Reading the Comics, June 13, 2018: Wild Squirrel Edition

I have another Reading the Comics post with a title that’s got nothing to do with the post. It has got something to do with how I spent my weekend. Not sure if I’ll ever get around to explaining that since there’s not much mathematical content to that weekend. I’m not sure whether the nonsense titles are any better than trying to find a theme in what Comic Strip Master Command has sent the past week. It takes time to pick something when anything would do, after all.

Scott Hilburn’s The Argyle Sweater for the 10th is the anthropomorphic numerals strip for the week. Also arithmetic symbols. The ÷ sign is known as “the division symbol”, although now and then people try to promote it as the “obelus”. They’re not wrong to call it that, although they are being the kind of person who tries to call the # sign the “octothorp”. Sometimes social media pass around the false discovery that the ÷ sign is a representation of a fraction, \frac{a}{b} , with the numbers replaced by dots. It’s a good mnemonic for linking fractions and division. But it’s wrong to say that’s what the symbol means. ÷ started being used for division in Western Europe in the mid-17th century, in competition with many symbols, including / (still in common use), : (used in talking about ratios or odds), – (not used in this context anymore, and just confusing if you do try to use it so). And ÷ was used in northern Europe to mean “subtraction” for several centuries after this.

Numeral 8, speaking to a numeral 4 on a motorcycle by a ramp at the edge of a canyon that has a giant division symbol island within it: 'I'd think twice. Even if you make it to the other side, you'll always be half the man I am.' Caption: 'Crossing the Great Divide.'
Scott Hilburn’s The Argyle Sweater for the 10th of June, 2018. I’m kind of curious how far in the comments one has to go before getting to a ‘jumping the shark’ comment but not so curious as to read the comments.

Tom Toles’s Randolph Itch, 2am for the 11th is a repeat; the too-short-lived strip has run through several cycles since I started doing these summaries. But it is also one of the great pie chart jokes ever and I have no intention of not telling people to enjoy it.

Randolph dreaming about his presentation; it shows a Pie Chart: Landed On Stage, 28%. Back wall, 13%. Glancing blow off torso, 22%. Hit podium, 12%. Direct hit in face, 25%. Several pies have been thrown, hitting the stage, back wall, his torso, the podium, his face. Corner illustration: 'I turn now to the bar graph.'
Tom Toles’s Randolph Itch, 2am for the 11th of June, 2018. I’m not sure when it did first run, past that it was in 2000, but I’ve featured it at least two times before, both of those in 2015, peculiarly. So in short I have no idea how GoComics picks its reruns for this strip.

Pie charts, and the also-mentioned bar charts, come to us originally from the economist William Playfair, who in the late 1700s and early 1800s devised nearly all the good ways to visualize data. But we know them thanks to Florence Nightingale. Among her other works, she recognized in these charts good ways to represent her studies about Crimean War medicine and about sanitation in India. Nightingale was in 1859 named the first woman in the Royal Statistical Society, and was named an honorary member of the American Statistical Association in 1874.

Esther: 'The first step of the assignment is to find a partner.' Nancy: 'What's the second step?' [ Worksheet: 'Find a partner. Solve: x^2 + y^2 = 3, 16 x^2 - 4y^2 = 0, for x and y ] Nancy, sitting beside Esther, talking to the teacher: 'Neither of us could find a partner.'
Olivia Jaimes’s Nancy for the 12th of June, 2018. Well, if you still need a partner you can probably find me hiding under the desk hoping I don’t have to talk to anybody, ever. For what that’s worth.

Olivia Jaimes’s Nancy for the 12th uses arithmetic as iconic for classwork nobody wants to do. Algebra, too; I understand the reluctance to start. Simultaneous solutions; the challenge is to find sets of values ‘x’ and ‘y’ that make both equations true together. That second equation is a good break, though. 16 x^2 - 4y^2 = 0 makes it easy to write what ‘y’ has to be in terms of ‘x’. Then you can replace the ‘y’ in the first equation with its expression in terms of ‘x’. In a slightly tedious moment, it’s going to turn out there’s multiple sets of answers. Four sets, if I haven’t missed something. But they’ll be clearly related to each other. Even attractively arranged.

x^2 + y^2 = 3 is an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a circle. This is if the coordinates are using the Cartesian coordinate system for the plane, which is such a common thing to do that mathematicians can forget they’re doing that. The circle has radius \sqrt{3} . So you can look at the first equation and draw a circle and write down a note that its radius is \sqrt{3} and you’ve got it. 16x^2 - 4y^2 = 0 looks like an equation that’s true if the numbers ‘x’ and ‘y’ are coordinates of the points on a hyperbola. Again in the Cartesian coordinate system. But I have to feel a little uncomfortable saying this. If the equation were (say) 16x^2 - 4y^2 = 1 then it’d certainly be a hyperbola, which mostly looks like a mirror-symmetric pair of arcs. But equalling zero? That’s called a “degenerate hyperbola”, which makes it sound like the hyperbola is doing something wrong. Unfortunate word, but one we’re stuck with.

The description just reflects that the hyperbola is boring in some way. In this case, it’s boring because the ‘x’ and ‘y’ that make the equation true are just the points on a pair of straight lines that go through the origin, the point with coordinates (0, 0). And they’re going to be mirror-images of each other around the x- and the y-axis. So it seems like a waste to use the form of a hyperbola when we could do just as well using the forms of straight lines to describe the same points. This hyperbola will look like an X, although it might be a pretty squat ‘x’ or a pretty narrow one or something. Depends on the exact equation.

So. The solutions for ‘x’ and ‘y’ are going to be on the points that are on both a circle centered around the origin and on an X centered around the origin. This is a way to see why I would expect four solutions. Also they they would look about the same. There’d be an answer with positive ‘x’ and positive ‘y’, and then three more answers. One answer has ‘x’ with the same size but a minus sign. One answer has ‘y’ with the same size but a minus sign. One has both ‘x’ and ‘y’ with the same values but minus signs.

[ A woman turns a row on a Rubik's cube. She speaks into her phone. ] ' If I move Jen's ortho to Friday, it conflicts with Sam's clarinet. But I can't move that to Monday because Tina has soccer! Ugh, how do I line this thing up?'
Dave Coverly’s Speed Bump for the 12th of June, 2018. This is one of those gimmicks I could see having a niche. Not so much as something someone could use, but as a mildly amusing joke present to give someone you like but don’t really know anything about when for some reason you can’t just give a book instead.

Sorry I wasn’t there to partner with.

Dave Coverly’s Speed Bump for the 12th is a Rubik’s Cube joke. Here it merges the idea with the struggles of scheduling anything anymore. I’m not sure that the group-theory operations of lining up a Rubik’s cube can be reinterpreted as the optimization problems of scheduling stuff. But there are all sorts of astounding and surprising links between mathematical problems. So I wouldn’t rule it out.

Kid: 'Gramma says lotteries are a tax for people who are bad at math.' Dad: 'In a manner of speaking.' Kid: 'What's the tax for people who are bad at reading?' Dad: 'Handicapped-parking fines.'
John Allen’s Nest Heads for the 13th of June, 2018. Not to get too cranky but I can’t figure out what the kid’s name is. I understand some cartoonists want dialogue that’s a bit more natural than someone saying each character’s name at least once per daily strip, but could a cast list please be put on the strip’s ‘About’ page at leaset?

John Allen’s Nest Heads for the 13th is a lotteries joke. I’m less dogmatic than are many mathematicians about the logic of participating in a lottery. At least in the ones as run by states and regional authorities the chance of a major payout are, yes, millions to one against. There can be jackpots large enough that the expectation value of playing becomes positive. In this case the reward for that unlikely outcome is so vast that it covers the hundreds of millions of times you play and lose. But even then, you have the question of whether doing something that just won’t pay out is worth it. My taste is to say that I shall do much more foolish things with my disposable income than buying a couple tickets each year. And while I would like to win the half-billion-dollar jackpot that would resolve all my financial woes and allow me to crush those who had me imprisoned in the Château d’If, I’d also be coming out ahead if I won, like, one of the petty $10,000 prizes.


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 and when it appeared on

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, June 4, 2018: Weezer’s Africa Edition

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

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

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

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

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

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

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

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

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

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

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

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

Reading the Comics, June 1, 2018: His First Name Is Tom For What That’s Worth Edition

And now I’ve got caught up with last week’s comics. I can get to readying for this coming Sunday looking at … so far … nine comic strips that made the preliminary cut. Whimper.

This time the name does mean something.

Thaves’s Frank and Ernest for the 31st complains about not being treated as a “prime number”. There’s a lot of linguistic connotation gone into this strip. The first is the sense that to be a number is to be stripped of one’s humanity, to become one of a featureless horde. Each number is unique, of course; Iva Sallay’s Find the Factors page each day starts with some of the features of each whole number in turn. But one might look at, oh, 84,644 and not something very different from 84,464.

Frank: 'The boss treats me like a number, and not a prime one.'
Thaves’s Frank and Ernest for the 31st of May, 2018. In the past I’ve gone out trying to find and print Thaves’s first name, on the grounds that I should fully credit people. I’m coming around on this, first because I keep forgetting his first name and looking it up every time is tiresome. But more important, if Thaves wants to be known simply as ‘Thaves’ what am I doing arguing that? Is there a different Thaves, possibly his evil twin, producing another comic strip named Frank and Ernest that I have to make clear I’m not talking about? So that’s my level of overthinking these captions right now.

And yet there’s the idea that there are prime numbers, celebrities within the anonymous counting numbers. The name even says it; a prime something is especially choice. And we speak of prime numbers as somehow being the backbone of numbers. This reflects that we find unique factorizations to be a useful thing to do. But being a prime number doesn’t make a number necessarily better. There are reasons most (European) currencies, before decimalization, divided their currency unit into 20 parts of 12 parts each. And nobody divided them into 19 parts of 13 parts each. As often happens, whether something is good depends on what you’re hoping it’s good for.

[ Movie showing the digits of Pi marching out of a flying saucer.] Guy in movie: 'What are they?' Woman in movie: 'They appear to be numbers.' Guy watching movie: 'I just love sci-pi movies.'
Nate Fakes’s Break of Day for the 1st of June, 2018. Sure, but what do you do for the sequel? No, τ is not a thing.

Nate Fakes’s Break of Day for the 1st of June is more or less the anthropomorphized numerals installment for the week. It’s also a bit of wordplay, so, good on them. There’s not so many movies about mathematics. Darren Aronofsky’s Pi, Ron Howard’s A Beautiful Mind, and Theodore Melfi’s Hidden Figures are the ones that come to mind, at least in American cinema. And there was the TV detective series Numbers. It seems odd that there wasn’t, like, some little studio prestige thing where Paul Muni played Évariste Galois back in the day. But a lot of the mathematical process isn’t cinematic. People scribbling notes, typing on a computer, or arguing about something you don’t understand are all hard to make worth watching. And the parts that anyone could understand — obsession, self-doubt, arguments over priority, debates about implications — are universal to any discovery or invention. Note that the movies listed are mostly about people who happen to be doing mathematics. You could change the specialties to, say, chemical engineering without altering the major plot beats. Well, Pi would need more alteration. But you could make it about any process that seems to offer reliable forecasting in a new field.

Bernice, whispering: 'Luann! Did you hear? Tiffany asked Aaron Hill to the dance but he turned her down! He said he's inviting 'someone else'!' Teacher: 'So if x is 1/4 y over 42.6 minus (Q^2 R)/19 ...' Bernice: 'And we know WHO that 'someone else' is, don't we?' [ Luann is wide-eyed with joy. ] Teacher: 'Can anyone tell me what 'R' is?' Luann: 'YES!' Teacher: 'Good! Come up here to the board, Luann.'
Greg Evans’s Luann Againn for the 1st of June, 2018. It originally ran the 1st of June, 1990.

Greg Evans’s Luann Againn for the 1st takes place in mathematics class. The subject doesn’t matter for the joke. It could be anything that doesn’t take much word-balloon space but that someone couldn’t bluff their way through.

