What’s the most probable date for Easter? What’s the least?


This is a nice chance to highlight one of my older pieces. I had been wondering about the most and the least likely dates for Easter. And I calculated several hundred years’ worth of Easters, to find when they’re most and least likely to happen.

The 22nd of March is the least probable date for Easter. That date was last Easter in 1818, and will next be Easter in 2285. The 12th of April, though? That’s one of the most likely dates for Easter. To say what is “the” most probable date for Easter requires some thought. First, what it means to talk about the chance of an algorithmically defined quantity. Second, what it means to look at Easter. The holiday is intended to happen early in the European spring. But the start of European spring is moving through the calendar. Someday we will abandon the Gregorian calendar, or radically change the calculation of Easter. This makes it harder to say how often each possible date turns up. But we can make some rough answers.

The 15th of April is the most probable date for Easter, if we look at a 532-year span. (There are astronomical reasons to look at 532 years.) If we look at a more limited stretch, 1925 to 2100, on the assumption that that’s the maximum spread of dates that anyone alive today can be expected to see, then we have ten dates equally common, the 12th of April among them.

Reading the Comics, April 4, 2020: Ruling Things Out Edition


This little essay should let me wrap up the rest of the comic strips from the past week. Most of them were casual mentions. At least I thought they were when I gathered them. But let’s see what happens when I actually write my paragraphs about them.

Darrin Bell and Theron Heir’s Rudy Park rerun for the 1st of April uses arithmetic as emblematic of things which we know with certainty to be true.

Thaves’s Frank and Ernest for the 2nd is a bit of wordplay, having Euclid and Galileo talking about parallel universes. I’m not sure that Galileo is the best fit for this, but I’m also not sure there’s another person connected who could be named. It’d have to be a name familiar to an average reader as having something to do with geometry. Pythagoras would seem obvious, but the joke is stronger if it’s two people who definitely did not live at the same time. Did Euclid and Pythagoras live at the same time? I am a mathematics Ph.D. and have been doing pop mathematics blogging for nearly a decade now, and I have not once considered the question until right now. Let me look it up.

It doesn’t make any difference. The comic strip has to read quickly. It might be better grounded to post Euclid meeting Gauss or Lobachevsky or Euler (although the similarity in names would be confusing) but being understood is better than being precise.

Stephan Pastis’s Pearls Before Swine for the 2nd is a strip about the foolhardiness of playing the lottery. And it is foolish to think that even a $100 purchase of lottery tickets will get one a win. But it is possible to buy enough lottery tickets as to assure a win, even if it is maybe shared with someone else. It’s neat that an action can be foolish if done in a small quantity, but sensible if done in enough bulk.

Chalkboard problem 10 - 7, with answers given and crossed out of 0, 5, 7, 4, 17, 9, 1, 2, and 70. Wavehead, to teacher: 'OK, the good news is we've ruled these out.'
Mark Anderson’s Andertoons for the 3rd of April, 2020. This is actually the first time I’ve mentioned this strip in two months. But any time I discuss a topic raised by Andertoons should appear at this link.

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. Wavehead has made a bunch of failed attempts at subtracting seven from ten, but claims it’s at least progress that some thing have been ruled out. I’ll go along with him that there is some good in ruling out wrong answers. The tricky part is in how you rule them out. For example, obvious to my eye is that the correct answer can’t be more than ten; the problem is 10 minus a positive number. And it can’t be less than zero; it’s ten minus a number less than ten. It’s got to be a whole number. If I’m feeling confident about five and five making ten, then I’d rule out any answer that isn’t between 1 and 4 right away. I’ve got the answer down to four guesses and all I’ve really needed to know is that 7 is greater than five but less than ten. That it’s an even number minus an odd means the result has to be odd; so, it’s either one or three. Knowing that the next whole number higher than 7 is an 8 says that we can rule out 1 as the answer. So there’s the answer, done wholly by thinking of what we can rule out. Of course, knowing what to rule out takes some experience.

Mark Parisi’s Off The Mark for the 4th is roughly the anthropomorphic numerals joke for the week. It’s a dumb one, but, that’s what sketchbooks are for.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is the Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th for the week. It shows in joking but not wrong fashion a mathematical physicist’s encounters with orbital mechanics. Orbital mechanics are a great first physics problem. It’s obvious what they’re about, and why they might be interesting. And the mathematics of it is challenging in ways that masses on springs or balls shot from cannons aren’t.

How To Learn Orbital Mechanics. Step 1: Gauge Difficulty. Person reading a text: 'It's Newtonian! Piece of cake. Just a bunch of circles and dots.' Step 2: Correction. 'OK, *ellipses* and dots.' Step 3: Concern. 'Oh, Christ, sometimes there are more than two dots.' Step 4: Pick an easier subject. 'I'm gonna go study quantum computing.' The textbook is in the trash.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th of April, 2020. This is actually the first time I’ve mentioned this strip ina week. But any time I discuss a topic raised in Saturday Morning Breakfast Cereal should appear at this link.

A few problems are very easy, like, one thing in circular orbit of another. A few problems are not bad, like, one thing in an elliptical or hyperbolic orbit of another. All our good luck runs out once we suppose the universe has three things in it. You’re left with problems that are doable if you suppose that one of the things moving is so tiny that it barely exists. This is near enough true for, for example, a satellite orbiting a planet. Or by supposing that we have a series of two-thing problems. Which is again near enough true for, for example, a satellite travelling from one planet to another. But these is all work that finds approximate solutions, often after considerable hard work. It feels like much more labor to smaller reward than we get for masses on springs or balls shot from cannons. Walking off to a presumably easier field is understandable. Unfortunately, none of the other fields is actually easier.

Pythagoras died somewhere around 495 BC. Euclid was born sometime around 325 BC. That’s 170 years apart. So Pythagoras was as far in Euclid’s past as, oh, Maria Gaetana Agnesi is to mine.

I did a little series looking into orbital mechanics, not necessarily ones that look like planetary orbits, a couple years ago. You might enjoy that. And I figure to have more mathematically-themed comic strips in the near future. Thanks for reading.

Reading the Comics, March 31, 2020: End March, Already, Edition


I think few will oppose me if I say the best part of March 2020 was that it ended. Let me close out nearly all my March business by getting through the last couple comic strips which mentioned some mathematics topic that month. I’ll still have my readership review, probably to post Friday, and then that finishes my participation in the month at last.

