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.

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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!

Reading the Comics, April 25, 2018: Coronet Blue Edition


You know what? Sometimes there just isn’t any kind of theme for the week’s strips. I can use an arbitrary name.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of April, 2018 would have gone in last week if I weren’t preoccupied on Saturday. The joke is aimed at freshman calculus students and then intro Real Analysis students. The talk about things being “arbitrarily small” turns up a lot in these courses. Why? Well, in them we usually want to show that one thing equals another. But it’s hard to do that. What we can show is some estimate of how different the first thing can be from the second. And if you can show that that difference can be made small enough by calculating it correctly, great. You’ve shown the two things are equal.

Delta and epsilon turn up in these a lot. In the generic proof of this you say you want to show the difference between the thing you can calculate and the thing you want is smaller than epsilon. So you have the thing you can calculate parameterized by delta. Then your problem becomes showing that if delta is small enough, the difference between what you can do and what you want is smaller than epsilon. This is why it’s an appropriately-formed joke to show someone squeezed by a delta and an epsilon. These are the lower-case delta and epsilon, which is why it’s not a triangle on the left there.

Mad scientist cackling at a man being crushed between giant delta and epsilon figure: 'And now, good doctor, we will see how you fit between this delta and this epsilon!' Caption: Soon, soon the calculus teacher would become arbitrarily small.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of April, 2018. I feel vaguely personally called out by the calculus teacher wearing cargo shorts, tall white socks, and sandals.

For example, suppose you want to know how long the perimeter of an ellipse is. But all you can calculate is the perimeter of a polygon. I would expect to make a proof of it look like this. Give me an epsilon that’s how much error you’ll tolerate between the polygon’s perimeter and the ellipse’s perimeter. I would then try to find, for epsilon, a corresponding delta. And that if the edges of a polygon are never farther than delta from a point on the ellipse, then the perimeter of the polygon and that of the ellipse are less than epsilon away from each other. And that’s Calculus and Real Analysis.

John Zakour and Scott Roberts’s Maria’s Day for the 22nd is the anthropomorphic numerals joke for this week. I’m curious whether the 1 had a serif that could be wrestled or whether the whole number had to be flopped over, as though it were a ruler or a fat noodle.

Maria at her desk challenges a giant number 4 to arm wrestling; she slams its 'arm' down easily. Other numerals flee as she yells out: 'Okay, anyone else wanna take me on? Huh? --- Yeah, didn't think so!' Reality: she's at her desk with a book and some paper and says, "Whew! This math homework was tough --- but I think I got it down.'
John Zakour and Scott Roberts’s Maria’s Day for the 22nd of April, 2018. I’m curious whether Zakour and Roberts deliberately put 2 and 3 to the left, with pain stars indicating they’ve been beaten already, while the bigger numbers are off to the side. Or was it just an arbitrary choice? The numbers are almost in order, left to right, except that the 7’s out of place. So maybe the ordering is just coincidence?

Anthony Blades’s Bewley for the 23rd offers advice for what to do if you’ve not got your homework. This strip’s already been run, and mentioned here. I might drop this from my reading if it turns out the strip is done and I’ve exhausted all the topics it inspires.

Bea: 'Aaaah! I forgot to do my maths homework!' Tonus: 'I did mine.' Bea: 'Can I copy yours?' Tonus: 'Of course you can. I didn't know the answers so I drew a picture of a scary dinosaur.' [ Silent penultimate panel. ] Bea: 'Better than nothing.' Tonus: 'Remember the big teeth. Big teeth make it scary.'
Anthony Blades’s Bewley for the 23rd of April, 2018. Whenever a comic strip with this setup begins I think of the time in geometry class when I realized I hadn’t done any homework and wondered if I could get something done in the time it took papers to be passed up. This in a class of twelve students. No, there was not, even though the teacher started from the other side of the classroom.

Dave Whamond’s Reality Check for the 23rd is designed for the doors of mathematics teachers everywhere. It does incidentally express one of those truths you barely notice: that statisticians and mathematicians don’t seem to be quite in the same field. They’ve got a lot of common interest, certainly. But they’re often separate departments in a college or university. When they do share a department it’s named the Department of Mathematics and Statistics, itself an acknowledgement that they’re not quite the same thing. (Also it seems to me it’s always Mathematics-and-Statistics. If there’s a Department of Statistics-and-Mathematics somewhere I don’t know of it and would be curious.) This has to reflect historical influence. Statistics, for all that it uses the language of mathematics and that logical rigor and ideas about proofs and all, comes from a very practical, applied, even bureaucratic source. It grew out of asking questions about the populations of nations and the reliable manufacture of products. Mathematics, even the mathematics that is about real-world problems, is different. A mathematician might specialize in the equations that describe fluid flows, for example. But it could plausibly be because they have interesting and strange analytical properties. It’d be only incidental that they might also say something enlightening about why the plumbing is stopped up.

[ Clown Statistician vs Clown mathematician. ] The Clown Statistician holds up a pie chart, ready to throw it. The mathematician holds up a pi symbol, ready to throw it. Corner squirrel's comment: 'There's always room for more pi.'
Dave Whamond’s Reality Check for the 23rd of April, 2018. I’m not sure I’ve laughed more at a dumb joke than I have at this in a long while.

Neal Rubin and Rod Whigham’s Gil Thorp for the 24th seems to be setting out the premise for the summer storyline. It’s sabermetrics. Or at least the idea that sports performance can be quantized, measured, and improved. The principle behind that is sound enough. The trick is figuring out what are the right things to measure, and what can be done to improve them. Also another trick is don’t be a high school student trying to lecture classmates about geometry. Seriously. They are not going to thank you. Even if you turn out to be right. I’m not sure how you would have much control of the angle your ball comes off the bat, but that’s probably my inexperience. I’ve learned a lot about how to control a pinball hitting the flipper. I’m not sure I could quantize any of it, but I admit I haven’t made a serious attempt to try either. Also, when you start doing baseball statistics you run a roughly 45% chance of falling into a deep well of calculation and acronyms of up to twelve letters from which you never emerge. Be careful. (This is a new comic strip tag.)