Mr Barrows: 'You're pretty good at numbers, Quincy. Are you going to work with figures when you grow up?' Quincy: 'I'm not sure yet, Mr Barrows. I'm either gonna be a very tall accountant or a very short basketball player.'
Ted Shearer’s Quincy for the 7th of April, 1979 and reprinted the 1st of June, 2018. I get why Quincy would figure he’d grow up to be a very tall accountant, but why does he just assume he’d be a very short basketball player? Isn’t it as easy to imagine you’ll grow up to be a typically-sized basketball player? Does he know something we don’t?

Ted Shearer’s Quincy for the 7th of April, 1979 has Quincy thinking what he’ll do with his head for figures. He sees accounting as plausible. Good for him. Society always needs accountants. And they probably do more of society’s mathematics than the mathematicians do.

Scientist type pointing to the blackboard full of arithmetic: 'Cutting-edge formula? No, that's the wi-fi password.'
Bill Abbott’s Spectickles for the 1st of June, 2018. So, is this all the characters that have to be typed in, or is it one of those annoying things where you have to solve the puzzle to get the password?

Bill Abbott’s Spectickles for the 1st features the blackboard-full-of-mathematics to represent the complicated. It shows off the motif that an advanced mathematical formula will be a long and complicated one. This has good grounds behind it. If you want to model something interesting that hasn’t been done before, chances are it’s because you need to consider many factors. And trying to represent them will be clumsily done. It takes reflection and consideration and, often, new mathematical tools to make a formula pithy. Famously, James Clerk Maxwell introduced his equations about electricity and magnetism as a set of twenty equations. By 1873 Maxwell, making some use of quaternions, was able to reduce this to eight equations. Oliver Heaviside, in the late 19th century, used the still-new symbols of vector mechanics. This let him make an attractive quartet. We still see that as the best way to describe electromagnetic fields. As with writing, much of mathematics is rewriting.

Reading the Comics, May 30, 2018: Spherical Photos Edition

Last week’s offerings from Comic Strip Master Command got away from me. Here’s some more of the strips that had some stuff worth talking about. I should have another installment this week. I’m back to nonsense edition names; sorry.

Lincoln Pierce’s Big Nate for the 29th of May is about the gambler’s fallacy. Everyone who learns probability learns about it. The fallacy builds on indisputable logic: your chance of losing at something eighteen times in a row is less than the chance of your losing at that thing seventeen times in a row. So it makes sense that if you’ve lost seventeen times in a row then you must be due.

And that’s one of those lies our intuition tells us about probability. What’s important to Nate here is not the chance he’s in an 18-at-bat losing streak. What’s important is the chance that he’s in an 18-at-bat losing streak, given that he’s already failed 17 times in a row. These are different questions. The chance of an 18th at-bat in a row being a failure (for him) is much larger than the chance of an 18-at-bat losing streak starting from scratch.

Nate: 'Time for me to break this 0-and-17 stretch.' Teddy: 'Exactly! You're due, Nate! You're due!' Francis: 'Not necessarily. The chances of Nate getting a hit aren't enhanced by the fact that he's gone five games without one.' Teddy: 'I lied. You're not due.' Francis: 'But miracles happen, so go for it.'
Lincoln Pierce’s Big Nate rerun for the 29th of May, 2018. The strip first ran the 18th of May, 2010. I’ve not heard anything about why Pierce has been away from the strip since the start of the year.

That said I can’t go along with Francis’s claim that the chance of Nate getting a hit isn’t enhanced by his long dry spell. We can, and often do, model stuff like at-bats as though they’re independent. That is, that the chance of getting a hit doesn’t depend on what came before. Doing it this way gives results that look like real sports matches do. But it’s very hard to quantify things like losing streaks or their opposite, hot hands. It’s hard to dismiss the evidence of people who compete, though. Everyone who does has known the phenomenon of being “in the zone”, where things seem easier. I was in it for two games out of five just last night at pinball league. (I was dramatically out of it for the other three. I nearly doubled my best-ever game of Spider-Man and still came in second place. And by so little a margin my opponent thought the bonus might make the difference. Such heartbreak.)

But there is a huge psychological component to how one plays at a game. Nate thinks differently about what he’s doing going up to bat after seventeen failures in a row than he would after, say, three home runs in a row. It’s hard to believe that this has no effect on how he plays, even if it’s hard to track down a consistent signal through the noise. Maybe it does wash out. Maybe sometimes striking out the first three at-bats in a game makes the batter give up on the fourth. Meanwhile other times it makes the batter focus better on the fourth, and there’s no pinning down which effect will happen. But I can’t go along with saying there’s no effect.

Melvin: 'Hold on now --- replacement? Who could you find to do all the tasks only Melvin can perform?' Rita: 'A macaque, in fact. Listen, if an infinite number of monkeys can write all the great works, I'm confident that one will more than cover for you.'
John Zakour and Scott Roberts’s Working Daze for the 29th of May, 2018. Earlier in the sequence they had the Zootopia sloth replacing Ed, but there’s no making that on topic for my blog here.

John Zakour and Scott Roberts’s Working Daze for the 29th is an infinite-monkeys joke. Well, given some reasonable assumptions we can suppose that sufficiently many monkeys on typewriters will compose whatever’s needed, given long enough. Figuring someone’s work will take fewer monkeys and less time is a decent probability-based insult.

Hazel, with mathematics book, asking a bored kid: 'Okay, now what's nine times eight?' Next panel: the kid's coming out and saying 'Next'; a sign reads, 'Need help with your homework? See Hazel 1 to 5 pm Saturdays'.
Ted Key’s Hazel rerun for the 30th of May, 2018. I can’t say when this first ran. I’m not sure what the kid’s name is, sorry.

Ted Key’s Hazel for the 30th has the maid doing a bit of tutoring work. That’s about all I can make of this either. Doesn’t seem like a lot of fun, but there is only so much to do with arithmetic computation like this. It’s convenient to know a times table by memory.

Accessories of Famous Teachers: Einstein's Chalkboard; Galileo's Compass; Confucius's Fortune Cookie; Socrates's Hemlock; Miss Othmar's Trombone.
Scott Hilburn’s The Argyle Sweater for the 30th of May, 2018. Are … Einstein, Galileo, and Confucius really famous teachers? Calling Socrates a teacher is a lesser stretch.

Scott Hilburn’s The Argyle Sweater for the 30th has a chalkboard full of mathematical symbols as iconic for deep thinking. And it’s even Einstein’s chalkboard. And it’s even stuff that could plausibly be on Einstein’s chalkboard at some point. Besides E = mc2 the other formulas are familiar ones from relativity. They’re about the ways our ideas of how much momentum or mass a thing has has to change if we see the thing in motion. (I’m a little less sure about that \Delta t expression, but I think I can work something out.) And as a bonus it includes the circle-drawing compass as Galileo might have used. Well, he surely used a compass; I’m just not sure that the model shown wouldn’t be anachronistic. As though that matters; fortune cookies, after all, are a 20th century American invention and we’re letting that pass.

Mathematical Fun Fact: For each of the possible espresso-to-milk ratios, there exists at least one Italian-sounding name: Just Milk; 1:3 'latte', 1:2 'Cappuccino', 1:1 'Antoccino', 2:1 'Macchiato', 3:1 'Antilatte', Just Espresso. Also: 1/c^2 'Relativisto'; (espresso + milk)/espresso = espresso/milk 'Phicetto'; i:1 'Imaginarati', pi:1 'Irratiognito'; 6.022*10^23 : 1, 'Avogadro'; lim_{milk->0} espresso/milk: 'Infiniccino'.
Zach Weinersmiths’s Saturday Morning Breakfast Cereal for the 30th of May, 2018. Kind of curious what sorts of drinks you get from putting in infinitesimals. (You get milk or espresso with a homeopathic bit of the other.)

Zach Weinersmiths’s Saturday Morning Breakfast Cereal for the 30th builds on a fun premise. Underneath the main line it gets into some whimsical ratios built on important numbers you’d never use for this sort of thing, such as π, and the imaginary unit \imath . The Golden Ratio makes an appearance too, sneaking a definition for φ in in terms of espresso and milk. Here’s a free question: is there a difference between the “infiniccino” and “just espresso” except for the way it’s presented? … Well, presentation can be an important part of a good coffee.

π is well-known. Not sure I have anything interesting to add to its legend. φ is an irrational number a bit larger than 1.6. I’m not sure if I’ve ever called it the Boba Fett of numbers, but I should have. It’s a cute enough number, far more popular than its importance would suggest. \imath is far more important. Suppose that there is some number, which we give that name, with the property that \imath^2 equals -1. Then we get complex-valued numbers, which let us solve problems we’d like to know but couldn’t do before. It’s a great advance.

The name tells you how dubiously people approached this number, when it was first noticed. I wonder if people would be less uneasy with “imaginary numbers” if it weren’t for being told how there’s no such thing as the square root of minus one for years before algebra comes along and says, well, yes there is. It’s hard to think of a way that, say, “negative four” is more real than \imath , after all, and people are mostly all right with -4. And I understand why people are more skeptical of -4 than they are of, say, 6. Still, I wonder how weird \imath would look if people weren’t primed to think it was weird.

Can We Tell Whether A Pinball Player Is Improving?

The question posed for the pinball league was: can we say which of the players most improved over the season? I had data. I had the rankings of each of the players over the course of eight league nights. I had tools. I’ve taken statistics classes.

Could I say what a “most improved” pinball player looks like? Well, I can give a rough idea. A player’s improving if their rankings increase over the the season. The most-improved person would show the biggest improvement. This definition might go awry; maybe there’s some important factor I overlooked. But it was a place to start looking.

So here’s the first problem. It’s the plot of my own data, my league scores over the season. Yes, league night 2 is dismal. I’d had to miss the night and so got the lowest score possible.

Blue dots, equally spaced horizontally, at the values: 467, 420, 472, 473, 472, 455, 479, and 462.
On the one hand, it’s my nightly finishes in the pinball league over the course of the season. On the other hand, it’s the constellation Delphinus.

Is this getting better? Or worse? The obvious thing to do is to look for a curve that goes through these points. Then look at what that curve is doing. The thing is, it’s always possible to draw a curve through a bunch of data points. As long as there’s not something crazy like there’s four data points for the same league night. As long as there’s one data point for each measurement you can always connect those points to some curve. Worse, you can always fit more than one curve through those points. We need to think harder.

Here’s the thing about pinball league night results. Or any other data that comes from the real world. It’s got noise in it. There’s some amount of it that’s just random. We don’t need to look for a curve that matches every data point. Or any data point particularly. What if the actual data is “some easy-to-understand curve, plus some random noise”?

It’s a good thought. It’s a dangerous thought. You need to have an idea of what the “real” curve should be. There’s infinitely many possibilities. You can bias your answer by choosing what curve you think the data ought to represent. Or by not thinking before you make a choice. As ever, the hard part is not in doing a calculation. It’s choosing what calculation to do.

That said there’s a couple safe bets. One of them is straight lines. Why? … Well, they’re easy to work with. But we have deeper reasons. Lots of stuff, when it changes, looks like it’s changing in a straight line. Take any curve that hasn’t got a corner or a jump or a break in it. There’s a straight line that looks close enough to it. Maybe not for long, but at least for some stretch. In the absence of a better idea of what ought to be right, a line is at least a starting point. You might learn something even if a line doesn’t fit well, and get ideas for why to look at particular other shapes.

So there’s good, steady mathematics business to be found in doing “linear regression”. That is, find the line that best fits a set of data points. What do we mean by “best fits”?

