. We can’t write it out in digits, I think. But you now have thought of that number, and while you may not know what its millionth decimal digit is, you also have no reason to care what that digit is. This is stepping into the troubled waters of algorithmic complexity.

Shady Shrew is selling fancy bubble-making wands. Shady says the crazy-shaped wands cost more than the ordinary ones because of the crazy-shaped bubbles they create. Even though Slylock Fox has enough money to buy an expensive wand, he bought the cheaper one for Max Mouse. Why?
Bob Weber Jr’s Slylock Fox and Comics for Kids for the 9th of January, 2017. Not sure why Shady Shrew is selling the circular wands at 50 cents. Sure, I understand wanting a triangle or star or other wand selling at a premium. But then why have the circular wands at such a cheap price? Wouldn’t it be better to put them at like six dollars, so that eight dollars for a fancy wand doesn’t seem that great an extravagance? You have to consider setting an appropriate anchor point for your customer base. But, then, Shady Shrew isn’t supposed to be that smart.

Bob Weber Jr’s Slylock Fox and Comics for Kids for the 9th is built on soap bubbles. The link between the wand and the soap bubble vanishes quickly once the bubble breaks loose of the wand. But soap films that keep adhered to the wand or mesh can be quite strangely shaped. Soap films are a practical example of a kind of partial differential equations problem. Partial differential equations often appear when we want to talk about shapes and surfaces and materials that tug or deform the material near them. The shape of a soap bubble will be the one that minimizes the torsion stresses of the bubble’s surface. It’s a challenge to solve analytically. It’s still a good challenge to solve numerically. But you can do that most wonderful of things and solve a differential equation experimentally, if you must. It’s old-fashioned. The computer tools to do this have gotten so common it’s hard to justify going to the engineering lab and getting soapy water all over a mathematician’s fingers. But the option is there.

Gordon Bess’s Redeye rerun from the 28th of August, 1970, is one of a string of confused-student jokes. (The strip had a Generic Comedic Western Indian setting, putting it in the vein of Hagar the Horrible and other comic-anachronism comics.) But I wonder if there are kids baffled by numbers getting made several different ways. Experience with recipes and assembly instructions and the like might train someone to thinking there’s one correct way to make something. That could build a bad intuition about what additions can work.

'I'm never going to learn anything with Redeye as my teacher! Yesterday he told me that four and one make five! Today he said, *two* and *three* make five!'
Gordon Bess’s Redeye rerun from the 28th of August, 1970. Reprinted the 9th of January, 2017. What makes the strip work is how it’s tied to the personalities of these kids and couldn’t be transplanted into every other comic strip with two kids in it.

Corey Pandolph’s Barkeater Lake rerun for the 9th just name-drops algebra. And that as a word that starts with the “alj” sound. So far as I’m aware there’s not a clear etymological link between Algeria and algebra, despite both being modified Arabic words. Algebra comes from “al-jabr”, about reuniting broken things. Algeria comes from Algiers, which Wikipedia says derives from `al-jaza’ir”, “the Islands [of the Mazghanna tribe]”.

Guy Gilchrist’s Nancy for the 9th is another mathematics-cameo strip. But it was also the first strip I ran across this week that mentioned mathematics and wasn’t a rerun. I’ll take it.

Donna A Lewis’s Reply All for the 9th has Lizzie accuse her boyfriend of cheating by using mathematics in Scrabble. He seems to just be counting tiles, though. I think Lizzie suspects something like Blackjack card-counting is going on. Since there are only so many of each letter available knowing just how many tiles remain could maybe offer some guidance how to play? But I don’t see how. In Blackjack a player gets to decide whether to take more cards or not. Counting cards can suggest whether it’s more likely or less likely that another card will make the player or dealer bust. Scrabble doesn’t offer that choice. One has to refill up to seven tiles until the tile bag hasn’t got enough left. Perhaps I’m overlooking something; I haven’t played much Scrabble since I was a kid.

Perhaps we can take the strip as portraying the folk belief that mathematicians get to know secret, barely-explainable advantages on ordinary folks. That itself reflects a folk belief that experts of any kind are endowed with vaguely cheating knowledge. I’ll admit being able to go up to a blackboard and write with confidence a bunch of integrals feels a bit like magic. This doesn’t help with Scrabble.

'Want me to teach you how to add and subtract, Pokey?' 'Sure!' 'Okay ... if you had four cookies and I asked you for two, how many would you have left?' 'I'd still have four!'
Gordon Bess’s Redeye rerun from the 29th of August, 1970. Reprinted the 10th of January, 2017. To be less snarky, I do like the simply-expressed weariness on the girl’s face. It’s hard to communicate feelings with few pen strokes.

Gordon Bess’s Redeye continued the confused-student thread on the 29th of August, 1970. This one’s a much older joke about resisting word problems.

Ryan North’s Dinosaur Comics rerun for the 10th talks about multiverses. If we allow there to be infinitely many possible universes that would suggest infinitely many different Shakespeares writing enormously many variations of everything. It’s an interesting variant on the monkeys-at-typewriters problem. I noticed how T-Rex put Shakespeare at typewriters too. That’ll have many of the same practical problems as monkeys-at-typewriters do, though. There’ll be a lot of variations that are just a few words or a trivial scene different from what we have, for example. Or there’ll be variants that are completely uninteresting, or so different we can barely recognize them as relevant. And that’s if it’s actually possible for there to be an alternate universe with Shakespeare writing his plays differently. That seems like it should be possible, but we lack evidence that it is.