[ With rain delaying (baseball) practice, Kevin Pelwecki expounds on his new favorite subject --- ] Kevin: 'Launch angle! You want the ball coming off the bat at 25 degrees.' Teammate: 'Anyone else notice we're taking math lessons --- from a guy who barely passed geometry?'
Neal Rubin and Rod Whigham’s Gil Thorp for the 24th of April, 2018. Are … both word balloons coming from the same guy? In the last panel there. I understand one guy starting and another closing a thought but that’s usually something you do with an established in-joke that anyone can feed and anyone else can finish. A spontaneous insult like this seems like it only needs the one person, but the word balloon tails are weird if they’re both from the same guy.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 25th feels a little like a slight against me. Well, no matter. Use the things that get you in the mood you need to do well. (Not a new comic strip tag because I’m filing it under ‘Randy Glasbergen’ which I guess I used before?)

Kid with guitar: 'I start every song by counting 1-2-3-4 because it reminds me of math. Math depresses me and that helps me sing the blues.'
Randy Glasbergen’s Glasbergen Cartoons rerun for the 25th of April, 2018. OK, but what’s his guitar plugged in to?

And What I’ve Been Reading


So here’s some stuff that I’ve been reading.

This one I saw through John Allen Paulos’s twitter feed. He points out that it’s like the Collatz conjecture but is, in fact, proven. If you try this yourself don’t make the mistake of giving up too soon. You might figure, like start with 12. Sum the squares of its digits and you get 5, which is neither 1 nor anything in that 4-16-37-58-89-145-42-20 cycle. Not so! Square 5 and you get 25. Square those digits and add them and you get 29. Square those digits and add them and you get 40. And what comes next?

This is about a proof of Fermat’s Theorem of Sums of Two Squares. According to it, a prime number — let’s reach deep into the alphabet and call it p — can be written as the sum of two squares if and only if p is one more than a whole multiple of four. It’s a proof by using fixed point methods. This is a fun kind of proof, at least to my sense of fun. It’s an approach that’s got a clear physical interpretation. Imagine picking up a (thin) patch of bread dough, stretching it out some and maybe rotating it, and then dropping it back on the board. There’s at least one bit of dough that’s landed in the same spot it was before. Once you see this you will never be able to just roll out dough the same way. So here the proof involves setting up an operation on integers which has a fixed point, and that the fixed point makes the property true.

John D Cook, who runs a half-dozen or so mathematics-fact-of-the-day Twitter feeds, looks into calculating the volume of an egg. It involves calculus, as finding the volume of many interesting shapes does. I am surprised to learn the volume can be written out as a formula that depends on the shape of the egg. I would have bet that it couldn’t be expressed in “closed form”. This is a slightly flexible term. It’s meant to mean the thing can be written using only normal, familiar functions. However, we pretend that the inverse hyperbolic tangent is a “normal, familiar” function.

For example, there’s the surface area of an egg. This can be worked out too, again using calculus. It can’t be written even with the inverse hyperbolic cotangent, so good luck. You have to get into numerical integration if you want an answer humans can understand.

My next mistake will be intentional, just to see how closely you are watching me.
Ashleigh Brilliant’s Pot-Shots rerun for the 15th of April, 2018. I understand people not liking Brilliant’s work but I love the embrace-the-doom attitude the strip presents.

Also, this doesn’t quite fit my Reading the Comics posts. But Ashleigh Brilliant’s Pot-Shots rerun for the 15th of April is something I’m going to use in future. I hope you find some use for it too.

Can I Still Get A B In This Class?


I’ve been informed by reliable sources that it’s near the end of the college semester for most United States colleges and universities. So let me bring back some of my very minor perennials. What Do I Need To Pass This Class? is my old and slightly overwritten description of how to get a particular course grade. It allows for any weightings of course work and final exam plus any extra credit that might be hanging around.

What Do I Need To Get An A In This Class? meanwhile is less flexible but maybe more useful at a glance. It’s simply tables of some common grade weightings and class averages, and shows what final exam score (if any) makes possible an A, B, C, D, or F. This is on the assumption that a 90 or above is an A, an 80 or above is a B, 70 or above is C, and 60 or above is a D.

And as ever the real answer is more fundamental. You should get more sleep. You should study a fair bit every day rather than cram at the end of the term. If you’re able, you should take notes by hand rather than by computer. You should talk with the instructor the moment you start feeling lost, rather than when you feel hopelessly lost. You should use the tutoring services the school offers. If you have special needs for exams or other class work, you should meet with the school’s office for that sort of thing. Your instructors can make accommodations but they need to know what you need, and the sooner the better. In short, don’t try to do it all in the final. But yeah, I know, you’re wondering too.

Reading the Comics, April 19, 2018: Late Because Of Pinball Edition


Hi, all. I apologize for being late in posting this, but my Friday and Saturday were eaten up by pinball competition. Pinball At The Zoo, particularly, in Kalamazoo, Michigan. There, Friday, I stepped up first thing and put in four games on the Classics, pre-1985, tournament bank and based on my entry scores was ranked the second-best player there. And then over the day my scores dwindled lower and lower on the list of what people had entered until, in the last five minutes of qualifying, they dropped off the roster altogether and I was knocked out. Meanwhile in the main tournament, I was never even close to making playoffs. But I did have a fantastic game of Bally/Midway’s World Cup Soccer, a game based on how much the United States went crazy for soccer that time we hosted the World Cup for some reason. The game was interrupted by one of the rubber straps around one of the kickers (the little triangular table just past the flippers that you would think would be called the bumpers) breaking, and then by the drain breaking in a way that later knocked the game entirely out of the competition. So anyway besides that glory I’ve been very busy trying to figure out what’s gone wrong and stepping outside to berate the fox squirrels out back, and that’s why I’m late with all this. I’m sure you relate.

Danielle Rabbit as a lion tamer whipping a 2. Danielle as orchestra conductor leading a 4 playing violin. As a puppet-master holding up an 8 and 3 as marionettes. Juggling the numerals 0 through 9. Nursing a 7. Then reality: Kevin saying, 'Danielle, thanks for doing our taxes.' Danielle: 'Well, you just have to know how to handle numbers.'
Bill Holbrook’s Kevin and Kell rerun for the 15th of April, 2018. The strip is this enormously tall format because at the time it originally ran (in 2012) the strip appeared in print in the Atlanta Journal-Constitution, sharing the page with Wiley Miller’s similarly-formatted Non Sequitur. The strip has since resumed more normal dimensions.