The mathematical community has an answer. I agree with it, surely to the comfort of the mathematical community. Here’s the premise. You have a bunch of data points, with a dependent variable ‘x’ and an independent variable ‘y’. So the data points are a bunch of points, \left(x_j, y_j\right) for a couple values of j. You want the line that “best” matches that. Fine. In my pinball league case here, j is the whole numbers from 1 to 8. x_j is … just j again. All right, as happens, this is more mechanism than we need for this problem. But there’s problems where it would be useful anyway. And for y_j , well, here:

j yj
1 467
2 420
3 472
4 473
5 472
6 455
7 479
8 462

For the linear regression, propose a line described by the equation y = m\cdot x + b . No idea what ‘m’ and ‘b’ are just yet. But. Calculate for each of the x_j values what the projection would be, that is, what m\cdot x_j + b . How far are those from the actual y_j data?

Are there choices for ‘m’ and ‘b’ that make the difference smaller? It’s easy to convince yourself there are. Suppose we started out with ‘m’ equal to 0 and ‘b’ equal to 472. That’s an okay fit. Suppose we started out with ‘m’ equal to 100,000,000 and ‘b’ equal to -2,038. That’s a crazy bad fit. So there must be some ‘m’ and ‘b’ that make for better fits.

Is there a best fit? If you don’t think much about mathematics the answer is obvious: of course there’s a best fit. If there’s some poor, some decent, some good fits there must be a best. If you’re a bit better-learned and have thought more about mathematics you might grow suspicious. That term ‘best’ is dangerous. Maybe there’s several fits that are all different but equally good. Maybe there’s an endless series of ever-better fits but no one best. (If you’re not clear how this could work, ponder: what’s the largest negative real number?)

Good suspicions. If you learn a bit more mathematics you learn the calculus of variations. This is the study of how small changes in one quantity change something that depends on it; and it’s all about finding the maxima or minima of stuff. And that tells us that there is, indeed, a best choice for ‘m’ and ‘b’.

(Here I’m going to hedge. I’ve learned a bit more mathematics than that. I don’t think there’s some freaky set of data that will turn up multiple best-fit curves. But my gut won’t let me just declare that. There’s all kinds of crazy, intuition-busting stuff out there. But if there exists some data set that breaks linear regression you aren’t going to run into it by accident.)

So. How to find the best ‘m’ and ‘b’ for this? You’ve got choices. You can open up DuckDuckGo and search for ‘matlab linear regression’ and follow the instructions. Or ‘excel linear regression’, if you have an easier time entering data into spreadsheets. If you’re on the Mac, maybe ‘apple numbers linear regression’. Follow the directions on the second or third link returned. Oh, you can do the calculation yourself. It’s not hard. It’s just tedious. It’s a lot of multiplication and addition and you know what? We’ve already built tools that know how to do this. Use them. Not if your homework assignment is to do this by hand, but, for stuff you care about yes. (In Octave, an open-source clone of Matlab, you can do it by an admirably slick formula that might even be memorizable.)

If you suspect that some shape other than a line is best, okay. Then you’ll want to look up and understand the formulas for these linear regression coefficients. That’ll guide you to finding a best-fit for these other shapes. Or you can do a quick, dirty hack. Like, if you think it should be an exponential curve, then try fitting a line to x and the logarithm of y. And then don’t listen to those doubts about whether this would be the best-fit exponential curve. It’s a calculation, it’s done, isn’t that enough?

Back to lines, back to my data. I’ll spare you the calculations and show you the results.

Blue dots, equally spaced horizontally, at the values: 467, 420, 472, 473, 472, 455, 479, and 462. Through them is a black line with slight positive slope. There are red circles along the line at the league night finishes.
Oh look, a shooting star going through Delphinus! It’s so beautiful.

Done. For me, this season, I ended up with a slope ‘m’ of about 2.48 and a ‘b’ of about 451.3. That is, the slightly diagonal black line here. The red circles are what my scores would have been if my performance exactly matched the line.

That seems like a claim that I’m improving over the season. Maybe not a compelling case. That missed night certainly dragged me down. But everybody had some outlier bad night, surely. Why not find the line that best fits everyone’s season, and declare the most-improved person to be the one with the largest positive slope?

Reading the Comics, May 29, 2018: Finding Reruns Edition

There were a bunch of mathematically-themed comic strips this past week. A lot of them are ones I’d seen before. One of them is a bit risque and I’ve put that behind a cut. This saves me the effort of thinking up a good nonsense name to give this edition, so there’s that going for me too.

Bill Amend’s FoxTrot Classics for the 24th of May ought to have run last Sunday, but I wasn’t able to make time to write about it. It’s part of a sequence of Jason tutoring Paige in geometry. She’s struggling with the areas of common shapes which is relatable. Many of these area formulas could be kept straight by thinking back to rectangles. The size of the area is equal to the length of the base times the length of the height. From that you could probably reason right away the area of a trapezoid. It would have the same area as a rectangle with a base of length the mean length of the trapezoid’s different-length sides. The parallelogram works like the rectangle, length of the base times the length of the height. That you can convince yourself of by imagining the parallelogram. Then imagine slicing a right triangle off one of its sides. Move that around to the other side. Put it together right and you have a rectangle. Already know the area of a rectangle. The triangle, then, you can get by imagining two triangles of the same size and shape. Rotate one of the triangles 180 degrees. Slide it over, so the two triangles touch. Do this right and you have a parallelogram and so you know the area. The triangle’s half the area of that parallelogram.

Paige: 'OK, let me see if I've got these area formulas memorized. For a triangle, it's 1/2 bh. For a trapezoid, it's 1/2 (a + b)h. And for a circle, it's pi r^2.' Jason: 'Yes! Yes! Yes! You got them all right! You're going to ace this test! I'm going to make $10!' Paige: 'I always get confused --- does h stand for my height or the triangle's? ... Just kidding.' Jason: 'WILL YOU QUIT TOYING WITH ME?!'
Bill Amend’s FoxTrot Classics for the 24th of May, 2018. It originally ran the 30th of May, 1996.

The circle, I don’t know. I think just remember that if someone says “pi” they’re almost certainly going to follow it with either “r squared” or “day”. One of those suggests an area; the other doesn’t. Best I can do.

Jeri: 'Arrrrhh'. Teena: 'Sup?' Jeri: 'I'm having issues with this math issue. It's the way they phrase these word things. They're like trick questions. I can never figure them out.' Teena: 'Here, let me see what you're having trouble with. ... 'Sarah sits next to Stephen, who is very good at algebra. This causes Sarah, who has issues with BOTH Steven AND algebra, to feel bad But if Sarah moves next to someone else, Steven will feel bad. How do you protect Sarah's and Steven's self-esteem?' Jeri: 'I'm not that comfortable with my answer.' Teena: 'Which is?' Jeri: 'Eleven.' Teena: 'It must be very interesting in your school.'
Allison Barrows’s PreTeena rerun for the 27th of May, 2018. It originally ran the 15th of February, 2004.

Allison Barrows’s PreTeena rerun for the 27th discusses self-esteem as though it were a good thing that children ought to have. This is part of the strip’s work to help build up the Old Person Complaining membership that every comics section community group relies on. But. There is mathematics in Jeri’s homework. Not mathematics in the sense of something particular to calculate. There’s just nothing to do there. But it is mathematics, and useful mathematics, to work out the logic of how to satisfy multiple requirements. Or, if it’s impossible to satisfy them all at once, then to come as near satisfying them as possible. These kinds of problems are considered optimization or logistics problems. Most interesting real-world examples are impossibly hard, or at least become impossibly hard before you realize it. You can make a career out of doing as best as possible in the circumstances.

Charles Schulz’s Peanuts rerun for the 27th features an extended discussion by Lucy about the nature of … well, she explicitly talks about “nothing”. Is she talking about zero? Probably; you have to get fairly into mathematics or philosophy to start worrying about the difference between the number zero and the idea of nothing. In Algebra, mathematicians learn to work with systems of things that work like numbers enough that you can add and subtract and multiply them together, without committing to the idea that they’re working with numbers. They will have something that works like zero, though, a “nothing” that can be added to or subtracted from anything without changing it. And for which multiplication turns something into that “nothing”.

Charlie Brown, at Lucy's Psychiatric Help 5 cents booth: 'I always wanted to go up to that little red-haired girl and talk to her, but I just couldn't. I couldn't start a conversation because I was such a nothing and she was something. If she had wanted to talk to me, it would have been easy because someone who is really something can just go right up to someone who is nothing, and just talk.' Lucy: 'I think your problem is mathematical, Charlie Brown.' Charlie Brown: 'Mathematical?' Lucy: 'If you add nothing and something, what do you get?' Charlie Brown: 'Something, I guess.' Lucy: 'Right ... now, if you subtract nothing from something, what do you get?' Charlie Brown: 'Something.' Lucy: 'Very good ... now, if you multiply something by nothing, what do you get?' Charlie Brown: 'Nothing.' Lucy: 'Five cents, please.' Charlie Brown: 'When you're a nothing, you have a hard time understanding anything!'
Charles Schulz’s Peanuts rerun for the 27th of May, 2018. It originally ran the 30th of May, 1971. This strip originally ran during a time when, in-continuity, the Little Red-Haired Girl had moved away and Charlie Brown was coping with having never spoken to her. At some point she moved back, possibly because Schulz felt he had done everything he could with that or possibly because he forgot she had moved away.

I’m with Charlie Brown in not understanding where Lucy was going with all this, though. Maybe she lost the thread herself.

Mark Anderson’sAndertoons for the 28th is Mark Anderson’sAndertoons for the week. Wavehead’s worried about the verbs of both squaring and rounding numbers. Will say it’s a pair of words with contrary alternate meanings that I hadn’t noticed before. I have always taken the use of “square” to reflect, well, if you had a square with sides of size 4, then you’d have a square with area of size 16. The link seems obvious and logical. So on reflection that’s probably not at all where English gets it from. I mean, not to brag or anything but I’ve been speaking English all my life. If I’ve learned anything about it, it’s that the origin is probably something daft like “while Tisquantum [Squanto] was in England he impressed locals with his ability to do arithmetic and his trick of multiplying one number by itself got nicknamed squantuming, which got shortened to squaning to better fit the meter in a music-hall song about him, and a textbook writer in 1704 thought that was a mistake and `corrected’ it to squaring and everyone copied that”. I’m not even going to venture a guess about the etymology of “rounding”.

On the board: 2^2 = 4, 3^2 = 9, 4^2 = 16. Wavehead: 'Wait, we're squaring numbers now? We just figured out how to round them!'
Mark Anderson’sAndertoons for the 28th of May, 2018. But why would the examples be written out before the students were told what the were doing?

Marguerite Dabaie and Tom Hart’s Ali’s House for the 28th sets up a homework-help session over algebra. Can’t say where exactly Maisa is going wrong. Her saying “x equals 30 but the train equals” looks like trouble to me. It’s often good practice to start by writing out what are the things in the problem that seem important. And what symbol one wants each to mean. And what one knows about the relationship between these things. It helps clarify why someone would want to do that instead of something else. This is a new comic strip tag and I don’t think I’ve ever had cause to discuss it before.

Maisa: 'Can you help me with my homework?' Sahib: 'Dad promised me a hamburger.' Maisa: 'You see - x equals 30 but the train equals ... ' Sahib: 'Dad never makes hamburgers ... mutter mutter mutter.' Maisa: 'Look, I really need help with this.' Sahib: 'My brain isn't set on pay attention my brain is set on burger!'
Marguerite Dabaie and Tom Hart’s Ali’s House for the 28th of May, 2018. Relatable.

Hilary Price’s Rhymes With Orange for the 29th is a Rubik’s Cube joke. I’ve counted that as mathematical enough, usually. The different ways that you can rotate parts of the cube form a group. This is something like what I mentioned in the Peanuts discussion. The different rotations you can do can be added to or subtracted from each other, the way numbers can. (Multiplication I’m wary about.)

Rubik's Headquarters. It's a three-by-three wireframe with tiny offices inside. Person looking in: 'My corner office ... gone! I hate when they do a management shuffle.' [ Title panel, Other person: 'Rumor has it they're going to an open-concept model.' ]
Hilary Price’s Rhymes With Orange for the 29th of May, 2018. Pity whoever gets the center office, bottom layer.