Connie Sun’s Connie to the for the 30th features the title character trying to explain what “exponential growth” is. She struggles. Appropriately, as it’s something we see very rarely in ordinary life.

They turn up in mathematics all the time. And mathematical physics, and such. Any process with a rate of change that’s proportional to the current amount of the thing tends to be exponential. This whether growing or decaying. Even circular motion, periodic motion, can be understood as exponential growth with imaginary numbers. So anyone doing mathematics gets trained to see, and expect, exponentials. They have great analytic properties, too. You can use them to solve differential equations. And differential equations are so much of science that it’s easy to forget they’re not.

In ordinary life, though? Well, yes, a lot of quantities will change at rates which depend on their current quantity. But in anything that’s been around a while, the quantity will usually be at, or near enough, an equilibrium. Some kind of balance. It may move away from that balance, but usually, it’ll move back towards it. (I am skipping some complicating factors. Don’t worry about them.) A mathematician will see the hidden exponentials in this. But to anyone else? The thing may start growing, but then it peters out and slows to a stop. Or it might collapse, but that change also peters out. Maybe it’ll hit a new equilibrium; maybe it’ll go back to the old. We rarely see something changing without the sorts of limits that tamp the change back down.

Connie, narrating: 'I recently tried to explain exponential growth to my parents, using an awkward mix of English and Chinese. The problem is that I'm rusty on the math, on top of the language barrier.' Her phone ;'You know how when a line on a graph curves up really sharply?? It's, like, a math thing . Cases are doubling every day or two! Okay, wait, let me look it up. [ Looking over a picture of the exponential growth curve. ] Uh, it's ... [ something ] in Chinese. Does that make sense? ... Yeah, so, I think what it means is that you should definitely STAY HOME.'
Connie Sun’s Connie to the for the 30th of March, 2020. Although I’ve mentioned this strip one time before, it’s not had any serious attention before. Well, this and future essays discussing something mentioned in Connie to the Wonnie should appear at this link.

Even the growth of infection rates for Covid-19 will not stay exponential forever, even if there were no public health measures responding to it. There can’t be more people infected than there are people in the world. At some point, the curve representing number of infected people versus time would stop growing more and more, and would level out, from a pattern called the logistic equation. But the early stages of this are almost indistinguishable from exponential growth.

Samson’s Dark Side of the Horse for the 29th is a comforting counting-sheep joke, with half-sized sheep counted as fractions of a whole sheep. Comforting little bit of business here.

Sam Hurts’s Eyebeam for the 30th describes one version of Zeno’s most famous paradox, and applies it to an event that already seems endless.

Zeno's Paradox: To get from point A to point B, you must first reach the halfway point. From there, you will have to cross a new halfway point. Etc. Etc. Etc. Etc. Etc. Etc. ... You will never run out of halfway points, so you can never arrive. Zeno's Kids: [ Zeno driving, with two kids in the back. ] Kids: 'Are we halfway there yet?'
Sam Hurts’s Eyebeam for the 30th of March, 2020. This is the first time in over two years that I’ve mentioned this strip. Essays featuring Eyebeam are gathered at this link.

Todd Clark’s Lola for the 30th has a student asking what the end of mathematics is. And learning how after algebra comes geometry, trigonometry, calculus, topology, and more. All fair enough, though I’m surprised to see it put for that that of course someone who does enough mathematics will do topology. (I only have a casual brush with it myself, mostly in service to other topics.) But it’s nice to have it acknowledged that, if you want, you can go on learning new mathematics fields, practically without limit.

Ashleigh Brilliant’s Pot-Shots for the 30th just declares infinity to be a favorite number. Is it a number? … We have to be careful what exactly we mean by number. Allow that we are careful, though. It’s certainly at least number-adjacent.

John Zakour and Scott Roberts’s Maria’s Day for the 31st has Maria hoping to get out of new schoolwork. So she gets a review of fractions instead. Typical.


There were some more mathematically-themed comic strips last week. I’ll get to them in an essay at this link, sometime soon. Thanks for reading.

Rjlipton’s thoughts on the possible ABC Conjecture proof


There is this thing called the abc Conjecture. It’s a big question in number theory, which is the part of mathematics where we learn we don’t understand anything about prime numbers. Nearly a decade ago Shinichi Mochizuki announced a proof. It’s been controversial. Most importantly, it’s not been well-understood.

It’s finally getting published in a proper journal. A lot of mathematics work is passed around as PDFs, usually on arXiv.org, these days. It’s good for sharing fresh thoughts. But journal publication usually means that the paper has been reviewed, critically, and approved by people who could tell whether the reasoning is sound. Mochizuki’s paper is somewhere around 500 to 600 pages (I’ve seen different figures), and by every report hard to understand even for number theory proofs. A proof is, more than mathematicians like to admit, really an argument that convinces other mathematicians that, if we wanted to spend the time, we could find a completely rigorous proof. With very long proofs, and very complicated proofs, the standard of being convincing gets tougher.

In this essay, Not As Easy As ABC, rjlipton discusses some of the conjecture, and the problems of Mochizuki’s paper. Not specifically about whether this proof is right, but about the general problem of how we can trust difficult proofs. So you may find that worth the read.

Reading the Comics, March 28, 2020: Closing A Week Edition


I know; I’m more than a week behind the original publication of these strips. The Playful Math Education Blog Carnival took a lot of what attention I have these days. I’ll get caught up again soon enough. Comic Strip Master Command tried to help me, by having the close of a week ago being pretty small mathematics mentions, too. For example:

Thaves’s Frank and Ernest for the 26th is the anthropomorphic numerals joke for the week. Also anthropomorphic letters, for a bonus.

Craig Boldman and Henry Scarpelli’s Archie for the 27th has Moose struggling in mathematics this term. This is an interesting casual mention; the joke, of Moose using three words to describe a thing he said he could in two, would not fit sharply for anything but mathematics. Or, possibly, a measuring class, but there’s no high school even in fiction that has a class in measuring.

Bud Blake’s Vintage Tiger for the 27th has Tiger and Hugo struggling to find adjective forms for numbers. We can giggle at Hugo struggling for “quadruple” and going for something that makes more sense. We all top out somewhere, though, probably around quintuple or sextuple. I have never known anyone who claimed to know what the word would be for anything past decuple, and even staring at the dictionary page for “decuple” I don’t feel confident in it.