Bill Holbrook’s Kevin and Kell rerun for the 15th is the anthropomorphic numerals strip for the week. Also the first of the anthropomorphic strips for the week. Calculating taxes has always been one of the compelling social needs for mathematics, arithmetic especially. If we consider the topic to be “accounting” then that might be the biggest use of mathematics in society. At least by humans; I’m not sure how to rate the arithmetic that computers do even for not explicitly mathematical tasks like sending messages back and forth. New comic strip tag for around here, too.

Fauna, to her brother Tucker: 'I learned a valuable lesson in trigonometry class today. The next time I sign up for a class, it will have nothing to do with numbers.'
Bill Schorr’s The Grizzwells for the 17th of April, 2018. Yeah, people say that, but then they get into Abstract Algebra and then they see any proof whatsoever that involves ideals of rings.

Bill Schorr’s The Grizzwells for the 17th sees Fauna not liking trigonometry class. I’m sympathetic. I remember it as seeming to be a lot of strange new definitions put to vague purposes. On the bright side, when you get into calculus trigonometry starts solving more problems than it creates. On the dim side, at least when I took it they tried to pass off “trigonometric substitution” as a thing we might need. (OK, it’s come in useful sometimes, but not as often as the presentation made it look.) Also a new comic strip tag.

A two-circle Venn diagram. In one circle: 'Eric's friends'. In the other: 'Eric's enemies'. In the intersection: 'Eric's cat'.
Eric the Circle for the 18th of April, 2018, this one by sdhardie. It’s a rerun, yes, although I don’t know just from when. The copyright date of 2012 suggests I’ve probably already covered this in a Reading the Comics post before. (If I have I can’t find it.)

Eric the Circle for the 18th, this one by sdhardie, is a joke in the Venn Diagram mode. The strip’s a little unusual for not having one of the circles be named Eric. Not a new comic strip tag.

A trophy room. Behind the adult are the heads of an elephant and a tiger . Behind the child are Maths Teacher Year 1 and Maths Teacher Year 2.
Ham’s Life on Earth for the 19th of April, 2018. I suppose that Ham is a pseudonym but I have no information about the cartoonist other than that I guess she’s not American.

Ham’s Life on Earth for the 19th leaves me feeling faintly threatened. Maybe it’s just me. Also not a new comic strip tag, somehow.

Mostly a list of '6 Daydreams That Will Immediately Improve Your Mood'. Relevant is #3, 'Oh hey professor who failed me in college math I'm doing pretty well thanks MATH SLAP.'
Lord Birthday’s Dumbwitch Castle for the 19th of April, 2018. I … I would swear when this comic first started appearing it was by a less absurd pseudonym. I don’t remember, though.

Lord Birthday’s Dumbwitch Castle for the 19th is a small sketch and mostly a list of jokes. This is the normal format for this strip, which tests the idea of what makes something a comic strip. I grant it’s a marginal inclusion, but I am tickled by the idea of a math slap so here you go. This one’s another new comic strip tag.

Reading the Comics, April 14, 2018: Friday the 13th Edition?


And now I can close out last week’s mathematically-themed comic strips. There was a bunch toward the end of the week. And I’m surprised that none of the several comics to appear on Friday the 13th had anything to do with the calendar. Or at least not enough for me to talk about them.

Julie Larson’s Dinette Set rerun for the 12th is a joke built on the defining feature of (high school) algebra. The use of a number whose value we hope to figure out isn’t it. Those appear from the start of arithmetic, often as an empty square or circle or a spot of ____ that’s to be filled out. We used to give these numbers names like “thing” or “heap” or “it” or the like. Something pronoun-like. The shift to using ‘x’ as the shorthand is a legacy of the 16th century, the time when what we see as modern algebra took shape. People are frightened by it, to suddenly see letters in the midst of a bunch of numbers. But it’s no more than another number. And it communicates “algebra” in a way maybe nothing else does.

Timmy: 'Can you help me on my summer math practice book, Grandpa? It says 2x - 5 = 3. So what is x?' Grandpa: 'Must be a misprint, cause last time I checked, x is NOT a number!' Dad: 'I'd show your teacher that typo so she can complain to the publisher.'
Julie Larson’s Dinette Set rerun for the 12th of April, 2018. Don’t be thrown by the side bits like the show on the TV or the rather oversized Nutty Professor DVD box. They’re just side jokes, not part of the main gag.

Ruben Bolling’s Tom the Dancing Bug rerun for the 12th is one of the God-Man stories. I’m delighted by the Freshman Philosophy-Major Man villain. The strip builds on questions of logic, and about what people mean by “omnipotence”. I don’t know how much philosophy of mathematics the average major takes. I suspect it’s about as much philosophy of mathematics as the average mathematics major is expected to take. (It’s an option, but I don’t remember anyone suggesting I do it, and I do feel the lost opportunity.) But perhaps later on Freshman Philosophy-Major Man would ask questions like what do we mean by “one” and “plus” and “equals” and “three”. And whether anything could, by a potent enough entity, be done about them. For the easiest way to let an omnipotent creature change something like that. WordPress is telling me this is a new tag for me. That can’t be right.

God-Man, the Super-Hero with Omnipotent Powers. This week: Danger int he Dorm! [ God-Man settles in to watch 'Two Weeks Notice' on TBS when ... ' God-Man: 'Sandra Bullock is just adorable!' Voice: 'Help!' God-Man: 'Aw, nuts ... that cry for help came from Mid-Central University! Fear not! I'm he --- YOU?'' FPMM: 'Ah, I knew you'd take the bait!' God-Man: 'You again ... ' FPMM: 'YES! Your arch-enemy --- Freshman Philosophy-Major Man!' God-Man: 'Arch-enemy. Right.' FPMM: 'And I can PROVE that you're NOT OMNIPOTENT! You can't make one plus one equal THREE! Ha! The logic and structure of the universe couldn't exist if 1 + 1 = 3!!' God-Man: 'Of course it can, you ninny.' FPMM, disappearing in a windy vortex: 'Whaaaaa?' God-Man: 'It's just a little different. ... What a pain! Well, if I hurry back, I can catch the end of Three Weeks Notice'. [ God-Man flies out past a streaming vortex of stuff. ]
Ruben Bolling’s Tom the Dancing Bug rerun for the 12th of April, 2018. Yes, basically every God-Man installment is the same strip, but it works for me every time. (It helps that there’s only a couple each year.)