And now here’s the strip that is unsuitable for reading at work, owing to the appearance of an undressed woman.

Continue reading “Reading the Comics, May 29, 2018: Finding Reruns Edition”

How May 2018 Treated My Mathematics Blog

And now the easiest post I write all month: my review of what my readership looked like the past 31 days. I have to admit once more I’m not satisfied with my writership. I didn’t get some projects going that I wanted; but that’s all right. I’ve got five big ideas in mind for the coming several months. Thinking up what to do is always the hard part, other than actually doing it. So that’s my part. Now on to your, the readers’, part. Here I pause while savoring my last moments of not knowing the response was bad.

Oh, how about that. It wasn’t bad. It was even good. Readership was back up in May, rising to 1,274 page views all told. This ties with January for the second-greatest number of readers so far this year. It’s a fair bit up from April’s 1,117. Down from March’s 1,779, but what wouldn’t be? The number of unique visitors rose too, to 837. That’s below March’s tantalizing 999, but up from April’s 731. I did post 12 pieces in May, compared to 11 in April, and 16 in March. I suspect that the number of posts published is the only thing in my control that can influence readership numbers.

A bar chart of readership figures, mostly circling around 1,000 readers per month with a spike two months ago.
Of course the real information is in the Views Per Visitor statistic, but I don’t know what it means.

I can say what people were looking for. The most popular post of the month was, once again, about the number of grooves on a record’s side. I think it’s been getting more popular lately. This shows the power of uploading a better picture of that Buggles album cover, I suppose. The five top posts of the month:

So it’s worth spending some time improving the graphics for my crushingly detailed examination of the area of trapezoids. Writing blogs always say use quality graphics for your articles and it turns out they’re so right.

I struggle still with reader engagement, and I understand that. A lot of what I write is in improv terms hard to advance. I need to be better at writing open things that encourage response. There were a mere 17 comments in May, improved from April’s 13 but still not much at all, especially compared to March’s 53. Which still isn’t great but is something. There were 73 things liked in May, the same number as in April. And way down from March’s 142.

What countries sent me readers? Mostly the United States, as always. But here’s the full roster:

Country Readers
United States 915
Canada 60
India 59
United Kingdom 36
Australia 21
Germany 12
Puerto Rico 11
Denmark 10
Philippines 10
Singapore 10
Malaysia 9
Slovenia 9
Indonesia 8
New Zealand 7
South Africa 7
Israel 6
Spain 6
Brazil 5
Hong Kong SAR China 5
Italy 5
Switzerland 5
Sweden 4
Vietnam 4
Greece 3
Ireland 3
Norway 3
Russia 3
Albania 2
Belgium 2
Ghana 2
Japan 2
Pakistan 2
Slovakia 2
Thailand 2
Turkey 2
Algeria 1
Austria 1
Brunei 1
Costa Rica 1
Egypt 1
European Union 1
Finland 1 (*)
France 1
Jamaica 1
Kuwait 1
Mauritius 1
Morocco 1
Netherlands 1
Poland 1
Qatar 1
Saudi Arabia 1
Serbia 1 (**)
South Korea 1
Sri Lanka 1
Saint Kitts & Nevis 1 (*)
Trinidad & Tobago 1
Ukraine 1

That’s 58 countries which sent me readers over the month. That’s three months in a row the total’s been 58 countries so I assume WordPress is just making these numbers up and figures 58 looks about right. Not suspiciously few, not suspiciously many. We’ll see.

There were 22 single-reader countries. That’s different at least; in April there were 14, and in March 15. Finland and Saint Kitts & Nevis were single-reader countries in April also. Serbia’s been single-reader for three months running now.

The Insights panel tells me that for 2018 so far I’ve had 66 posts, and have accumulated a total of 443 likes and 161 comments. There’s 55,677 total words. This means I published 10,836 total words over the month, which is more than I did in April. I thought I was tired. My year’s average right now is 843.6 words per post; at the end of April that was 830.4. My posts for May alone averaged 903 words. The April posts averaged 772. I knew I was getting more verbose. There’s 2.4 comments and 6.7 likes on average for the post. At the end of April this was 3.5 comments and 6.9 likes per post.

The month officially starts with 62,824 pages viewed from a tracked 30,339 unique visitors. I’ve officially got 759 WordPress visitors, who’re following through their Readers page. I’d be glad if you joined them: you can use the button at the upper-right corner of this page to follow via WordPress. You can also see me as @Nebusj on Twitter. And if you’d prefer you can follow the RSS feed for my posts. If you do that I get absolutely no information about what you read or how interesting you find it, and that’s fine by me.

We ended up putting 38 goldfish back in the pond. The reader with long-term memory may remember we brought 52 in. The fish had a hard winter, one afflicted by water quality issues and feeding issues. We’re trying to recover emotionally, and to work out a plan for better fish care next winter.

Who’s The Most Improved Pinball Player?

My love just completed a season as head of a competitive pinball league. People find this an enchanting fact. People find competitive pinball at all enchanting. Many didn’t know pinball was still around, much less big enough to have regular competitions.

Pinball’s in great shape compared to, say, the early 2000s. There’s one major manufacturer. There’s a couple of small manufacturers who are well-organized enough to make a string of games without (yet) collapsing from not knowing how to finance game-building. Many games go right to private collections. But the “barcade” model of a hipster bar with a bunch of pinball machines and, often, video games is working quite well right now. We’re fortunate to live in Michigan. All the major cities in the lower part of the state have pretty good venues and leagues in or near them. We’re especially fortunate to live in Lansing, so that most of these spots are within an hour’s drive, and all of them are within two hours’ drive.

Ah, but how do they work? Many ways, but there are a couple of popular ones. My love’s league uses a scheme that surely has a name. In this scheme everybody plays their own turn on a set of games. Then they get ranked for each game. So the person who puts up the highest score on the game Junkyard earns 100 league points. The person who puts up the second-highest score on Junkyard earns 99 league points. The person with the third-highest score on Junkyard earns 98 league points. And so on, like this. If 20 people showed up for the day, then the poor person who bottoms out earns a mere 81 league points for the game.

This is a relative ranking, yes. I don’t know any competitive-pinball scheme that uses more than one game that doesn’t rank players relative to each other. I’m not sure how an alternative could work. Different games have different scoring schemes. Some games try to dazzle with blazingly high numbers. Some hoard their points as if giving them away cost them anything. A score of 50 million points? If you had that on Attack From Mars you would earn sympathetic hugs and the promise that life will not always be like that. (I’m not sure it’s possible to get a score that low without tilting your game away.) 50 million points on Lord of the Rings would earn a bunch of nods that yeah, that’s doing respectably, but there’s other people yet to play. 50 million points on Scared Stiff would earn applause for the best game anyone had seen all year. 50 million points on The Wizard of Oz would get you named the Lord Mayor of Pinball, your every whim to be rapidly done.

And each individual manifestation of a table is different. It’s part of the fun of pinball. Each game is a real, physical thing, with its own idiosyncrasies. The flippers are a little different in strength. The rubber bands that guard most things are a little harder or softer. The table is a little more or less worn. The sensors are a little more or less sensitive. The tilt detector a little more forgiving, or a little more brutal. Really the least unfair way to rate play is comparing people to each other on a particular table played at approximately the same time.

It’s not perfectly fair. How could any real thing be? It’s maddening to put up the best game of your life on some table, and come in the middle of the pack because everybody else was having great games too. It’s some compensation that there’ll be times you have a mediocre game but everybody else has a lousy one so you’re third-place for the night.

Back to league. Players earn these points for every game played. So whoever has the highest score of all on, say, Attack From Mars gets 100 league points for that regardless of whatever they did on Junkyard. Whoever has the best score on Iron Maiden (a game so new we haven’t actually played it during league yet, and that somehow hasn’t got an entry on the Internet Pinball Database; give it time) gets their 100 points. And so on. A player’s standings for the night are based on all the league points earned on all the tables played. For us that’s usually five games. Five or six games seems about standard; that’s enough time playing and hanging out to feel worthwhile without seeming too long.

So each league night all the players earn between (about) 420 and 500 points. We have eight league nights. Add the scores up over those league nights and there we go. (Well, we drop the lowest nightly total for each player. This lets them miss a night for some responsibility, like work or travel or recovering from sickness or something, without penalizing them.)

As we got to the end of the season my love asked: is it possible to figure out which player showed the best improvement over time?

Well. I had everybody’s scores from every night played. And I’ve taken multiple classes in statistics. Why would I not be able to?

Reading the Comics, May 23, 2018: Nice Warm Gymnasium Edition

I haven’t got any good ideas for the title for this collection of mathematically-themed comic strips. But I was reading the Complete Peanuts for 1999-2000 and just ran across one where Rerun talked about consoling his basketball by bringing it to a nice warm gymnasium somewhere. So that’s where that pile of words came from.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for this installment. It has Wavehead suggest a name for the subtraction of fractions. It’s not by itself an absurd idea. Many mathematical operations get specialized names, even though we see them as specific cases of some more general operation. This may reflect the accidents of history. We have different names for addition and subtraction, though we eventually come to see them as the same operation.

On the board, 3/5 - 1/4. Wavehead, to teacher: 'You should call it sub-*fraction*. You can use that --- that's a freebie.'
Mark Anderson’s Andertoons for the 21st of May, 2018. I’m not sure the girl in class needs to be quite so horrified by this suggestion. On the other hand, she sees a lot of this kind of stuff in class.

In calculus we get introduced to Maclaurin Series. These are polynomials that approximate more complicated functions. They’re the best possible approximations for a region around 0 in the domain. They’re special cases of the Taylor Series. Those are polynomials that approximate more complicated functions. But you get to pick where in the domain they should be the best approximation. Maclaurin series are nothing but a Taylor series; we keep the names separate anyway, for the reasons. And slightly baffling ones; James Gregory and Brook Taylor studied Taylor series before Colin Maclaurin did Maclaurin series. But at least Taylor worked on Taylor series, and Maclaurin on Macularin series. So for a wonder mathematicians named these things for appropriate people. (Ignoring that Indian mathematicians were poking around this territory centuries before the Europeans were. I don’t know whether English mathematicians of the 18th century could be expected to know of Indian work in the field, in fairness.)

In numerical calculus, we have a scheme for approximating integrals known as the trapezoid rule. It approximates the areas under curves by approximating a curve as a trapezoid. (Any questions?) But this is one of the Runge-Kutta methods. Nobody calls it that except to show they know neat stuff about Runge-Kutta methods. The special names serve to pick out particularly interesting or useful cases of a more generally used thing. Wavehead’s coinage probably won’t go anywhere, but it doesn’t hurt to ask.

Skippy: 'Look at 'im. The meanest kid on the block. He's got a grudge on the school teacher 'cause she made him stop copyin' answers out of his arithmetic. So he tore out the front of the book an' says 'What good is it without the last part?'
Percy Crosby’s Skippy for the 22nd of May, 2018. It was originally run, looks like, the 12th of February, 1931.

Percy Crosby’s Skippy for the 22nd I admit I don’t quite understand. It mentions arithmetic anyway. I think it’s a joke about a textbook like this being good only if it’s got the questions and the answers. But it’s the rare Skippy that’s as baffling to me as most circa-1930 humor comics are.

Lecturer presenting a blackboard full of equations, titled, 'Mathematical Proof that God does not exist'. In the audience is God.
Ham’s Life on Earth for the 23rd of May, 2018. How did the lecturer get stuff on the top of the board there?

Ham’s Life on Earth for the 23rd presents the blackboard full of symbols as an attempt to prove something challenging. In this case, to say something about the existence of God. It’s tempting to suppose that we could say something about the existence or nonexistence of God using nothing but logic. And there are mathematics fields that are very close to pure logic. But our scary friends in the philosophy department have been working on the ontological argument for a long while. They’ve found a lot of arguments that seem good, and that fall short for reasons that seem good. I’ll defer to their experience, and suppose that any mathematics-based proof to have the same problems.