Hilary Price’s Rhymes With Orange for the 28th uses a blackboard full of calculations as shorthand for real insight into science. From context they’re likely working on some physics problem and it’s quite hard to do that without mathematics, must agree.

Ham’s Life On Earth for the 28th uses E = mc^2 as a milestone in a child’s development.

John Deering’s Strange Brew for the 28th name-drops slide rules, which, yeah, have mostly historical or symbolic importance these days. There might be some niche where they’re particularly useful (besides teaching logarithms), but I don’t know of it.


And what of the strips from last week? I’ll discuss them in an essay at this link, soon, I hope. Take care, please.

Reading the Comics, March 25, 2020: Regular Old Mathematics Mentions Edition


I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.

Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.

Marcus, signalling a pitch: 'Two ... plus ... two ... minus .. one ... point ... three ... ' Jason, to Peter: 'If teams want to steal our signals, they're welcome to try.' Marcus: 'Can I just use a slash for 'divided by'?'.
Bill Amend’s FoxTrot for the 22nd of March, 2020. Essays mentioning either the new-run, Sunday, strips or the rerun, weekday, FoxTrot strips are gathered at this link.

Jerry Scott and Jim Borgman’s Zits for the 23rd has Jeremy preparing for a calculus test. Could be any subject.

James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself of his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.

Tom Batiuk’s Funky Winkerbean Vintage strip for the 24th, originally run the 13th of May, 1974, is wordplay about acute triangles.

Hector D Cantú and Carlos Castellanos’s Baldo for the 25th has Gracie work out a visual joke about plus signs. Roger Price, name-checked here, is renowned for the comic feature Droodles, extremely minimalist comic panels. He also, along with Get Smart’s Leonard Stern, created Mad Libs.

Man wrapped in flame, standing before God: 'Oh, come on! Grant me that I was within an order of magnitude of believing in the correct number of deities!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th of March, 2020. It is quite common for me to write about this strip. You can see me explaining Saturday Morning Breakfast Cereal at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.


There’s another half-dozen strips from last week that at least mention mathematics. I’ll at least mention them soon, in an essay at this link. Thank you.

The Playful Math Education Blog Carnival #136


Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.

Playful Math Education Blog Carnival banner, showing a coati dressed in bright maroon ringmaster's jacket and top hat, with multiplication and division signs sitting behind atop animal-training podiums; a greyscale photograph audience is in the far background.
Banner art again by Thomas K Dye, creator of Newshounds, Infinity Refugees, Something Happens, and his current comic strip, Projection Edge. You can follow him on Patreon and read his comic strip nine months ahead of its worldwide publication. The banner art was commissioned several weeks ago when I expected I would be in a more playful mood this week.

In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.

Kreig’s writing drew the attention of another kind contributor to my harvesting. Symmetry and Multiplying Negative Numbers explores one of those confusing things about negative numbers: how can a negative number times a negative number be positive? One way to understand this is to represent arithmetic operations as geometric operations. Particularly, we can see negation as a reflection.

That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …

This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.

Fermat’s Two-Squares Theorem, discussed here, is not the famous one about a^n + b^2 = c^n . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.

What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.

Iva Sallay also brought up Jenna Laib’s Making Meaning with Arrays: More Preschooler Division which similarly sees numerical truths revealed through geometric reasoning. Here, particularly, by the problem of baking muffins and thinking through how to divide them up. A key piece here, for a particular child’s learning, was being able to pick up and move things around. Often in shifting between arithmetic and geometry we suppose that we can rearrange things without effort. As adults it’s easy to forget that this is an abstraction that we need to learn.

Sharing of food, in this case cookies, appears in Helena Osana’s Mathematical thinking begins in the early years with dialogue and real-world exploration. Mathematic, Osana notes, is primarily about thinking. An important part in mathematics education is working out how the thinking children most like to do can also find mathematics.

I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?

Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.

Speaking of the game, David Coffey’s Creating Positive Change in Math Class was written in anticipation of the standardized tests meant to prove out mathematics education. Coffey gets to thinking about how to frame teaching to more focus on why students should have a skill, and how they can develop it. How to get students to feel involved in their work. Even how to get students to do homework more reliably. Coffey’s scheduled to present at the Michigan Council of Teachers of Mathematics conference in Grand Rapids this July. This if all starts going well. And this is another post I know of thanks to Goldenoj.

These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.

Mathematical study starts small, though. Often it starts with games. There are many good ones, not least Iva Sallay’s Find the Factors puzzles.

Besides that, Dads Worksheets has provided a set of Math Word Search Puzzles. It’s a new series from people who create worksheets for many grade levels and many aspects of mathematics. They’re on Twitter as @dadsworksheets.

Mr Wu, of the Singapore Math Tuition blog, has also begun a new series of recreational mathematics puzzles. He lays out the plans for this, puzzles aimed at children around eight to ten years old. One of the early ones is the Stickers Math Question. A more recent one is The Secret of the Sweets (Sweet Distribution Problem). Mr Wu can be found on Twitter as @mathtuition88.

Denise Gaskins, on Twitter as @letsplaymath, and indefatigable coordinator for this carnival, offers the chance to Play Math with Your Kids for Free. This is an e-book sampler of mathematics gameplay.

I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.

The Guys and Good Health blog presented Happy Pi Day on the 14th, with — in a move meant to endear the blog to me — several comic strips. This includes one from Grant Snider, who draws lovely strips. I’m sad that his Incidental Comics has left GoComics.com, so I can’t feature it often during my Reading the Comics roundups anymore.

Virtual Brush Box, meanwhile, offers To Celebrate Pi Day, 10 Examples of Numbers and 10 Examples of Math Involved with Horses which delights me by looking at π, and mathematics, as they’re useful in horse-related activities. This may be the only blog post written specifically for me and my sister, and I am so happy that there is the one.

There’s a bit more, a bit of delight. It was my greatest surprise in looking for posts for this month. That is poetry. I mean this literally.

Whimsy-Mimsy wrote on Pi Day a haiku.

D Avery, on Shift N Shake, wrote the longer Another Slice of Pi Day, the third year of their composing poems observing the day.

Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.

Blind On The Light Side similarly wrote Pi poems, including a Pi-Ku, for March 12, 2020. These poems don’t reach long enough to deal with the zero-syllable line, but we can forgive someone not wanting to go on that long.