Mike Thompson’s Grand Avenue for the 13th is another resisting-the-story-problem joke, attacking the idea that a person would have ten apples. People like to joke about story problems hypothesizing people with ridiculous numbers of pieces of fruit. But ten doesn’t seem like an excessive number of apples to me; my love and I could eat that many in two weeks without trying hard. The attempted diversion would work better if it were something like forty watermelons or the like.

Teacher: 'If Sally had ten apples and ... ' Gabby: 'Oh, come on! Who goes around with ten apples?' Teacher: 'It's a math problem.' Gabby: 'No, it's a psychological problem. There's a problem with someone who feels the need to carry around so many apples.' Teacher: 'You see to have great insight into other people's problems.' Gabby: 'They don't call me a problem child for nothing!'
Mike Thompson’s Grand Avenue for the 13th of April, 2018. And I realize this is like the complaint I raised about the Grand Avenue earlier this week. But Gabby is assuming that Sally is carrying around ten apples, when the problem hasn’t said anything of the sort. Ten isn’t a ridiculous number of apples to carry to start with, but to simply have them in one’s possession? That’s just not peculiar.

Mark Tatulli’s Heart of the City for the 13th has Heart and Dean complaining about their arithmetic class. I rate it as enough to include here because they go into some detail about things. I find it interesting they’re doing story problems with decimal points; that seems advanced for what I’d always taken their age to be. But I don’t know. I have dim memories of what elementary school was like, and that was in a late New Math-based curriculum.

Heart: 'Ugh, could you believe all those crazy word problems Mr Basner dumped on us today? I got a headache fro all the figuring!' Dean: 'Yeah, multiplication, division, adding, and subtracting. My brain is flip-flopping from all the numbers and decimal points. Well, we've got a Friday night and two whole days to undo the damage.' Heart: 'Oh, magical glowing box full of endless, empty entertainment, take us away!'
Mark Tatulli’s Heart of the City for the 13th of April, 2018. One of the things I do appreciate about Heart of the City is that while Dean is a nerd and mostly likes school, he’s not one-note about it and gets as tired of it as anyone else does. Nerd kids in comic strips have a tendency to take their pro-school agenda a bit far.

Nick Galifianakis’s Nick and Zuzu for the 13th is a Venn diagram joke, the clearest example of one we’ve gotten in a while. I believe WordPress when it tells me this is a new tag for the comic strip.

Nick(?), looking at a Venn diagram: 'Nice. I've neer seen spite and integrity overlap.'
Nick Galifianakis’s Nick and Zuzu for the 13th of April, 2018. First, this is some of the nicest grey-washing I’ve seen in these Reading the Comics posts in a while. Second, my experience is that spite with integrity is some of the most fun and delightful that spite ever gets to be. The integrity lets you add a layer of smugness to the spite. And if anyone protests, you get to feel smugly superior to them, too.

Mark Anderson’s Andertoons for the 14th is the Mark Anderson’s Andertoons for the week. It starts at least with teaching ordinal numbers. In normal English that’s the adjective form of a number. Ordinal numbers reappear in the junior or senior year of a mathematics major’s work, as they learn enough set theory to be confused by infinities. In this guise they describe the sizes of sets of things. And they’re introduced as companions to cardinal numbers, which also describe the sizes of sets of things. They’re different, in ways that I feel like I always forget in-between reading books about infinitely large sets. The kids don’t need to worry about this yet.

On the blackboard: 'Ordinal numbers: 1st, 2nd, 3rd'. Kid: 'You forgot Participant.'
Mark Anderson’s Andertoons for the 14th of April, 2018. The kid appears often enough I feel like I should know his name. Or assign one in case the strip doesn’t have a canonical name for him. I’ll take nominations if anyone wants to offer them.

Reading the Comics, April 11, 2018: Obscure Mathematical Terms Edition


I’d like to open today’s installment with a trifle from Thomas K Dye. He’s a friend, and the cartoonist behind the long-running web comic Newshounds, its new spinoff Infinity Refugees, and some other projects.

Dye also has a Patreon, most recently featuring a subscribers-only web comic. And he’s good enough to do the occasional bit of spot art to spruce up my work here.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 9th of April, 2018 is, for me, relatable. I think I’ve read off this anecdote before. The first time I took Real Analysis I was completely lost. Getting me slightly less lost was borrowing a library book on Real Analysis from the mathematics library. The book was in French, a language I can only dimly read. But the different presentation and, probably, the time I had to spend parsing each sentence helped me get a basic understanding of the topic. So maybe trying algebra upside-down isn’t a ridiculous idea.

Archie: 'I can't make any sense out of this algebra!' Jughead: 'Er, Arch! Your book is upside-down!' Archie: 'Yeah, I know! I already tried it the other way, and it didn't make sense then either!'
Henry Scarpelli and Craig Boldman’s Archie rerun for the 9th of April, 2018. Finally, an artistic explanation for putting the name of the book being read on house left!

Lincoln Pierce’s Big Nate rerun for the 9th presents an arithmetic sequence, which is always exciting to work with, if you’re into sequences. I had thought Nate was talking about mathematics quizzes but I see that’s not specified. Could be anything. … And yes, there is something cool in finding a pattern. Much of mathematics is driven by noticing, or looking for, patterns in things and then describing the rules by which new patterns can be made. There’s many easy side questions to be built from this. When would quizzes reach a particular value? When would the total number of points gathered reach some threshold? When would the average quiz score reach some number? What kinds of patterns would match the 70-68-66-64 progression but then do something besides reach 62 next? Or 60 after that? There’s some fun to be had. I promise.