Paige: 'I keep forgetting ... what's the cosine of 60 degrees?' Jason: 'Well, let's see. If I recall correctly ... 1 - (pi/3)^2/2! + (pi/3)^4/4! - (pi/3)^6/6! + (pi/3)^8/8! - (pi/3)^10/10! + (pi/3)^12/12! - (and this goes on a while, up to (pi/3)^32/32! - ... )' Paige: 'In case you've forgotten, I'm not paying you by the hour.' Jason: '1/2'.
Bill Amend’s FoxTrot Classics for the 23rd of May, 2018. It originally ran the 29th of May, 1996.

Bill Amend’s FoxTrot Classics for the 23rd deploys a Maclaurin series. If you want to calculate the cosine of an angle, and you know the angle in radians, you can find the value by adding up the terms in an infinitely long series. So if θ is the angle, measured in radians, then its cosine will be:

\cos\left(\theta\right) = \sum_{k = 0}^{\infty} \left(-1\right)^k \frac{\theta^k}{k!}

60 degrees is \frac{\pi}{3} in radians and you see from the comic how to turn this series into a thing to calculate. The series does, yes, go on forever. But since the terms alternate in sign — positive then negative then positive then negative — you have a break. Suppose all you want is the answer to within an error margin. Then you can stop adding up terms once you’ve gotten to a term that’s smaller than your error margin. So if you want the answer to within, say, 0.001, you can stop as soon as you find a term with absolute value less than 0.001.

For high school trig, though, this is all overkill. There’s five really interesting angles you’d be expected to know anything about. They’re 0, 30, 45, 60, and 90 degrees. And you need to know about reflections of those across the horizontal and vertical axes. Those give you, like, -30 degrees or 135 degrees. Those reflections don’t change the magnitude of the cosines or sines. They might change the plus-or-minus sign is all. And there’s only three pairs of numbers that turn up for these five interesting angles. There’s 0 and 1. There’s \frac{1}{2} and \frac{\sqrt{3}}{2} . There’s \frac{1}{\sqrt{2}} and \frac{1}{\sqrt{2}} . Three things to memorize, plus a bit of orienteering, to know whether the cosine or the sine should be the larger size and whether they should positive or negative. And then you’ve got them all.

You might get asked for, like, the sine of 15 degrees. But that’s someone testing whether you know the angle-addition or angle-subtraction formulas. Or the half-angle and double-angle formulas. Nobody would expect you to know the cosine of 15 degrees. The cosine of 30 degrees, though? Sure. It’s \frac{\sqrt{3}}{2} .

Michael: 'It's near the end of the school year. You should ease up on the homework. I've learned more than enough this year.' Teacher: 'Oh, sure. How does a 50-percent cut sound?' Michael: 'Why cut it by just one-third?' Teacher: 'You're not helping your case.'
Mike Thompson’s Grand Avenue for the 23rd of May, 2018. I don’t know why the kid and the teacher are dressed the same. I’m honestly not sure if they’re related.

Mike Thompson’s Grand Avenue for the 23rd is your basic confused-student joke. People often have trouble going from percentages to decimals to fractions and back again. Me, I have trouble in going from percentage chances to odds, as in, “two to one odds” or something like that. (Well, “one to one odds” I feel confident in, and “two to one” also. But, say, “seven to five odds” I can’t feel sure I understand, other than that the second choice is a perceived to be a bit more likely than the first.)

… You know, this would have parsed as the Maclaurin Series Edition, wouldn’t it? Well, if only I were able to throw away words I’ve already written and replace them with better words before publishing, huh?

Without Tipping My Hand To My Plans For The Next Couple Weeks

I wanted to get this out of the way before I did it:

And the supplemental reading:

Why Stuff Can Orbit, featuring a dazed-looking coati (it's a raccoon-like creature from Latin America) and a starry background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Reading the Comics, May 18, 2018: Quincy Doesn’t Make The Cut Edition

I hate to disillusion anyone but I lack hard rules about what qualifies as a mathematically-themed comic strip. During a slow week, more marginal stuff makes it. This past week was going slow enough that I tagged Wednesday’s Quincy rerun, from March of 1979 for possible inclusion. And all it does is mention that Quincy’s got a mathematics test due. Fortunately for me the week picked up a little. It cheats me of an excuse to point out Ted Shearer’s art style to people, but that’s not really my blog’s business.

Also it may not surprise you but since I’ve decided I need to include GoComics images I’ve gotten more restrictive. Somehow the bit of work it takes to think of a caption and to describe the text and images of a comic strip feel like that much extra work.

Roy Schneider’s The Humble Stumble for the 13th of May is a logic/geometry puzzle. Is it relevant enough for here? Well, I spent some time working it out. And some time wondering about implicit instructions. Like, if the challenge is to have exactly four equally-sized boxes after two toothpicks are moved, can we have extra stuff? Can we put a toothpick where it’s just a stray edge, part of no particular shape? I can’t speak to how long you stay interested in this sort of puzzle. But you can have some good fun rules-lawyering it.

Dad: A guy showed me a brain teaser down at the coffee shop. Watch.' Molly: Ooh, coolie! I'm good at these!' Dad: 'OK, you've got 5 equal-sized boxes here ... moving only 2 toothpicks, make it into FOUR equal-size boxes.' (It's three matchstick boxes in the top row, and two underneath, with the rightmost of the top row above the leftmost of the bottom row.) Dad: 'Heh-heh! THAT ought to keep you busy for a while!' Molly: 'I'll have it in a minute.' Silent final panel, Molly there, bloodshot eyes, late at night.
Roy Schneider’s The Humble Stumble rerun for the 13th of May, 2018. This originally ran the 18th of August, 2006, but I wasn’t doing mathematics blogs back then. Also, Molly there is me with any mathematics puzzle, which is why I panic whenever someone brings one to me. This is a new tag for the comic strip.

Jeff Harris’s Shortcuts for the 13th is a children’s informational feature about Aristotle. Aristotle is renowned for his mathematical accomplishments by many people who’ve got him mixed up with Archimedes. Aristotle it’s harder to say much about. He did write great texts that pop-science writers credit as giving us the great ideas about nature and physics and chemistry that the Enlightenment was able to correct in only about 175 years of trying. His mathematics is harder to summarize though. We can say certainly that he knew some mathematics. And that he encouraged thinking of subjects as built on logical deductions from axioms and definitions. So there is that influence.

A panel full of jokes, activities, and trivia relating to Aristotle. There's no way for me to summarize it all (which includes a word search and a maze as activities) in the space available.
Jeff Harris’s Shortcuts for the 13th of May, 2018. That demonstration of Aristotle’s syllogisms is the same one I see when I search DuckDuckGo for ‘aristotle mathematics’ so it must come right from his texts that I’ve never read! That’s how citations work, right?

Dan Thompson’s Brevity for the 15th is a pun, built on the bell curve. This is also known as the Gaussian distribution or the normal distribution. It turns up everywhere. If you plot how likely a particular value is to turn up, you get a shape that looks like a slightly melted bell. In principle the bell curve stretches out infinitely far. In practice, the curve turns into a horizontal line so close to zero you can’t see the difference once you’re not-too-far away from the peak.

Baseball manager warning the player, 'Watch out, he's got a wicked curve'. The pitcher is a classic hand-style bell with clapper, and also arms and a glove and ball.
Dan Thompson’s Brevity for the 15th of May, 2018. I am curious whether there’s any significance to Thompson’s uniforms, particularly the player having a ‘B’ camp and a ‘U’ shoulder patch. I don’t think there’s an obvious relevance to the statistics jokes being made.

Jason Chatfield’s Ginger Meggs for the 16th I assume takes place in a mathematics class. I’m assuming the question is adding together four two-digit numbers. But “what are 26, 24, 33, and 32” seems like it should be open to other interpretations. Perhaps Mr Canehard was asking for some class of numbers those all fit into. Integers, obviously. Counting numbers. Compound numbers rather than primes. I keep wanting to say there’s something deeper, like they’re all multiples of three (or something) but they aren’t. They haven’t got any factors other than 1 in common. I mention this because I’d love to figure out what interesting commonality those numbers have and which I’m overlooking.

Teacher: 'Meggs! Pop quiz: what are 26, 24, 33, and 32?' Ginger Meggs, after a panel of silent thought: 'Your last four payslips?'
Jason Chatfield’s Ginger Meggs for the 16th of May, 2018. Little surprised Ginger didn’t name cricketeers with those uniform numbers, trusting that cricket players have uniform numbers.

Ed Stein’s Freshly Squeezed for the 17th is a story problem strip. Bit of a passive-aggressive one, in-universe. But I understand why it would be formed like that. The problem’s incomplete, as stated. There could be some fun in figuring out what extra bits of information one would need to give an answer. This is another new-tagged comic.

Nate, the son: 'We're supposed to do today's homework with our parents.' Mom: 'Okay.' Nate: '1. If there are 28 kids in a class, and the education budget is cut by $465 million, how many will be in the class next year?' Dad: 'Taking parental involvement to the next level.' Nate: '2. If the teacher's insurance doesn't cover nervous breakdowns ... '
Ed Stein’s Freshly Squeezed rerun for the 17th of May, 2018. This originally ran the 5th of May, 2011 and maybe I even featured it then. … No, it doesn’t look like I did. Well, I can only imagine how very well this appeal to the parents of the school district under guise of homework went over!

Henry Scarpelli and Craig Boldman’s Archie for the 19th name-drops calculus, credibly, as something high schoolers would be amazed to see one of their own do in their heads. There’s not anything on the blackboard that’s iconically calculus, it happens. Dilton’s writing out a polynomial, more or less, and that’s a fit subject for high school calculus. They’re good examples on which to learn differentiation and integration. They’re a little more complicated than straight lines, but not too weird or abstract. And they follow nice, easy-to-summarize rules. But they turn up in high school algebra too, and can fit into geometry easily. Or any subject, really, as remember, everything is polynomials.

Archie: 'It's amazing how Dilton can do calculus in his head!' Reggie: 'Yeah, I suppose! I guess I'll settle for being the school's most admired athlete and greatest sex symbol!' Jughead: 'It's amazing how Reggie does all that in *his* head, too!'
Henry Scarpelli and Craig Boldman’s Archie rerun for the 19th of May, 2018. And yeah, C^2 + x + 1) isn’t really a coherent expression. It’s either missing a ( mark or, if the C is the open-parentheses, then it’s got nothing-in-particular squared. Also I am so bothered to have close-parentheses and open-parentheses out of order that last sentence. You have no idea.

Mark Anderson’s Andertoons for the 19th is Mark Anderson’s Andertoons for the week. Glad that it’s there. Let me explain why it is proper construction of a joke that a Fibonacci Division might be represented with a spiral. Fibonacci’s the name we give to Leonardo of Pisa, who lived in the first half of the 13th century. He’s most important for explaining to the western world why these Hindu-Arabic numerals were worth learning. But his pop-cultural presence owes to the Fibonacci Sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, and so on. Each number’s the sum of the two before it. And this connects to the Golden Ratio, one of pop mathematics’ most popular humbugs. As the terms get bigger and bigger, the ratio between a term and the one before it gets really close to the Golden Ratio, a bit over 1.618.

Business group looking at a slide showing the golden spiral. Speaker: 'And, as you can see, the Fibonacci division is right on track.'
Mark Anderson’s Andertoons for the 19th of May, 2018. I wonder which direction it’s moving in.

So. Draw a quarter-circle that connects the opposite corners of a 1×1 square. Connect that to a quarter-circle that connects opposite corners of a 2×2 square. Connect that to a quarter-circle connecting opposite corners of a 3×3 square. And a 5×5 square, and an 8×8 square, and a 13×13 square, and a 21×21 square, and so on. Yes, there are ambiguities in the way I’ve described this. I’ve tried explaining how to do things just right. It makes a heap of boring words and I’m trying to reduce how many of those I write. But if you do it the way I want, guess what shape you have?