As a last note, I have joined Mathstodon, the Mastodon instance with a mathematics theme. You can follow my shy writings there as @nebusj@mathstodon.xyz, or follow a modest number of people talking, largely, about mathematics. Mathstodon is a mathematically-themed microblogging site. On WordPress, I do figure to keep reading the comics for their mathematics topics. And sometime this year, when I feel I have the energy, I hope to do another A to Z, my little glossary project.

And this is what I have to offer. I hope the carnival has brought you some things of interest, and some things of delight. And, if I may, please consider this Grant Snider cartoon, Hope.

Life Through A Mathematician’s Eyes is scheduled to host the 137th installment of the Playful Math Education Blog Carnival, at the end of April. I look forward to seeing it. Good luck to us all.

One last call for the Playful Math Education Blog Carnival


I hope to publish the March 2020 Playful Math Education Blog Carnival tomorrow. If you’ve recently seen any web site that shares and explains some aspect of mathematics or mathematics education that interested or delighted you, please, share it with me, so I can share it with more people. If you do, please, let me know of your own projects, besides that, so I can mention that to this month’s audience. Thank you.

Reading the Comics, March 21, 2020: Pragmatic Calculations Edition


There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.

And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …

Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.

But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?

Kid: 'A number is divisible by 3 if the sum of its digits is divisible by 3. But what if the number is so big there's too many digits to add up easily?' Frazz: 'If it's that big, the 1 or 2 left over isn't going to matter much.' Kid: 'Why don't they teach THAT kind of math more in school?' Frazz: 'I guess there's only jobs for so many songwriters, cartoonists, and janitors.'
Jef Mallett’s Frazz for the 20th of March, 2020. Essays featuring some topic raised by Frazz should be gathered at this link.

This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.

There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.

Speaker at a podium: 'If one person kills someone, 50% of the people involved are victims. If 99 people kill someone, 1% of the people involved are victims. The percent of victims is given by V = the limit of K/x as x approachs infinity, where K is people killed and x is the number of people killed. Thus, for sufficiently large x, murder is a victimless crime. So, the bigger we make a war, the more ethical it becomes!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of March, 2020. I have many essays that mention something raised by this comic strip. The many things Saturday Morning Breakfast Cereal has given me to write about are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.

This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)

Mamet: 'I figure I have about 14,000 remaining days of life. So what's the big deal if I want to spend ONE of those days goofing off? That still leaves me with 13,00 days!' Cobb: 'Maybe you could spend a couple of those days learning math.' Mamet: 'Wait, make that 12,000. I'll need one day to PLAN the goof-off day.'
Jeff Corriveau’s Deflocked for the 20th of March, 2020. I’m surprised to learn this is a new tag for me. I’ve discussed the strip, it appears, only twice before, in 2012 and in 2015, before I tagged strips by name. All right. Well, this and future appearances by Deflocked will be at this link.

Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.


This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.

Reading the Comics, March 17, 2020: Random Edition


I thought last week’s comic strips mentioning mathematics in detail were still subjects easy to describe in one or two paragraphs each. I wasn’t quite right. So here’s a half of a week, even if it is a day later than I had wanted to post.

John Zakour and Scott Roberts’s Working Daze for the 15th is a straggler Pi Day joke, built on the nerd couple Roy and Kathy letting the date slip their minds. This is a very slight Pi Day reference but I feel the need to include it for completeness’s sake. It reminds me of the sequence where one year Schroeder forgot Beethoven’s birthday, and was devastated.

Sue: 'So, Roy, what big fun did you and Kathy have for Pi Day this year?' Roy, caught by surprise, freezes, and then turns several colors in succession before he starts to cry. Ed, to Sue: 'Hard to say which is worse for him, that you forgot, or that you remembered.'
John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2020. Essays featuring Working Daze, which often turns up in Pi Day events, are at this link. And generally essays tied to Pi Day are at this link.

Lincoln Peirce’s Big Nate for the 15th is a wordy bit of Nate refusing the story problem. Nate complains about a lack of motivation for the characters in it. But then what we need for a story problem isn’t the characters to do something so much as it is the student to want to solve the problem. That’s hard work. Everyone’s fascinated by some mathematical problems, but it’s hard to think of something that will compel everyone to wonder what the answer could be.

At one point Nate wonders what happens if Todd stops for gas. Here he’s just ignoring the premise of the question: Todd is given as travelling an average 55 mph until he reaches Saint Louis, and that’s that. So this question at least is answered. But he might need advice to see how it’s implied.

Quiz: 'Many lives in Los Angeles. Todd lives in Boston. They plan to meet in St Louis, which is 1,825 miles from Los Angeles and 1,192 miles from Boston. If Mandy takes a train travelling a constant 80 mph and Todd drives a car at a constant 55 mph, which of them will reach St Lous first?' Nate's answer: 'That depends. Who ARE these people? Are they a couple? Is this romance? If it is, wouldn't Todd drive way faster than 55 mph? He'd be all fired up to see Many, right? And wouldn't Mandy take a plane and get to St Louis in like three hours? Especially if she hasn't seen Todd in a while? But we don't know how long since they've been together because you decided not to tell us! Plus anything can happen while they're traveling. What if Todd stops for gas and the cashier is a total smoke show and he's like, Mandy Who? I can't answer until I have some real intel on these people. I can't believe you even asked the question.' Out loud, 'Also, Todd and Mandy are dorky names.' Teacher: 'This isn't what I meant by show your work.'
Lincoln Peirce’s Big Nate for the 15th of March, 2020. Essays with something mentioned by either Big Nate or the 1990s-repeats Big Nate: First Class are gathered at this link.

So this problem is doable by long division: 1825 divided by 80, and 1192 divided by 55, and see what’s larger. Can we avoid dividing by 55 if we’re doing it by hand? I think so. Here’s what I see: 1825 divided by 80 is equal to 1600 divided by 80 plus 225 divided by 80. That first is 20; that second is … eh. It’s a little less than 240 divided by 80, which is 3. So Mandy will need a little under 23 hours.

Is 23 hours enough for Todd to get to Saint Louis? Well, 23 times 55 will be 23 times 50 plus 23 times 5. 23 times 50 is 22 times 50 plus 1 times 50. 22 times 50 is 11 times 100, or 1100. So 23 times 50 is 1150. And 23 times 5 has to be 150. That’s more than 1192. So Todd gets there first. I might want to figure just how much less than 23 hours Mandy needs, to be sure of my calculation, but this is how I do it without putting 55 into an ugly number like 1192.