Nate: 'Four quizzes ago, I got a 70. Three quizzes ago, I got a 68. Two quizzes ago, I got a 66, and last quiz I got a 64! See the pattern?' Francis: 'The pattern of academic incompetence?' Nate: 'No, the way it keeps decreasing by twos! Isn't that COOL?'
Lincoln Pierce’s Big Nate rerun for the 9th of April, 2018. Trick question: there’s infinitely many sequences that would start 70, 68, 66, 64. But when we extrapolate this sort of thing we tend to assume that it’ll be some simple sequence. These are often arithmetic — each term increasing or decreasing by the same amount — or geometric — each term the same multiple of the one before. They don’t have to be. These are just easy ones to look for and often turn out well, or at least useful.

Mike Thompson’s Grand Avenue for the 10th is one of the resisting-the-teacher’s-problem style. The problem’s arithmetic, surely for reasons of space. The joke doesn’t depend on the problem at all.

Teacher: 'Gabby, can you solve the problem?' [ '33 x 22' on the blackboard. ] Gabby: 'No, thank you. You're the adult, so I'll let you solve the problem. Why do you need a kid? Adults are able to solve problems on their own.' [ Gabby sits outside the Principal's office, thinking ] 'Looks like he solved his problem after all.'
Mike Thompson’s Grand Avenue for the 10th of April, 2018. My grudge against Grand Avenue is well-established and I fear it will make people think I am being needlessly picky at this. But Gabby’s protest would start from a logical stance if the teacher asked “Would you solve the problem?” Then she’d have reason to argue that adults should be able to solve the problem. “Can” you doesn’t reflect on who ought to solve arithmetic problems.

Dave Whamond’s Reality Check for the 10th similarly doesn’t depend on what the question is. It happens to be arithmetic, but it could as easily be identifying George Washington or picking out the noun in a sentence.

Dog reading an exam: 'Do you know the square root of 81? Do you? Do you? Yes, you do!'
Dave Whamond’s Reality Check for the 10th of April, 2018. I keep wanting to think the exam is playing on the pun between K-9 and canine but it’s not quite there.

Leigh Rubin’s Rubes for the 10th riffs on randomness. In this case it’s riffing on the unpredictability and arbitrariness of random things. Random variables are very interesting in certain fields of mathematics. What makes them interesting is that any specific value — the next number you generate — is unpredictable. But aggregate information about the values is predictable, often with great precision. For example, consider normal distributions. (A lot of stuff turns out to be normal.) In that case we can be confident that the values that come up most often are going to be close to the arithmetic mean of a bunch of values. And that there’ll be about as many values greater than the mean as there are less than the mean. And this will be only loosely true if you’ve looked at a handful of values, at ten or twenty or even two hundred of them. But if you looked at, oh, a hundred thousand values, these truths would be dead-on. It’s wonderful and it seems to defy intuition. It just works.

Door to the Randomness Research Institute. Sign hanging on the doorknob: 'Be Back In: (Your Guess Is As Good As Ours.)'
Leigh Rubin’s Rubes for the 10th of April, 2018. My guess, in the absence of other information, would be “back in about as long as the last time we were out”. In surprisingly many cases your best plausible guess about what the next result should be is whatever the last result was.

John Atkinson’s Wrong Hands for the 10th is the anthropomorphic numerals joke for the week. It’s easy to think of division as just making numbers smaller: 4 divided by 6 is less than either 4 or 6. 1 divided by 4 is less than either 1 or 4. But this is a bad intuition, drawn from looking at the counting numbers that don’t look boring. But 4 divided by 1 isn’t less than either 1 or 4. Same with 6 divided by 1. And then when we look past counting numbers we realize that’s not always so. 6 divided by ½ gives 12, greater than either of those numbers, and I don’t envy the teachers trying to explain this to an understandably confused student. And whether 6 divided by -1 gives you something smaller than 6 or smaller than -1 is probably good for an argument in an arithmetic class.

'The Great Divide'. Numeral 6, looking at an obelus, and speaking to a 4 and a 1; 'It's the guy from division. Looks like we're downsizing'.
John Atkinson’s Wrong Hands for the 10th of April, 2018. Oh yeah, remember a couple months ago when the Internet went wild about how ÷ was a clever way of representing fractions, with the dots representing the numerator and denominator? … Yeah, that wasn’t true, but it’s a great mnemonic.

Zach Weinersmith, Chris Jones and James Ashby’s Snowflakes for the 11th has an argument about predicting humans mathematically. It’s so very tempting to think people can be. Some aspects of people can. In the founding lore of statistics is the astonishment at how one could predict how many people would die, and from what causes, over a time. No person’s death could be forecast, but their aggregations could be. This unsettles people. It should: it seems to defy reason. It seems to me even people who embrace a deterministic universe suppose that while, yes, a sufficiently knowledgeable creature might forecast their actions accurately, mere humans shouldn’t be sufficiently knowledgeable.

Priti: 'Did you know that all human culture can be represented with GRAPHS?!' Sloan: 'Doubtful. Here. Read Machiavelli, Durkheim, and Montesquieu.' Priti: 'I see a lot of French and a lack of graphs.' Sloan: 'Not everything can be represented graphical [sic]. Plus it's full of CITATIONS! Wonderful, wonderful citations!' Priti: 'So, you don't think your behavior can be predicted mathematically?' Sloan: 'Correct.' Priti: 'Predictable'.
Zach Weinersmith, Chris Jones and James Ashby’s Snowflakes for the 11th of April, 2018. So when James Webb, later of NASA fame, was named Under-Secretary of State in 1949 one of his projects was to bring more statistical measure to foreign affairs. He had done much to quantify economic measures, as head of the Bureau of the Budget. But he wasn’t able to overcome institutional skepticism (joking about obvious nonsense like “Bulgaria is down a point!”), and spent his political capital instead on a rather necessary reorganization of the department. That said, I would not trust the wildly enthusiastic promises of any pop mathematics book proclaiming human cultures can be represented by any simple numerical structure.

No strips are tagged for the first time this essay. Just noticing.

Someone Else’s Homework: Some More Thoughts


I wanted to get back to my friend’s homework problem. And a question my friend had about the question. It’s a question I figure is good for another essay.