And that is why this is a correctly-formed joke about the Fibonacci Division.

Reading the Comics, May 12, 2018: New Nancy Artist Edition

And now, closer to deadline than I like, let me wrap up last week’s mathematically-themed comic strips. I had a lot happening, that’s all I can say.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th is another tragic moment in the mathematics department. I’m amused that white lab coats are taken to read as “mathematician”. There are mathematicians who work in laboratories, naturally. Many interesting problems are about real-world things that can be modelled and tested and played with. It’s hardly the mathematics-department uniform, but then, I’m not sure mathematicians have a uniform. We just look like academics is all.

A wall of the Mathematics Department has fallen in. A guy in lab coat says, 'Quick --- someone call the square root of 829,921!!'
Glenn McCoy and Gary McCoy’s The Flying McCoys for the 10th of May, 2018. I suppose the piece of chalk serves as a mathematician’s professional badge, but it would be odd for a person walking in to the room to happen to have a piece. I mean, there’s good reason he might, since there’s never enough chalk in the right places and it has to be stolen from somewhere. But that’s a bit too much backstory for a panel like this.

It also shows off that motif of mathematicians as doing anything with numbers in a more complicated way than necessary. I can’t imagine anyone in an emergency trying to evoke 9-1-1 by solving any kind of puzzle. But comic strip characters are expected to do things at least a bit ridiculously. I suppose.

Mark Litzler’s Joe Vanilla for the 11th is about random numbers. We need random numbers; they do so much good. Getting them is hard. People are pretty lousy at picking random numbers in their head. We can say what “lousy” random numbers look like. They look wrong. There’s digits that don’t get used as much as the others do. There’s strings of digits that don’t get used as much as other strings of the same length do. There are patterns, and they can be subtle ones, that just don’t look right.

Person beside a sign, with the numbers 629510, 787921 and 864370 crossed out and 221473 at the bottom. Caption: 'Performance Art: Random Number Generator'.
Mark Litzler’s Joe Vanilla for the 11th of May, 2018. Bonus: depending on how you want to group a string of six numbers there’s as many as eleven random numbers to select there.

And yet we have a terrible time trying to say what good random numbers look like. Suppose we want to have a string of random zeroes and ones: is 101010 better or worse than 110101? Or 000111? Well, for a string of digits that short there’s no telling. It’s in big batches that we should expect to see no big patterns. … Except that occasionally randomness should produce patterns. How often should we expect patterns, and of what size? This seems to depend on what patterns we’ve found interesting enough to look for. But how can the cultural quirks that make something seem interesting be a substantial mathematical property?

Nancy: 'Don't you hate when you sit down at a computer and can't remember what you were going to do. For the life of me I can't recall what I wanted to do when I sat down.' Teacher: 'Nice try, Nancy, but you still have to take the countywide math test.' (Two other rows of students are on similar computers.)
Olivia Jaimes’s Nancy for the 11th of May, 2018. … Or has the Internet moved on from talking about Nancy already? Bear in mind, I still post to Usenet, so that’s how out of touch I am.

Olivia Jaimes’s Nancy for the 11th uses mathematics-assessment tests for its joke. It’s of marginal relevance, yes, but it does give me a decent pretext to include the new artist’s work here. I don’t know how long the Internet is going to be interested in Nancy. I have to get what attention I can while it lasts.

Scott Hilburn’s The Argyle Sweater for the 12th is the anthropomorphic-geometry joke for the week. Unless there was one I already did Sunday that I already forgot. Oh, no, that was anthropomorphic-numerals. It’s easy to see why a circle might be labelled irrational: either its radius or its area has to be. Both can be. The triangle, though …

Marriage Counsellor: 'She says you're very close-minded.' Triangle: 'It's called 'rational'. But she's all 'pi this' and 'pi that'. Circle: 'It's a constant struggle, doctor.'
Scott Hilburn’s The Argyle Sweater for the 12th of May, 2018. Will admit that I hadn’t heard of Heronian Triangles before I started poking around this, and I started to speculate whether it was even possible for all three legs of a triangle to be rational and the area also be rational. So you can imagine what I felt like when I did some searching and found the 5-12-13 right triangle, since that’s just the other Pythagorean Triplet you learn after the 3-4-5 one. Oh, I guess also the 3-4-5 one.

Well, that’s got me thinking. Obviously all the sides of a triangle can be rational, and so its perimeter can be too. But … the area of an equilateral triangle is \frac{1}{2}\sqrt{3} times the square of the length of any side. It can have a rational side and an irrational area, or vice-versa. Just as the circle has. If it’s not an equilateral triangle?

Can you have a triangle that has three rational sides and a rational area? And yes, you can. Take the right triangle that has sides of length 5, 12, and 13. Or any scaling of that, larger or smaller. There is indeed a whole family of triangles, the Heronian Triangles. All their sides are integers, and their areas are integers too. (Sides and areas rational are just as good as sides and areas integers. If you don’t see why, now you see why.) So there’s that at least. The name derives from Heron/Hero, the ancient Greek mathematician whom we credit with that snappy formula that tells us, based on the lengths of the three sides, what the area of the triangle is. Not the Pythagorean formula, although you can get the Pythagorean formula from it.

Still, I’m going to bet that there’s some key measure of even a Heronian Triangle that ends up being irrational. Interior angles, most likely. And there are many ways to measure triangles; they can’t all end up being rational at once. There are over two thousand ways to define a “center” of a triangle, for example. The odds of hitting a rational number on all of them at once? (Granted, most of these triangle centers are unknown except to the center’s discoverer/definer and that discoverer’s proud but baffled parents.)

Paul: 'Claire, this online business program looks good.' Claire: 'Yeah, I saw that one. But I think it's too intense. I mean, look at this. They make you take two courses in statistics and probability. What are the odds I'd ever need that? ... Oh, wait ... '
Carla Ventresca and Henry Beckett’s On A Claire Day rerun for the 12th of May, 2018. If I make it out right this originally ran the 14th of May, 2010. I forget whether I’ve featured this here already. Likely will drop it from repeats given how hard it is to write much about it. Shame, too, as I’ve just now added that tag to the roster here.

Carla Ventresca and Henry Beckett’s On A Claire Day for the 12th mentions taking classes in probability and statistics. They’re the classes nobody doubts are useful in the real world. It’s easy to figure probability is more likely to be needed than functional analysis on some ordinary day outside the university. I can’t even compose that last sentence without the language of probability.

I’d kind of agree with calling the courses intense, though. Well, “intense” might not be the right word. But challenging. Not that you’re asked to prove anything deep. The opposite, really. An introductory course in either provides a lot of tools. Many of them require no harder arithmetic work than multiplication, division, and the occasional square root. But you do need to learn which tool to use in which scenario. And there’s often not the sorts of proofs that make it easy to understand which tool does what. Doing the proofs would require too much fussing around. Many of them demand settling finicky little technical points that take you far from the original questions. But that leaves the course as this archipelago of small subjects, each easy in themselves. But the connections between them are obscured. Is that better or worse? It must depend on the person hoping to learn.

Someone Else’s Homework: A Probability Question

My friend’s finished the last of the exams and been happy with the results. And I’m stuck thinking harder about a little thing that came across my Twitter feed last night. So let me share a different problem that we had discussed over the term.

It’s a probability question. Probability’s a great subject. So much of what people actually do involves estimating probabilities and making judgements based on them. In real life, yes, but also for fun. Like a lot of probability questions, this one is abstracted into a puzzle that’s nothing like anything anybody does for fun. But that makes it practical, anyway.

So. You have a bowl with fifteen balls inside. Five of the balls are labelled ‘1’. Five of the balls are labelled ‘2’. Five of the balls are labelled ‘3’. The balls are well-mixed, which is how mathematicians say that all of the balls are equally likely to be drawn out. Three balls are picked out, without being put back in. What’s the probability that the three balls have values which, together, add up to 6?

My friend’s instincts about this were right, knowing what things to calculate. There was part of actually doing one of these calculations that went wrong. And was complicated by my making a dumb mistake in my arithmetic. Fortunately my friend wasn’t shaken by my authority, and we got to what we’re pretty sure is the right answer.

Reading the Comics, May 8, 2018: Insecure http Edition

Last week had enough mathematically-themed comics for me to split the content. Usually I split the comics temporally, and this time I will too. What’s unusual is that somewhere along the week the URLs that GoComics pages provide switched from http to https. https is the less-openly-insecure version of the messaging protocol that sends web pages around. It’s good practice; we should be using https wherever possible. I don’t know why they switched that on, and why switch it on midweek. I suppose someone there knew what they were doing.

Tom Wilson’s Ziggy for the 6th of May uses mathematical breakthroughs as shorthand for inspiration. In two ways, too, one with a basically geometric figure and one with a bunch of equations. The geometric figure doesn’t seem to have any significance to me. The equations … that’s a bit harder. They’re probably nonsense. But it’s hard to look at ‘a’ and not see acceleration; the letter is often used for that. And it’s hard to look at ‘v’ and not see velocity. ‘x’ is often a position and ‘t’ is often a time. ‘xf – xi‘ looks meaningful too. It almost begs to be read as “position, final, minus position, initial”. “tf – ti” almost begs to be read as “time, final, minus time, initial”. And the difference in position divided by a difference in time suggests a velocity.

People at Inspiration Point all saying Eureka. one things of an arithmetic formula, one of a geometric proof, one of a bar of music. Ziggy thinks of a vacuum cleaner.
Tom Wilson’s Ziggy for the 6th of May, 2018. I’m also curious whether the geometric figure means anything. But the spray of “x3 – 1” and “x2” and all don’t seem to fit a pattern to me.

So here’s something peculiar inspired by looking at the units that have to follow. If ‘v’ is velocity, then it’s got units of distance over time. \left(\frac{av}{V}\right)^2 and \left(\frac{av}{I}\right)^2 would have units of distance-squared over time-squared. At least unless ‘a ‘or ‘V’ or ‘I’ are themselves measurements. But the square root of their sum then gets us back to distance over time. And then a distance-over-time divided by … well, distance-over-time suggests a pure number. Or something of whatever units ‘R’ carries with it.

So this equation seems arbitrary, and of course the expression doesn’t need to make sense for the joke. But it’s odd that the most-obvious choice of meanings for v and x and t means that the symbols work out so well. At least almost: an acceleration should have units of distance-over-time-squared, and this has units of (nothing). But I may have guessed wrong in thinking ‘a’ meant acceleration here. It might be a description of how something in one direction corresponds to something in another. And that would make sense as a pure number. I wonder whether Wilson got this expression from from anything, or if any readers recognize something that I should have seen right away.

Monty: 'Exactly ONE month of school left, Mrs Lola!' Lola: 'How 'bout that, Monty.' Monty: 'So, subtracting weekends ... that's, um, let's see. Carry the 2, add the 6 ... only 47 days!' Lola: 'Your folks got you signed up for math camp?' Monty: 'How'd you know?'
Todd Clark’s Lola for the 7th of May, 2018. I’m not sure whether Monty means the 6th or the 7th of June is the last day of school, too, but either way I’m pretty sure that’s at least a week and maybe closer to two weeks before we ever got out of school. But we also never started before US Labor Day and it feels indecent when I see schools that do.

Todd Clark’s Lola for the 7th jokes about being bad at mathematics. The number of days left to the end of school isn’t something that a kid should have trouble working out. However, do remember the first rule of calculating the span between two dates on the calendar: never calculate the span between two dates on the calendar. There is so much that goes wrong trying. All right, there’s a method. That method is let someone else do it.