Cow: 'What're you doing?' Billy: 'I'm devising a system to win the lottery! Plugging in what I know about chaos theory and numerical behavior in nonlinear dynamical systems should give me the winning picks.' (Silent penultimate panel.) Cow: 'You're just writing down a bunch of numbers.' Billy: 'Maybe.'
Mark Leiknes’s Cow and Boy repeat for the 17th of March, 2020. The too-rare appearances of Cow and Boy Reruns in my essays are here.

Mark Leiknes’s Cow and Boy repeat for the 17th sees the Boy, Billy, trying to beat the lottery. He throws at it the terms chaos theory and nonlinear dynamical systems. They’re good and probably relevant systems. A “dynamical system” is what you’d guess from the name: a collection of things whose properties keep changing. They change because of other things in the collection. When “nonlinear” crops up in mathematics it means “oh but such a pain to deal with”. It has a more precise definition, but this is its meaning. More precisely: in a linear system, a change in the initial setup makes a proportional change in the outcome. If Todd drove to Saint Louis on a path two percent longer, he’d need two percent more time to get there. A nonlinear system doesn’t guarantee that; a two percent longer drive might take ten percent longer, or one-quarter the time, or some other weirdness. Nonlinear systems are really good for giving numbers that look random. There’ll be so many little factors that make non-negligible results that they can’t be predicted in any useful time. This is good for drawing number balls for a lottery.

Chaos theory turns up a lot in dynamical systems. Dynamical systems, even nonlinear ones, often have regions that behave in predictable patterns. We may not be able to say what tomorrow’s weather will be exactly, but we can say whether it’ll be hot or freezing. But dynamical systems can have regions where no prediction is possible. Not because they don’t follow predictable rules. But because any perturbation, however small, produces changes that overwhelm the forecast. This includes the difference between any possible real-world measurement and the real quantity.

Obvious question: how is there anything to study in chaos theory, then? Is it all just people looking at complicated systems and saying, yup, we’re done here? Usually the questions turn on problems such as how probable it is we’re in a chaotic region. Or what factors influence whether the system is chaotic, and how much of it is chaotic. Even if we can’t say what will happen, we can usually say something about when we can’t say what will happen, and why. Anyway if Billy does believe the lottery is chaotic, there’s not a lot he can be doing with predicting winning numbers from it. Cow’s skepticism is fair.

T-Rex: 'Dromiceiomimus, pick a number between one and a hundred thousand million.' Dromiceiomimus: '17?' T-Rex: 'Gasp! That's the number I was thinking of!' Dromiceiomimus: 'Great! Do I win something?' T-Rex: 'You just came out on a one in a hundred thousand million chance and you want a prize? It's not enough to spit in the face of probability itself?' Utahraptor: 'It's not THAT unlikely she'd chose your number. We're actually pretty bad at random number generation and if you ask folks to pick a number in a range, some choices show up more often than others. It's not that unlikely you'd both land on the same number!' T-Rex: 'But *I* didn't choose 17 randomly! It's ... the number of times I have thought about ice cream today, I'm not even gonna lie.'
Ryan North’s Dinosaur Comics for the 17th of March, 2020. Essays that mention something brought up in Dinosaur Comics are gathered at this link.

Ryan North’s Dinosaur Comics for the 17th is one about people asked to summon random numbers. Utahraptor is absolutely right. People are terrible at calling out random numbers. We’re more likely to summon odd numbers than we should be. We shy away from generating strings of numbers. We’d feel weird offering, say, 1234, though that’s as good a four-digit number as 1753. And to offer 2222 would feel really weird. Part of this is that there’s not really such a thing as “a” random number; it’s sequences of numbers that are random. We just pick a number from a random sequence. And we’re terrible at producing random sequences. Here’s one study, challenging people to produce digits from 1 through 9. Are their sequences predictable? If the numbers were uniformly distributed from 1 through 9, then any prediction of the next digit in a sequence should have a one chance in nine of being right. It turns out human-generated sequences form patterns that could be forecast, on average, 27% of the time. Individual cases could get forecast 45% of the time.

There are some neat side results from that study too, particularly that they were able to pretty reliably tell the difference between two individuals by their “random” sequences. We may be bad at thinking up random numbers but the details of how we’re bad can be unique.


And I’m not done yet. There’s some more comic strips from last week to discuss and I’ll have that post here soon. Thanks for reading.

Reading the Comics, March 20, 2020: Running from the Quiz Edition


I’m going to again start the week with the comics that casually mentioned mathematics. Later in the week I’ll have ones that open up discussion topics. I just don’t want you to miss a comic where a kid doesn’t want to do a story problem.

John Graziano’s Ripley’s Believe It or Not for the 15th mentions the Swiss mint issuing a tiny commemorative coin of Albert Einstein. I mention just because Einstein is such a good icon for mathematical physics.

Ashleigh Brilliant’s Pot-Shots for the 16th has some wordplay about multiplication and division. I’m not sure it has any real mathematical content besides arithmetic uniting multiplication and division, though.

Mark Pett’s Mr Lowe rerun for the 17th has the students bored during arithmetic class. Fractions; of course it would be fractions.

Justin Boyd’s Invisible Bread for the 18th> has an exhausted student making the calculation of they’ll do better enough after a good night’s sleep to accept a late penalty. This is always a difficult calculation to make, since you make it when your thinking is clouded by fatigue. But: there is no problem you have which sleep deprivation makes better. Put sleep first. Budget the rest of your day around that. Take it from one who knows and regrets a lot of nights cheated of rest. (This seems to be the first time I’ve mentioned Invisible Bread around here. Given the strip’s subject matter that’s a surprise, but only a small one.)

John Deering’s Strange Brew for the 18th is an anthropomorphic-objects strip, featuring talk about mathematics phobia.

One of Gary Larson’s The Far Side reruns for the 19th is set in a mathematics department, and features writing a nasty note “in mathematics”. There are many mathematical jokes, some of them written as equations. A mathematician will recognize them pretty well. None have the connotation of, oh, “Kick Me” or something else that would belong as a prank sign like that. Or at least nobody’s told me about them.