But I also had second thoughts about the answer I gave. Not that it’s wrong, but that it could be better. Also that I’m not doing as well in spelling “range” as I had always assumed I would. This is what happens when I don’t run an essay through Hemmingway App to check whether my sentences are too convoluted. I also catch smaller word glitches.

Let me re-state the problem: Suppose you have a function f, with domain of the integers Z and rage of the integers Z. And also you know that f has the property that for any two integers ‘a’ and ‘b’, f(a + b) equals f(a) + f(b). And finally, suppose that for some odd number ‘c’, you know that f(c) is even. The challenge: prove that f is even for all the integers.

Like I say, the answer I gave on Tuesday is right. That’s fine. I just thought of a better answer. This often happens. There are very few interesting mathematical truths that only have a single proof. The ones that have only a single proof are on the cutting edge, new mathematics in a context we don’t understand well enough yet. (Yes, I am overlooking the obvious exception of ______ .) But a question so well-chewed-over that it’s fit for undergraduate homework? There’s probably dozens of ways to attack that problem.

And yes, you might only see one proof of something. Sometimes there’s an approach that works so well it’s silly to consider alternatives. Or the problem isn’t big enough to need several different proofs. There’s something to regret in that. Re-thinking an argument can make it better. As instructors we might recommend rewriting an assignment before turning it in. But I’m not sure that encourages re-thinking the assignment. It’s too easy to just copy-edit and catch obvious mistakes. Which is valuable, yes. But it’s good for communication, not for the mathematics itself.

So here’s my revised argument. It’s much cleaner, as I realized it while showering Wednesday morning.

Give me an integer. Let’s call it m. Well, m has to be either an even or an odd number. I’m supposing nothing about whether it’s positive or negative, by the way. This means what I show will work whether m is greater than, less than, or equal to zero.

Suppose that m is an even number. Then m has to equal 2*k for some integer k. (And yeah, k might be positive, might be negative, might be zero. Don’t know. Don’t care.) That is, m has to equal k + k. So f(m) = f(k) + f(k). That’s one of the two things we know about the function f. And f(k) + f(k) is is 2 * f(k). And f(k) is an integer: the integers are the function’s rage range). So 2 * f(k) is an even integer. So if m is an even number then f(m) has to be even.

All right. Suppose that m isn’t an even integer. Then it’s got to be an odd integer. So this means m has to be equal to c plus some even number, which I’m going ahead and calling 2*k. Remember c? We were given information about f for that element c in the domain. And again, k might be positive. Might be negative. Might be zero. Don’t know, and don’t need to know. So since m = c + 2*k, we know that f(m) = f(c) + f(2*k). And the other thing we know about f is that f(c) is even. f(2*k) is also even. f(c), which is even, plus f(2*k), which is even, has to be even. So if m is an odd number, then f(m) has to be even.

And so, as long as m is an integer, f(m) is even.

You see why I like that argument better. It’s shorter. It breaks things up into fewer cases. None of those cases have to worry about whether m is positive or negative or zero. Each of the cases is short, and moves straight to its goal. This is the proof I’d be happy submitting. Today, anyway. No telling what tomorrow will make me think.

Someone Else’s Homework: A Solution


I have a friend who’s been taking mathematical logic. While talking over the past week’s work they mentioned a problem that had stumped them. But they’d figured it out — at least the critical part — about a half-hour after turning it in. And I had fun going over it. Since the assignment’s already turned in and I don’t even know which class it was, I’d like to share it with you.

So here’s the problem. Suppose you have a function f, with domain of the integers Z and rage of the integers Z. And also you know that f has the property that for any two integers ‘a’ and ‘b’, f(a + b) equals f(a) + f(b). And finally, suppose that for some odd number ‘c’, you know that f(c) is even. The challenge: prove that f is even for all the integers.

If you want to take a moment to think about that, please do.

A Californian rabbit (white body, grey ears and nose and paws) eating a pile of vegetables. In the background is the sunlit outside in the window, with a small rabbit statue silhouetted behind the rabbit's back.
So you can ponder without spoilers here’s a picture of the rabbit we’re fostering for the month, who’s having lunch. The silhouette behind her back is of a little statue decoration and not some outsider trying to lure our foster rabbit to freedom outside, so far as we know. (Don’t set domesticated rabbits outside. It won’t go well for them. And domesticated rabbits aren’t native to North America, I mention for the majority of my readers who are.)

So here’s my thinking about this.

First thing I want to do is show that f(1) is an even number. How? Well, if ‘c’ is an odd number, then ‘c’ has to equal ‘2*k + 1’ for some integer ‘k’. So f(c) = f(2*k + 1). And therefore f(c) = f(2*k) + f(1). And, since 2*k is equal to k + k, then f(2*k) has to equal f(k) + f(k). Therefore f(c) = 2*f(k) + f(1). Whatever f(k) is, 2*f(k) has to be an even number. And we’re given f(c) is even. Therefore f(1) has to be even.

Now I can prove that if ‘k’ is any positive integer, then f(k) has to be even. Why? Because ‘k’ is equal to 1 + 1 + 1 + … + 1. And so f(k) has to equal f(1) + f(1) + f(1) + … + f(1). That is, it’s k * f(1). And if f(1) is even then so is k * f(1). So that covers the positive integers.

How about zero? Can I show that f(0) is even? Oh, sure, easy. Start with ‘c’. ‘c’ equals ‘c + 0’. So f(c) = f(c) + f(0). The only way that’s going to be true is if f(0) is equal to zero, which is an even number.

By the way, here’s an alternate way of arguing this: 0 = 0 + 0. So f(0) = f(0) + f(0). And therefore f(0) = 2 * f(0) and that’s an even number. Incidentally also zero. Submit the proof you like.

What’s not covered yet? Negative integers. It’s hard not to figure, well, we know f(1) is even, we know f(a + b) if f(a) + f(b). Shouldn’t, like, f(-2) just be -2 * f(1)? Oh, it so should. I don’t feel like we have that already proven, though. So let me nail that down. I’m going to use what we know about f(k) for positive ‘k’, and the fact that f(0) is 0.