Mutt: 'You want to know what I bought you for Christmas? Think in the number ten!' Jeff: 'Ten? Done!' Mutt: 'Then divide it by two!' Jeff: 'Yes!' Mutt: 'Now you must take away five!' Jeff: 'Yes!' Mutt: 'How much is left?' Jeff: 'Nothing!' (Mutt leaves, while Jeff ponders '?'.)
Bud Fisher’s Mutt and Jeff rerun for the 7th of May, 2018. No idea when the original was from and the word balloons have been relettered with a computer typeface. (Look at the K’s or E’s.) The copyright is given as Aedita S de Beaumont, rather than Bud Fisher or any of the unnamed assistants who actually wrote and drew the strip by this point. Beaumont had married Fisher in 1925 and while they separated after a month they never divorced, so on Fisher’s death Beaumont inherited the rights. Some strips have the signature Pierre S de Beaumont, her son and it happens founder of the Brookstone retail stores. Every bit of this seems strange but I keep looking it over and it seems like I have it right.

Bud Fisher’s Mutt and Jeff for the 7th uses the form of those mathematics-magic games. You know, the ones where you ask someone to pick a number, then do some operations, and then tell you the result. From that you reverse-engineer the original number. They’re amusing enough tricks even if they are all basically the same. It’s instructive to figure out how they work. Replace your original number with symbols and follow the steps then. If you just need the number itself you can replace that with ‘x’. If you need the digits of the number then you’d replace it with something like “10*a + b”, to represent the numerals “ab”. Here, yeah, Mutt’s just being arbitrarily mean.

Robot 55: 'EXTERMINATE ALL LIFE!' Oliver, dressed as a robot: 'Quick, Jorge, act like a robot!' Jorge, dressed like a robot: '20 times 30 equals a million.' Robot 44: 'LIFE EMANATING FROM THIS DIRECTION.' (And approaches the kids.) Oliver: 'Just do the robot dance!' Jorge: 'That's ridiculous, Oliver. Who'd actually program a robot to dance?' (The robots laser-blast a flower.) Jorge, twitching: o/` BOOP BOOP BOOP-BE-BOOP! O/`
Paul Gilligan and Kory Merritt’s Poptropica rerun for the 7th of May, 2018. Sad to say the comic seems to have lapsed into perpetual rerun; I enjoyed the silly adventure and the illustration style.

Paul Gilligan and Kory Merritt’s Poptropica for the 7th depicts calculating stuff as the way to act like a robot. Can’t deny; calculation is pretty much what we expect computers to do. It may hide. It may be done so abstractly it looks like we’re playing Mini Metro instead. This is a new comics tag. I’m sad to say this might be the last use of that tag. Poptropica is fun, but it doesn’t touch on mathematics much at all.

Written on a wood fence: 'Kindergarten teachers know how to make the little things count'.
Gene Mora’s Graffiti for the 8th of May, 2018. I don’t know whether this is a rerun. The copyright date is new but so much about this comic’s worldview is from 1978 at the latest.

Gene Mora’s Graffiti for the 8th mentions arithmetic, albeit obliquely. It’s meant to be pasted on the doors of kindergarten teachers and who am I to spoil the fun?

Anthropomorphic 3/5: 'Honey, what's wrong?' Anthropomorphic 1/4: 'Sour son is leaving the faith! He said he's converting to decimals!'
Scott Hilburn’s The Argyle Sweater for the 9th of May, 2018. I like the shout-out to Archimedes in the background art, too. Archimedes, though, didn’t use fractions in the way we’d recognize them. He’d write out a number as a combination of ratios of some reference number. So he might estimate the length of something being as to the length of something else as 19 is to 7, or something like that. This seems like a longwinded and cumbersome way to write out numbers, or much of anything, and makes one appreciate his indefatigability as much as his insight.

Scott Hilburn’s The Argyle Sweater for the 9th is the anthropomorphic-numerals joke for this week. Converting between decimals and fractions has been done since decimals got worked out in the late 16th century. There’s advantages to either representation. To my eyes the biggest advantage of fractions is they avoid hypnotizing people with the illusion of precision. 0.25 reads as more exact than 1/4. We can imagine it being 0.2500000000000000 and think we know the quantity to any desired precision. 1/4 reads (to me, anyway) as being open to the possibility we’re rounding off from 0.998 out of 4.00023.

Another advantage fractions do have is flexibility. There are infinitely many ways to express the same number as a fraction. In decimals, there are at most two. If you’re trying to calculate something that would be more easily done with a denominator of 30 than of 5, you’re free to do that. Decimals can have advantages in computing, certainly, especially if you’re already set up to manipulate digits. And you can tell at a glance whether, say, 14/29th is greater or less than 154/317th. In case you ever find reason to wonder, I mean. I’m not saying either is always the right way to go.

Someone Else’s Homework: A Postscript

My friend aced the mathematics final. Not due to my intervention, I’d say; my friend only remembered one question on the exam being much like anything we had discussed recently. Though it was very like one of those, a question about the probability of putting together a committee with none, one, two, or more than two members of particular subgroups. And that one we didn’t even work through; I just confirmed my friend’s guess about what calculation to do. Which is good since that particular calculation is a tedious one that I didn’t want to do. No, my friend aced it by working steadily through the whole term. And yes, asking me for tutoring a couple times, but that’s all right. Small, steady work adds up, in mathematics as with so much else.

Meanwhile may I draw your attention over to my humor blog where last night I posted a bit of silliness about number divisibility. Because I can’t help myself, it does include a “quick” test for whether a number could be divided by 21. It’s in the same spirit as tests for whether a number can be divided by 3 or 9 (add the digits add see whether that sum’s divisible by 3 or 9) or 11 (add or subtract digits, in alternate form, and see whether that sum is divisible by 11). The process I give is correct, which is not to say that anyone would ever use it. Even if they did they’d be better off testing for divisibility by both 3 and 7. And I don’t think I’d use an add-the-digits scheme for 7 either.

Someone Else’s Homework: Was It Hard? An Umbrella Search

I wanted to follow up, at last, on this homework problem a friend had.

The question: suppose you have a function f. Its domain is the integers Z. Its rage range is also the integers Z. You know two things about the function. First, for any two integers ‘a’ and ‘b’, you know that f(a + b) equals f(a) + f(b). Second, you know there is some odd number ‘c’ for which f(c) is even. The challenge: prove that f is even for all the integers.

My friend asked, as we were working out the question, “Is this hard?” And I wasn’t sure what to say. I didn’t think it was hard, but I understand why someone would. If you’re used to mathematics problems like showing that all the roots of this polynomial are positive, then this stuff about f being even is weird. It’s a different way of thinking about problems. I’ve got experience in that thinking that my friend hasn’t.

All right, but then, what thinking? What did I see that my friend didn’t? And I’m not sure I can answer that perfectly. Part of gaining mastery of a subject is pattern recognition. Spotting how some things fit a form, while other stuff doesn’t, and some other bits yet are irrelevant. But also part of gaining that mastery is that it becomes hard to notice that’s what you’re doing.

But I can try to look with fresh eyes. There is a custom in writing this sort of problem, and that drove much of my thinking. The custom is that a mathematics problem, at this level, works by the rules of a Minute Mystery Puzzle. You are given in the setup everything that you need to solve the problem, yes. But you’re also not given stuff that you don’t need. If the detective mentions to the butler how dreary the rain is on arriving, you’re getting the tip to suspect the houseguest whose umbrella is unaccounted for.

(This format is almost unavoidable for teaching mathematics. At least it seems unavoidable given the number of problems that don’t avoid it. This can be treacherous. One of the hardest parts in stepping out to research on one’s own is that there’s nobody to tell you what the essential pieces are. Telling apart the necessary, the convenient, and the irrelevant requires expertise and I’m not sure that I know how to teach it.)

The first unaccounted-for umbrella in this problem is the function’s domain and range. They’re integers. Why wouldn’t the range, particularly, be all the real numbers? What things are true about the integers that aren’t true about the real numbers? There’s a bunch of things. The highest-level things are rooted in topology. There’s gaps between one integer and its nearest neighbor. Oh, and an integer has a nearest neighbor. A real number doesn’t. That matters for approximations and for sequences and series. Not likely to matter here. Look to more basic, obvious stuff: there’s even and odd numbers. And the problem talks about knowing something for an odd number in the domain. This is a signal to look at odds and evens for the answer.

The second unaccounted-for umbrella is the most specific thing we learn about the function. There is some odd number ‘c’, and the function matches that integer ‘c’ in the domain to some even number f(c) in the range. This makes me think: what do I know about ‘c’? Most basic thing about any odd number is it’s some even number plus one. And that made me think: can I conclude anything about f(1)? Can I conclude anything about f at the sum of two numbers?

Third unaccounted-for umbrella. The less-specific thing we learn about the function. That is that for any integers ‘a’ and ‘b’, f(a + b) is f(a) + f(b). So see how this interacts with the second umbrella. f(c) is f(some-even-number) + f(1). Do I know anything about f(some-even-number)?

Sure. If I know anything about even numbers, it’s that any even number equals two times some integer. Let me call that some-integer ‘k’. Since some-even-number equals 2*k, then, f(some-even-number) is f(2*k), which is f(k + k). And by the third umbrella, that’s f(k) + f(k). By the first umbrella, f(k) has to be an integer. So f(k) + f(k) has to be even.

So, f(c) is an even number. And it has to equal f(2*k) + f(1). f(2*k) is even; so, f(1) has to be even. These are the things that leapt out to me about the problem. This is why the problem looked, to me, easy.

Because I knew that f(1) was even, I knew that f(1 + 1), or f(2), was even. And so would be f(2 + 1), that is, f(3). And so on, for at least all the positive integers.

Now, after that, in my first version of this proof, I got hung up on what seems like a very fussy technical point. And that was, what about f(0)? What about the negative integers? f(0) is easy enough to show. It follows from one of those tricks mathematics majors are told about early. Somewhere in grad school they start to believe it. And that is: adding zero doesn’t change a number’s value, but it can give you a more useful way to express that number. Here’s how adding zero helps: we know c = c + 0. So f(c) = f(c) + f(0) and whether f(c) is even or odd, f(0) has to be even. Evens and odds don’t work any other way.

After that my proof got hung up on what may seem like a pretty fussy technical point. That amounted to whether f(-1) was even or odd. I discussed this with a couple people who could not see what my issue with this was. I admit I wasn’t sure myself. I think I’ve narrowed it down to this: my questioning whether it’s known that the number “negative one” is the same thing as what we get from the operation “zero minus one”. I mean, in general, this isn’t much questioned. Not for the last couple centuries.

You might be having trouble even figuring out why I might worry there could be a difference. In “0 – 1” the – sign there is a binary operation, meaning, “subtract the number on the right from the number on the left”. In “-1” the – sign there is a unary operation, meaning, “take the additive inverse of the number on the right”. These are two different – signs that look alike. One of them interacts with two numbers. One of them interacts with a single number. How can they mean the same thing?

With some ordinary assumptions about what we mean by “addition” and “subtraction” and “equals” and “zero” and “numbers” and stuff, the difference doesn’t matter much. We can swap between “-1” and “0 – 1” effortlessly. If we couldn’t, we probably wouldn’t use the same symbol for the two ideas. But in the context of this particular question, could we count on that?

My friend wasn’t confident in understanding what the heck I was getting on about. Fair enough. But some part of me felt like that needed to be shown. If it hadn’t been recently shown, or used, in class, then it had to go into this proof. And that’s why I went, in the first essay, into the bit about additive inverses.

This was me over-thinking the problem. I got to looking at umbrellas that likely were accounted for.

My second proof, the one thought up in the shower, uses the same unaccounted-for umbrellas. In the first proof, the second unaccounted-for umbrella seemed particularly important. Knowing that f(c) was odd, what else could I learn? In the second proof, it’s the third unaccounted-for umbrella that seemed key. Knowing that f(a + b) is f(a) + f(b), what could I learn? That right away tells me that for any even number ‘d’, f(d) must be even.

Call this the fourth unaccounted-for umbrella. Every integer is either even or odd. So right away I could prove this for what I really want to say is half of the integers. Don’t call it that. There’s not a coherent way to say the even integers are any fraction of all the integers. There’s exactly as many even integers as there are integers. But you know what I mean. (What I mean is, in any finite interval of consecutive integers, half are going to be even. Well, there’ll be at most two more odd integers than there are even integers. That’ll be close enough to half if the interval is long enough. And if we pretend we can make bigger and bigger intervals until all the integers are covered … yeah. Don’t poke at that and do not use it at your thesis defense because it doesn’t work. That’s what it feels like ought to work.)