Tauhid Bondia’s Crabgrass for the 20th sees Kevin trying to find luck ahead of the mathematics quiz.

Bob Weber Jr and Jay Stephens’s Oh, Brother! for the 20th similarly sees Bud fearing a mathematics test.


Thanks for reading. And, also, please remember that I’m hosting the Playful Math Education Blog Carnival later this month. Please share with me any mathematics stuff you’ve run across that teaches or entertains or more.

Reading the Comics, March 14, 2020: Pi Day Edition


Pi Day was observed with fewer, and fewer on-point, comic strips than I had expected. It’s possible that the whimsy of the day has been exhausted. Or that Comic Strip Master Command advised people that the educational purposes of the day were going to be diffused because of the accident of the calendar. And a fair number of the strips that did run in the back half of last week weren’t substantial. So here’s what did run.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 12th has a parent complaining about kids being allowed to use calculators to do mathematics. The rejoinder, asking how good they were at mathematics anyway, is a fair one.

Bill Watterson’s Calvin and Hobbes rerun for the 13th sees Calvin avoiding his mathematics homework. The strip originally ran the 16th of March, 1990.

And now we get to the strips that actually ran on the 14th of March.

Gracie, to her father: 'If I had $1.39 for every time I've struggled with a mathematics problem ... I'd have ... ' (She taps on a calculator) '6.23 cents.'
Hector D Cantú and Carlos Castellanos’s Baldo for the 14th of March, 2020. Essays with some mention of Baldo are gathered at this link.

Hector D Cantú and Carlos Castellanos’s Baldo is a slightly weird one. It’s about Gracie reflecting on how much she’s struggled with mathematics problems. There are a couple pieces meant to be funny here. One is the use of oddball numbers like 1.39 or 6.23 instead of easy-to-work-with numbers like “a dollar” or “a nickel” or such. The other is that the joke is .. something in the vein of “I thought I was wrong once, but I was mistaken”. Gracie’s calculation indicates she thinks she’s struggled with a math problem a little under 0.045 times. It’s a peculiar number. Either she’s boasting that she struggles very little with mathematics, or she’s got her calculations completely wrong and hasn’t recognized it. She’s consistently portrayed as an excellent student, though. So the “barely struggles” or maybe “only struggles a tiny bit at the start of a problem” interpretation is more likely what’s meant.

Mark Parisi’s Off the Mark is a Pi Day joke that actually features π. It’s also one of the anthropomorphic-numerals variety of jokes. I had also mistaken it for a rerun. Parisi’s used a similar premise in previous Pi Day strips, including one in 2017 with π at the laptop.

An anthropomorphic pi at a laptop, facing a web page demanding, 'Enter your full name'. It's gotten through 26 digits past the decimal.
Mark Parisi’s Off The Mark for the 14th of March, 2020. Other essays featuring something raised by Off The Mark, including a fair number of Pi Day jokes, are at this link.

π has infinitely many decimal digits, certainly. Of course, so does 2. It’s just that 2 has boring decimal digits. Rational numbers end up repeating some set of digits. It can be a long string of digits. But it’s finitely many, and compared to an infinitely long and unpredictable string, what’s that? π we know is a transcendental number. Its decimal digits go on in a sequence that never ends and never repeats itself fully, although finite sequences within it will repeat. It’s one of the handful of numbers we find interesting for reasons other than their being transcendental. This though nearly every real number is transcendental. I think any mathematician would bet that it is a normal number, but we don’t know that it is. I’m not aware of any numbers we know to be normal and that we care about for any reason other than their normality. And this, weirdly, also despite that we know nearly every real number is normal.

At the ATM, a pie with arms enters a pin. An onlooking doughnut says '3.14? Please tell me that's not really your pin.'
Dave Whamond’s Reality Check for the 14th of March, 2020. Essays that show off something from a Reality Check panel are at this link.

Dave Whamond’s Reality Check plays on the pun between π and pie, and uses the couple of decimal digits of π that most people know as part of the joke. It’s not an anthropomorphic numerals joke, but it is circling that territory.

Loose sketch of Albert Einstein, accompanied by the quote, 'Only two things are infinite: the universe and human stupidity, and I'm not sure about the former', along with a note wishing him a happy birthday.
Michael Cavna’s Warped for the 14th of March, 2020. The rare appearances here of Warped are gathered at this link.

Michael Cavna’s Warped celebrates Albert Einstein’s birthday. This is of marginal mathematics content, but Einstein did write compose one of the few equations that an average lay person could be expected to recognize. It happens that he was born the 14th of March and that’s, in recent years, gotten merged into Pi Day observances.


I hope to start discussing this week’s comic strips in some essays starting next week, likely Sunday. Thanks for reading.

How February 2020 Treated My Mathematics Blog


Oh, yes, so. I did intend to review my readership around here last month. It’s just that things got in the way. Most of them not related to the Covid-19 pandemic; it’s much more been personal matters and my paying job and such. If someone is interested in paying me to observe that I had readers WordPress records as coming merely from the European Union, drop me a note. We can work something out. Heck, slip me ten bucks and I’ll write an essay on any mathematics topic I don’t feel wholly incompetent to discuss. Or wait around for the 2020 Mathematics A-to-Z, coming whenever I do feel up to it.

Also, do please remember that I’m hosting the Playful Math Education Blog Carnival at the end of this month. If you’ve spotted anything on the web — blog, static web site, video, podcast — that enlightened you about some field of mathematics, please let me know. And let me know of your own projects. It’ll be fun.

Now to see what my readership was like back in February, impossibly long ago as that does seem to be.

Bar chart of four and a half years of monthly readership statistics, with the readership having returned to a level after a huge spike in October.
I mean, yes, the anomaly in October makes the rest of my readership look like nothing, but if you look at the apparent floor I’ve had since July and compare it to one or two or three years ago I’m clearly managing to get a nice steady influx of readers and bots that WordPress doesn’t recognize as such.

I posted 11 things in February. January had been 10. There were 1,419 page views in February. That’s just about what January was. It’s below the twelve-month running average of 2,060.3 page views. This looks dire, but it’s about the same as January’s readership. And the twelve-month average does have that anomalous October spike messing things up. If we pretend that October didn’t happen, well, that mean was something like 1460 page views.