So give me any negative integer; I’m going call it ‘-k’. Its additive inverse is ‘k’, which is a positive number. -k + k = 0. And so f(-k + k) = f(-k) + f(k) = f(0). So, f(-k) + f(k) = 0, and f(-k) = -f(k). If f(k) is even — and it is — then f(-k) is also even.

So there we go: whether ‘k’ is a positive, zero, or negative integer, f(k) is even. All the integers are either positive, zero, or negative. So f is even for any integer.

I’ve got some more thoughts about this problem.

Reading the Comics, April 2018: Another Normal Week Edition


And for another week running the pace of mathematically-themed comic strips has been near normal. There’s nowhere near enough to split the essay into two pieces, which is fine. There is some more work involved in including images for all the strips I discuss and this pace better fits the time I could make for writing this week. Will admit I’m scared of what’s going to happen when I have a busy week and Comic Strip Master Command orders more comics for me. I admit this isn’t an inspired name for the Edition. But the edition names are mostly there so people have a chance of telling whether they’ve read an installment before. The date alone doesn’t do it. A couple of words will. Maybe I should give up on meaningful names if there isn’t an obvious theme for the week. It’s got to be at least as good to name something “Coronet Blue Edition” as to name it “Lots Of Andertoons Edition”.

Frank Cho’s Liberty Meadows rerun for the 1st riffs on quantum computers. You’ve maybe seen much talk about them in pop science columns and blogs. They require a bunch of stuff that gets talked about as if it were magical. Quantum mechanics, obviously, the biggest bit of magic in popular science today. Complex-valued numbers, which make for much more convenient mathematical descriptions. Probability, which everyone thinks they understand and which it turns out nobody does. Vector spaces and linear algebra, which mathematics (and physics) majors get to know well. The mathematics of how a quantum computer computes is well-described as this sort of matrix and vector work. Quantum computing promises to be a really good way to do problems where the best available approach is grinding it out: testing every possibility and finding the best ones. No part of making a quantum computer is easy, though, so it’s hard to say when we’ll have the computing power to make a version of SimCity with naturally curving roads. (This is a new tag for my Reading the Comics essays, but I’ve surely featured the strip some before.)

Frank: 'What are you doing in my room?' Ralph, in spacesuit gear and in front of a swirling vortex of light: 'Your room as the best electrical outlet to power my quantum computer.' Frank: 'Quantum computer?' Ralph: 'You wouldn't understand.' Frank: 'Try me, monkey boy.' Ralph: 'All computers and electronic systems are based on the binary principle. They operate using two states, on and off. The quantum computer utilizes the fundamental nature of subatomic reality. Instead of operating in two states it operates on a multitude of states between on and off. It doesn't calculate serially like a binary computer. It performs operations simultaneously across each state, across each different reality, if you will. Each quantum state is another universe, another time. Since there are multiple quantum states, there are, theoretically, multiple universes coexisting side by side. This quantum computer makes teleportation and time travel possible.' [Awkward pause.] Frank: 'OK, uh, just don't mess with my Star Wars collection.' Ralph: 'I knew you wouldn't understand.' [Alley Oop pops in.]
Frank Cho’s Liberty Meadows rerun for the 1st of April, 2018. First, good cameo. Second, this rerun’s being from around 2000 means quantum computers have been fit subjects for newspaper jokes about two decades now, and I didn’t realize that. And yeah, in the penultimate panel Cho says ‘with apologies and respect to V T Hamlin. (Hamlin created Alley Oop, and you can read my thoughts about the current strip on this link.) Cartoonists always write ‘apologies to’ when they use another artist’s characters and I don’t know how the convention started. Certainly not for cameos like this where it’s not like Oop does something that could damage his character.

Niklas Eriksson’s Carpe Diem for the 2nd is a mathematics-education-these-days joke. The extremely small child talking about counting-without-a-calculator as a subject worth studying. People are always complaining that people don’t do arithmetic well enough in their heads. I understand the frustration, considering last week I stymied a cashier at a Penn Station by giving $22.11 for my $11.61 order. I don’t know why he put in my payment as $20; why not let the machine designed to do this work, do the work? He did fine working out that I should get $10 in bills back but muddled up the change. As annoyances go it ranks up there with the fast food cashier asking my name for the order and entering it as “Joeseph”.

Kid: 'Yup, 'counting without a calculator' is a subject in its own right these days.'
Niklas Eriksson’s Carpe Diem for the 2nd of April, 2018. I’m kind of distracted trying to work out the perspective between the kid and the adult. Either the kid’s standing pretty far away or is really tiny and is standing on a chair.

Lard’s World Peace Tips for the 4th mentions the Möbius Strip. It’s got to be the most famous exotic piece of geometry to have penetrated the popular culture. It’s also a good shape to introduce geometry students to a “non-orientable” surface. Non-orientable means about what you’d imagine. There’s not a way to put coordinates on it that don’t get weird. For example, try drawing an equator on the surface of the strip. Any curve along the surface that doesn’t run off the edges will do. The curve just has to meet itself. It looks like this divides the strip into two pieces. Fine, then; which of these two pieces is “north” and which is “south” of this equator? There’s not a way to do that. You get surprising results if you try.

Waiter: 'Here's one for you.' Lard: 'Yes?' Waiter: 'Why did the chicken cross the Mobius strip?' Lard: 'To get to the same side? At least that's what the chicken told me ... ' [ LATER ] The waiter is chasing the chicken along a Mobius strip: 'Come back here! You ruined my punchline!'
Lard’s World Peace Tips for the 4th of April, 2018. Until transcribing the strip for the alt-text here I didn’t realize it was a chicken, and not Lard, being chased in that final panel.

Karen Montague-Reyes’s Clear Blue Water rerun for the 5th has Eve deploying a mathematical formula. She’s trying to describe the way that perception of time changes over the course of events. It’s not a bad goal. Many things turn out to be mathematically describable. I don’t see what the equation is supposed to even mean, but then, I haven’t seen the model she developed that implies this equation. (This is not a new tag and I’m surprised by that.)