But that I could cover the even integers in the domain with one quick sentence was a hint. The hint was, look for some thing similar that would cover the odd integers in the domain. And hey, that second unaccounted-for umbrella said something about one odd integer in the domain. Add to that one of those boring little things that a mathematician knows about odd numbers: the difference between any two odd numbers is an even number. ‘c’ is an odd number. So any odd number in the domain, let’s call it ‘d’, is equal to ‘c’ plus some even number. And f(some-even-number) has to be even and there we go.

So all this is what I see when I look at the question. And why I see those things, and why I say this is not a hard problem. It’s all in spotting these umbrellas.

Reading the Comics, May 5, 2018: Does Anyone Know Where The Infinite Hotel Comes From Edition

With a light load of mathematically-themed comic strips I’m going to have to think of things to write about twice this coming week. Fortunately, I have plans. We’ll see how that works out for me. So far this year I’m running about one-for-eight on my plans.

Mort Walker and Dik Browne’s Hi and Lois for the 1st of November, 1960 looks pretty familiar somehow. Having noticed what might be the first appearance of “the answer is twelve?” in Peanuts I’m curious why Chip started out by guessing twelve. Probably just coincidence. Possibly that twelve is just big enough to sound mathematical without being conspicuously funny, like 23 or 37 or 42 might be. I’m a bit curious that after the first guess Sally looked for smaller numbers than twelve, while Chip (mostly) looked for larger ones. And I see a logic in going from a first guess of 12 to a second guess of either 4 or 144. The 32 is a weird one.

Teacher: 'Chip, do you know the answer to number five?' Chip: 'Is it twelve? No, wait ... it's four. Or is it 32 ... it's either that or 144. No, wait a second ... I'll get it.' Teacher: 'I'm sure you will --- we're nearly out of numbers!'
Mort Walker and Dik Browne’s Hi and Lois for the 1st of November, 1960 and reprinted the 30th of April, 2018. Yeah, sure, eleven years later Charles Schulz would do basically the same joke. But comics snarkers get all smug when they notice that The Argyle Sweater is using the same premise as a Far Side from 1983 or something.

Tom Toles’s Randolph Itch, 2 am for the 30th of April, 2018 is on at least its third appearance around here. I suppose I have to retire the strip from consideration for these comics roundups. It didn’t run that long, sad to say, and I think I’ve featured all its mathematical strips. I’ll go on reading, though, as I like the style and Toles’s sense of humor.

Randolph, thinking in bed: 'Algebra pretty much put pirates out of business.' Pirate teacher: 'If ax^2 + bx + c = 0, what is x?' And a pirate sweats. Footer joke: '15 men on a dead man's chest, you ho ho, and a bottle of rum equals what?'
Tom Toles’s Randolph Itch, 2 am for the 30th of April, 2018. I think I’ve mentioned not knowing whether the legendary X on pirate maps is related to the use of X as the thing-to-be-found in algebra. My understanding is the X on a pirate map thing is mostly a matter of storytelling rather than something anyone really did.

Mark Tatulli’s Heart of the City for the 3rd of May is a riff on the motivation problem. For once, not about the motivation of the people in story problems to do what they do. It’s instead about why the student should care what the story people do. And, fair enough, really. It’s easy to calculate something you’d like to know the answer to. But give the teacher or textbook writer a break. There’s nothing that’s interesting to everybody. No, not even what minimum grade they need on this exam to get an A in the course. After a moment of clarity in fifth grade I never cared what my scores were. I just did my work and accepted the assessment. My choice not to worry about my grades had more good than bad results, but I admit, there were bad results too.

Heart: 'This homework is ridiculous! How are we supposed to know this? OK, so Julio has eleven apples! Why is it any of my business how many he shares with Allisa and Wesley? Why should I care how he divides them or what fraction he keeps for himself?' Dean: 'They're just math word problems. I don't think we have to explain motivation.' Heart: 'I guess I just can't get past why exactly a kid has eleven apples in the first place!'
Mark Tatulli’s Heart of the City for the 3rd of May, 2018. How does Heart know that Julio is a kid? Adults will go out and buy a dozen apples without that seeming strange, and we’ll eat as many as four of them before forgetting they’re on the counter and letting them rot by accident.

John McNamee’s Pie Comic for the 4th of May riffs on some ancient story-problems built on infinite sets. I don’t know the original source. I assume a Martin Gardiner pop-mathematics essay. I don’t know, though, and I’m curious if anyone does know.

[ You arrive at the Infinite Hotel ... but the concierge says *all* infinity of their rooms are booked. Luckily, you know MATH. 'Infinity = Infinity + 1'. You explain that if you just ask each guest to move one room over ... ] There's a large man in diapers with a sock puppet on his hand in the first door. [ Yeah ... math's not solving that. ] You drive away.
John McNamee’s Pie Comic for the 4th of May, 2018. Also you know how long it’d take to re-code everybody’s door access card? It’d be like forever.

Often I see these kinds of problem as set at the Hilbert Hotel. This references David Hilbert, the late-19th/early-20th century mastermind behind the 20th century’s mathematics field. They try to challenge people’s intuitions about infinitely large sets. Ponder a hotel with one room for each of the counting numbers. Suppose it’s full. How many guests can you add to it? Can you add infinitely many more guests, and still have room for them all? If you do it right, and if “infinitely many more guests” means something particular, yes. If certain practical points don’t get in the way. I mean practical for a hotel with infinitely many rooms.

This is a new-tag comic.

Medieval monks, talking about the one who's written down E = mc^2. 'God only knows what it means. This guy isn't all that swift.'
Dave Whamond’s Reality Check for the 4th of May, 2018. Yeah, I heard the sequel to A Canticle for Leibowitz was disappointing.

Dave Whamond’s Reality Check for the 4th is a riff on Albert Einstein’s best-known equation. He had some other work, granted. But who didn’t?

Reading the Comics, April 28, 2018: Friday Is Pretty Late Edition

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

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

Crock to the two prisoners in lockboxes: 'Guess the number I'm thinking and I'll set you free.' First prisoner: '4,796,034,621,322.' Crock: 'Sorry, it's nine.' Second prisoner: 'What made you guess THAT number?' First prisoner: 'It was the first one to pop into my head.'
Bill Rechin’s Crock rerun for the 26th of April, 2018. Going ahead and guessing there’s another Crock with the same setup, except the prisoner guesses nine, and Crock says it was 4,796,034,621,322, and then in the final panel we see that Crock really had thought nine and lied.

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

On the board: 51 - 26, with the 51 rewritten as 4 with a borrowed 11. Wavehead: 'So we're just borrowing 10 no questions asked? What about a credit check? What's the interest rate?'
Mark Anderson’s Andertoons for the 26th of April, 2018. This is Mark Anderson’s Andertoons for the week.

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

Larry: 'Your shirt is on all wrong.' Toby: 'It was bound to happen.' Larry: 'What? Why?' Toby: 'There's FOUR different ways a shirt can go on! That gives me only, like, a 20% chance any time I put it on.'
Greg Cravens’s The Buckets for the 26th of April, 2018. I’m not sure Larry (the father)’s disbelief at his kid figuring putting the shirt on all wrong was bound to happen. It’s a mistake we all make; accepting the inevitability of that doesn’t seem that wrong.

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

Sally, at her schooldesk: 'The answer is twelve! It isn't? How about six? Four? Nine? Two? Ten? ... Do you have the feeling that I'm guessing?'
Charles Schulz’s Peanuts rerun for the 27th of April, 2018. This strip first ran the 30th of April, 1971. It also was rerun the 25th of April, 2003, with a different colorization scheme for some reason.

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

Venn Diagram. One circle's labelled 'Venn Diagrams'; the second 'Jokes'. The intersection is 'Lazy Cartoonists'.
Maria Scrivan’s Half-Full for the 28th of April, 2018. Hey, cartoonists deserve easy days at work too. And there’s not always a convenient holiday they can have the cast just gather around and wish everyone a happy instance of.

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

Guy: 'Relax. Half the time, job interviewers don't even read your resume. They just see how long it is.' Mathematician: 'Really?' Guy: 'Yeah. Where are you going?' Mathematician: 'To make a Mobius strip.' Interviewer: 'Wow! I've never met someone with *infinite* skills and work experience.' Mathematician: 'I don't like to brag.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of April, 2018. If I had seen this strip in 2007 maybe I would’ve got that tenure-track posting instead of going into the world of technically being an extant mathematics blog.

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

How April 2018 Treated My Mathematics Blog

People were far less interested in the number of grooves on a record’s side this past month. That’s what I take away from the readership figures around here for April, as WordPress reports. There were, it appears, some 1,117 pages viewed in April, from 731 unique visitors. That’s well down from March’s 1,779 views and 999 visitors. But March was clearly an outlier; February saw 1,062 page views from 611 visitors. This is four months in a row with at least a thousand page views, so everything seems consistent.

The number of likes fell to 73, down from 142. This seems like a lot of drop, but considering there were 102 likes in February and 112 in January … yeah, that’s a bit lower. Hm. Comments were down, too, with a mere 13 posted in April. There were 53 in March, 30 in February, those are much more engaged numbers. It’s my doing, I know; I had a month of mostly writing about comics and that’s fun, but it’s not much to discuss. What’s to say, “That wasn’t really a student making fun of the story problem!”? Nah. Also I’m abashed to realize I had only eleven posts in April; March had a healthier count of 16.

Statistics chart showing a big spike in March and a return to the roughly twelve-month normal for April 2018.
Definitely more normal than the March 2018 figures.

So what were people reading? One perennial and then a bunch of mostly new stuff:

The Insights panel tells me I’ve gotten to 44,841 total words published this year so far, with 135 total comments and 370 total likes. So, 8,494 words over the month. I’m currently averaging 830.4 words per post, 3.5 comments per post, and 6.9 likes per post. Words and likes are slightly up from March; comments are down a bit.

As I make it out 58 countries sent me readers this past month. That’s the same as March, and up from February’s 54. They’re these:

Country Readers
United States 687
United Kingdom 84
Canada 59
India 38
Australia 21
Singapore 18
Philippines 17
Brazil 16
South Africa 16
Ireland 11
Spain 11
Turkey 11
Puerto Rico 8
Denmark 7
France 7
Afghanistan 6
Italy 6
Netherlands 5
Peru 5
Slovenia 5
Sweden 5
Germany 4
Israel 4
New Zealand 4
Poland 4
Ukraine 4
Mongolia 3
Russia 3
South Korea 3
United Arab Emirates 3
Algeria 2
Argentina 2
Belgium 2
Bulgaria 2
Egypt 2
Hong Kong SAR China 2
Indonesia 2
Japan 2
Lebanon 2
Lithuania 2
Malaysia 2
Norway 2
Romania 2
Switzerland 2
Armenia 1
Czech Republic 1
Finland 1
Gibraltar 1
Iraq 1
Kenya 1
Luxembourg 1
Nigeria 1
Palestinian Territories 1
Senegal 1
Serbia 1 (*)
St. Kitts & Nevis 1
Tunisia 1
Vietnam 1

That’s 14 single-reader countries, down one from March and down two from February. Serbia was a single-reader country in March; nowhere else was. May starts with 61,549 pages viewed from 29,502 admitted unique visitors.

I’d appreciate it if you did follow NebusResearch regularly. I haven’t restored the e-mail postings, although if I go another month or two without anything suspicious turning up in the comments I might try it. But you can follow on your WordPress Reader, by using the button at the upper right corner of the page. Here’s the RSS feed, if you’d rather read the way you like without WordPress being able to trace you. And if you don’t mind Twitter you can follow me as @Nebusj there. Watch as I give the tally of how many goldfish we’re getting back out to the backyard pond!