There were 991 unique visitors in February. That’s again rather below the twelve-month running average of 1401.1 unique visitors. But again if we pretend there was no October, then the running average was something like 950 unique visitors, so things aren’t all that dire. Just that the occasional taste of popularity spoils you for ages to come.

A mere 36 things got likes here in February, below the running average of 64.1 and I’m not working out what that is with October included. Most of that readership spike didn’t convert to likes or comments anyway. Those were well-liked months but they were also ones that got something posted every single day. There were 12 comments in February, roughly in line with the 13.8 comments running average.

Per post, all these figures look a bit better. There were 129 views per posting, just over the 116.6 running average. There were 90.1 unique visitors per posting, above the running average of 78.6. There were 3.3 likes per posting, below the anemic average of 4.1. There were even 1.1 comments per posting, technically above the average of 0.9. If I could just post something four times per day that October peak would be merely an average month.

The most popular postings in February were mostly the usual suspects. Just one surprised me with its appearance:

The most popular thing written in February were two equally-popular Reading the Comics posts, Symbols Edition and 90s Doonesbury Edition.

There were 210 pages that got any views at all in February, close to the 218 of January. 108 of them got more than one view, just about the same as January’s 102. 25 pages got at least ten views. The previous couple months saw 23 and 27 posts that popular.

67 countries or country-like entities sent me any readers at all in February. That’s up from 63 in January and 60 in December. 19 of them were single-view countries, up from January’s 15 and December’s 18. Here’s the roster:

Mercator-style map of the world with the United States in darkest pink. Most of the Americas, Europe, Asia, and Oceana are in a more uniform pink. Mediterranean Africa and a few scattered other countries are also represented, though not Mongolia or most of the former Soviet republics.
Aw, my Scandinavian readers are all moving on to other stuff.
Country Readers
United States 851
Philippines 85
India 57
Canada 53
United Kingdom 41
Germany 35
Australia 26
Finland 23
Singapore 23
Brazil 19
Thailand 14
Denmark 13
Hungary 13
Hong Kong SAR China 10
South Africa 10
Russia 9
Japan 8
Netherlands 8
New Zealand 8
Vietnam 8
Mexico 7
Indonesia 6
Poland 6
Malaysia 5
Belgium 4
France 4
Italy 4
Sweden 4
Austria 3
Colombia 3
Greece 3
Jamaica 3
Uganda 3
Ukraine 3
Algeria 2
Azerbaijan 2
China 2
Cyprus 2
Ghana 2
Israel 2
Kenya 2
Nigeria 2
Portugal 2
Slovenia 2
Spain 2
Switzerland 2
Turkey 2
United Arab Emirates 2
American Samoa 1 (**)
Argentina 1
Bulgaria 1
Cambodia 1 (*)
Croatia 1
Dominican Republic 1
Egypt 1
European Union 1
Ireland 1
Libya 1
Lithuania 1
Northern Mariana Islands 1
Peru 1
Puerto Rico 1
Saudi Arabia 1 (**)
Slovakia 1 (**)
South Korea 1 (*)
Sri Lanka 1
Taiwan 1

Cambodia and South Korea were single-view countries in January also. American Samoa, Saudi Arabia, and Slovakia have been single-view countries for three months.

In February I posted 9,699 words by WordPress’s counter. That’s 881.7 words per posting. For the year my average post, as of the start of the month, was 755.1 words per post. Some months are talky. I had started the month with 100,432 page views, just missing out on being number 100,000 myself. And these came from a logged 54,920 unique visitors. And I had posted a total of 1,424 things from the dawn of time to the 1st of March, which by some strange fluke was itself fifty thousand years ago.

Well, I hope to keep posting and to review March a little closer to the 1st of April, which looks to be about fifty thousand years in the future. To follow along with me, add the feed https://nebusresearch.wordpress.com/feed/ to your RSS reader. If you need an RSS reader, sign up for a free account on Dreamwidth or Livejournal; you can put RSS feeds on your Friends page. Or if you prefer a more old-fashioned way that shows up in my statistics here, use the “Follow Nebusresearch” button at the upper right corner of this page and follow it through your WordPress account. Or follow my fallow @Nebusj account on Twitter where new posts get announced at least. As ever, thank you for reading. Be well, please.

Reading the Comics, March 11, 2020: Half Week Edition


There were a good number of comic strips mentioning mathematical subjects last week, as you might expect for one including the 14th of March. Most of them were casual mentions, though, so that’s why this essay looks like this. And is why the week will take two pieces to finish.

Jonathan Lemon and Joey Alison Sayer’s Little Oop for the 8th is part of a little storyline for the Sunday strips. In this the young Alley Oop has … travelled in time to the present. But different from how he does in the weekday strips. What’s relevant about this is Alley Oop hearing the year “2020” and mentioning how “we just got math where I come from” but being confident that’s either 40 or 400. Which itself follows up a little thread in the Sunday strips about new numbers on display and imagining numbers greater than three.

Venn Diagram with two bubbles. The left is 'Day after Daylight Savings [sic] Start'; the right is 'Monday'. The intersection has an arrow from it pointing to a travel cup of coffee.
Maria Scrivan’s Half Full for the 9th of March, 2020. Essays featuring some topic raised by Half Full appear at this link.

Maria Scrivan’s Half Full for the 9th is the Venn Diagram strip for the week.

Paul Trap’s Thatababy for the 9th is a memorial strip to Katherine Johnson. She was, as described, a NASA mathematician, and one of the great number of African-American women whose work computing was rescued from obscurity by the book and movie Hidden Figures. NASA, and its associated agencies, do a lot of mathematical work. Much of it is numerical mathematics: a great many orbital questions, for example, can not be answered with, like, the sort of formula that describes how far away a projectile launched on a parabolic curve will land. Creating a numerical version of a problem requires insight and thought about how to represent what we would like to know. And calculating that requires further insight, so that the calculation can be done accurately and speedily. (I think about sometime doing a bit about the sorts of numerical computing featured in the movie, but I would hardly be the first.)

Eulogy strip, as drawn by the baby, celebrating Katherine Johnson, NASA mathematician 1918 - 2020. It shows a child's drawing of her, and of a Mercury capsule, with formulas describing a ballistic trajectory making the motion trail of the capsule.
Paul Trap’s Thatababy for the 9th of March, 2020. My essays featuring something raised by Thatababy are at this link.