Eve: 'Remember when I told you I'd figured out how to slow down time?' Manny: '... by getting pregnant?' Eve: 'Exactly! Well, here it is. Eve's theory of pregativity! Ta-da!' Manny: 'Oh dear ... T = pt + 1y^2 - 0 ... Huh?' Eve: 'It explains time! How it slows down in pregnancy, then zooms to hyperspeed during baby's first year, resulting in a net gain of zero! I want a patent!' Manny: 'This makes NO sense whatsoever.' Eve: 'Well, not to a layman, no.'
Karen Montague-Reyes’s Clear Blue Water rerun for the 5th of April, 2018. I’m about 60% sure Eve is just describing Soap Opera Rapid Aging Syndrome here, which carries over to the comics. (Remember over in Rex Morgan, M.D. that June Morgan carried her latest child for like two years.)

Dan Thompson’s Brevity for the 6th is some mathematics wordplay, built on the abacus. I’m not sure there’s more to say about this, past that you can do much more on an abacus. You can, at least. I keep reading directions about how to multiply with it and then I look at mine and I feel helpless.

Chinese real-estate agent: 'In this room, you'll notice the lovely stone abacuses.' Potential homebuyer: 'We just love granite counters!'
Dan Thompson’s Brevity for the 6th of April, 2018. My father’s trained me to be skeptical of granite counters, although I don’t remember why. In any case in our kitchen we’re keeping the counter as is, to respect the history of a house that’s nine decades old this year and that we hope to be in when it reaches its centennial. And because we like ourselves too much to inflict countertop-replacement work on us.

Bil Keane and Jeff Keane’s Family Circus for the 7th is a kids-mispronouncing-a-mathematics-word strip. I have even less to say about this. It’s a normal week.

Dolly to her mother: 'I'm having trouble with eagles in school --- One plus one eagles two, two plus two eagles four'.
Bil Keane and Jeff Keane’s Family Circus for the 7th of April, 2018. This is probably a rerun; most Family Circus strips are these days. No idea when from exactly; most of the identifiable reruns have been from the 70s. Also, so far as this goes, she isn’t demonstrating problems with eaglity.

How March 2018 Treated My Mathematics Blog


Well, one thing I know to post this week is my review of what my readership was like in March. Let me go see what WordPress will tell me about that.

Huh.

Not at all sure what happened there but it looks like I might’ve just had my best month ever. WordPress tells me there were 1,779 page views in March, way up from February’s 1,062 and January’s 1,274. Also it tells me this came from what I’m sure is a record 999 unique visitors and now that’s going to drive me crazy for like ever. There were 611 unique visitors in February and 670 in January. I am not positive but think my previous records were in March 2016 (1,557 views) and April 2016 (757 visitors). That’s on 16 essays posted, up from the 13 in February and 14 in January.

A bar chart showing the 1,779 page views and 999 visitors from March 2018, and lower numbers for other months going back to November 2015.
Is this self-indulgent? No; I’ve learned that people are much more interested in posts when there’s any picture, however unimportant, attached. This is self-serving, an important difference.

Had 53 comments made around here in March, my best since the glory days of early 2016. February saw 30 and January 39 comments and oh I did my best to keep caught up, but it’s hard. There were 143 things liked over the month; that’s up from February’s 102 and January’s 112. Greatest number since August 2017 and my last round of A To Z work.

I don’t know precisely what drew so many readers in, as in, why many people were looking for this. But I know what they were looking for. The most popular, by far, essay this month drew 279 page views. I have to guess some forum found the answer to years of argument and posted a link to settle the issue. The top five:

Insights for the year tell me that (as of the 3rd of April, anyway) I’ve had 44 total posts, with 120 total comments and 301 total likes. There’s 36,347 words posted so far in the year, and an average of 826 words per post. I’m averaging 2.7 comments per post, and averaging 6.8 likes per post. This is dangerous stuff to consider: at the start of March I averaged 2.8 comments per post, but a mere 6.7 likes. In fairness, there’s some comments I need to respond to and just haven’t had the chance; Easter and a pinball event ate up a lot of time.

So what countries are sending me readers, suspecting or otherwise? This bunch:

Country Readers
United States 1,278
Canada 72
United Kingdom 52
India 42
Philippines 37
Singapore 28
Austria 24
Switzerland 21
Brazil 20
Hong Kong SAR China 20
Sweden 20
South Africa 18
Australia 16
Denmark 14
Romania 11
Italy 7
Norway 7
Germany 5
South Korea 5
Algeria 4
Belgium 4
Ireland 4
Spain 4
Thailand 4
Argentina 3
Czech Republic 3
Malaysia 3
New Zealand 3
Poland 3
Puerto Rico 3
Saudi Arabia 3
Egypt 2
Estonia 2
European Union 2
Finland 2
Kenya 2
Kuwait 2
Netherlands 2
Pakistan 2
Portugal 2
Qatar 2
Russia 2
Turkey 2
United Arab Emirates 2
Belize 1
Croatia 1
Ecuador 1
France 1
Greece 1
Israel 1 (*)
Japan 1
Kyrgyzstan 1
Laos 1
Latvia 1
Lebanon 1
Mexico 1
Serbia 1
Ukraine 1
Venezuela 1

That’s 58 countries, up from February’s 54. There’s 15 single-reader countries, down one from February. Israel’s keeps me from having a clean break in the single-reader country streak; there was just the one reader from there in February too. April starts with a logged 60,445 visits, from an admitted 28,781 unique visitors.

If you’d like to follow NebusResearch regularly, please do. There’s a button at the upper-right of the page to add this to your WordPress Reader page. You can also follow me as @Nebusj on Twitter, where I routinely post announcements of new essays here and on my humor blog. (The humor blog normally posts between 7 and 9 pm Eastern Time; the mathematics blog, typically, between 1 and 3 pm Eastern Time.) If you’d rather use your RSS reader here’s the feed for that.

If you’d like posts e-mailed to you as they’re made … I’m sorry, I can’t take signups for that just now. I noticed a weird and large number of signups from people, from addresses that were a bunch of random words followed by four digits and all from outlook.com. I don’t know what angle they’re working but that’s got to be some spammer nonsense going on. So that’s turned off for a while at least. If you’re one of the nearly four people who’ve taken out e-mail subscriptions hold on to those accounts! They’re sure to be worth something someday. It’s not necessary to bag them in mylar just yet, but feel free to do that if you think it’ll be fun.