I also had thought the Mathematical Moments from the American Mathematical Society had posted an interview with her last year. I was mistaken but in, I think, a forgivable way. In the episode “Winning the Race”, posted the 12th of June, they interviewed Christine Darden, another of the people in the book, though not (really) the movie. Darden joined NASA in the late 60s. But the interview does talk about this sort of work, and how it evolved with technology. And, of course, mentions Johnson and her influence.

Graham Harrop’s Ten Cats for the 9th is another strip mentioning Albert Einstein and E = mc2. And using the blackboard full of symbols to represent deep thought.

Patrick Roberts’s Todd the Dinosaur for the 10th showcases Todd being terrified of fractions. And more terrified of story problems. I can’t call it a false representation of the kinds of mathematics that terrify people.

Teacher: 'All right, class, please take out your math books!' Todd: 'Teacher, this isn't gonna be fractions, is it?' Teacher: 'No, Todd, no fractions.' Todd: 'Whewwww!' Teacher: 'Now listen carefully, class. Train A leaves Chicago at 7:00 am, and ... ' (Todd, screaming in panic, runs out crashing through the wall and over the horizon.)
Patrick Roberts’s Todd the Dinosaur for the 10th of March, 2020. Essays that discuss something mentioned in a Todd the Dinosaur should be gathered at this link.

Stephen Beals’s Adult Children for the 11th has a character mourning that he took calculus as he’s “too stupid to be smart”. Knowing mathematics is often used as proof of intelligence. And calculus is used as the ultimate of mathematics. It’s a fair question why calculus and not some other field of mathematics, like differential equations or category theory or topology. Probably it’s a combination of slightly lucky choices (for calculus). Calculus is old enough to be respectable. It’s often taught as the ultimate mathematics course that people in high school or college (and who aren’t going into a mathematics field) will face. It’s a strange subject. Learning it requires a greater shift in thinking about how to solve problems than even learning algebra does. And the name is friendly enough, without the wordiness or technical-sounding language of, for example, differential equations. The subject may be well-situated.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 11th has the pacing of a logic problem, something like the Liar’s Paradox. It’s also about homework which happens to be geometry, possibly because the cartoonists aren’t confident that kids that age might be taking a logic course.


I’ll have the rest of the week’s strips, including what Comic Strip Master Command ordered done for Pi Day, soon. And again I mention that I’m hosting this month’s Playful Math Education Blog Carnival. If you have come across a web site with some bit of mathematics that brought you delight and insight, please let me know, and mention any creative projects that you have, that I may mention that too. Thank you.

Getting Ready for Pi Day, and also the Playful Math Blog Carnival


So the first bit of news: I’m hosting the Playful Math Education Blog Carnival later this month. This is a roaming blog link party, sharing blogs that delight or educate, or ideally both, about mathematics. As mentioned the other day Iva Sallay of Find the Factors hosted the 135th of these. My entry, the 136th, I plan to post sometime the last week of March.

And I’ll need help! If you’ve run across a web site, YouTube video, blog post, or essay that discusses something mathematical in a way that makes you grin, please let me know, and let me share it with the carnival audience.


This Saturday is March 14th, which we’ve been celebrating as Pi Day. I remain skeptical that it makes a big difference in people’s view of mathematics or in their education. But an afternoon spent talking about mathematics with everyone agreeing that, for today, we won’t complain about how hard it always was or how impossible we always found it, is pleasant. And that’s a good thing. I don’t know how much activity there’ll be for it, since the 14th is a weekend day this year. And the Covid-19 problem has got all the schools in my state closed through to April, so any calendar relevance is shattered.

But I have some things in the archive anyway. Last year I gathered Six Or Arguably Four Things For Pi Day, a collection of short essays about ways to calculate π well or poorly, and about some of the properties we’re pretty sure that π has, even if we can’t prove it. Also this fascinating physics problem that yields the digits of π.

And the middle of March often brings out Comic Strip Master Command. It looks like I’ve had at least five straight Pi Day editions of Reading the Comics, although most of them cover strips from more than just the 14th of March. From the past:

What will 2020 offer? There’s no guessing about anything in 2020 anymore, really. But when I get to look at the Pi Day comic strips for 2020 my essay on them should appear at this link. Thanks ever for reading. And for letting me know about sites that would be good for this month’s Carnival.

Find the Factors hosted the Playful Math Education Blog Carnival this month


I apologize that obligations have kept me from writing some things that I mean to. So let me just point you to Iva Sallay, whose Find the Factor recreational math puzzle page hosted the 135th Playful Math Education Blog Carnival this past month. The Blog Carnival is a fun roaming thing that I’ve hosted once, and do hope to host again. It’s a curated collection of other mathematics sites that are fun or interesting or hopefully both together.

Reading the Comics, March 7, 2020: Everybody Has Tests Edition


It was another pretty quiet week for mathematically-themed comic strips. Most of what did mention my subject just presented it as a subject giving them homework or quizzes or exams. But let’s look over what is here.

Morrie Turner’s Wee Pals rerun for the 3rd is an example of this, with one of the kids mourning his arithmetic grade. The strip previously ran the 3rd of March, 2015.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 4th similarly has mathematics homework under review. And, you know, one of those mistakes that’s obvious if you do a quick “sanity check”, thinking over whether your answer could make sense.

Ted Shearer’s Quincy for the 5th is the most interesting strip of the week, since it suggests an actual answerable mathematics problem. How much does a professional basketball player earn per dribble? The answer requires a fair bit of thought, like, what do you mean by “a professional basketball player”? There’s many basketball leagues around the world; even if we limit the question to United States-and-Canada leagues, there’s a fair number of minor leagues. If we limit it to the National Basketball Association there’s the question of whether the salary is the minimum union contract guarantee, or the mean salary, or the median salary. It’s exciting to look at the salary of the highest-paid players, too, of course.

Quincy, playing with a basketball: 'Li'l Bo, some pro basketball players have million-dollar contracts.' Bo; 'Boy! That's a lot of money for playing a game.' Quincy: 'I figure they're gettin' about a dollar a dribble.'
Ted Shearer’s Quincy for the 5th of March, 2020. It looks to have originally run the 9th of January, 1981. I don’t get to discuss the strip often, but when I do, Quincy appears in essays at this link.

Working out the number of dribbles per year is also a fun estimation challenge. Even if we pick a representative player there’s no getting an exact count of how many dribbles they’ve made over a year, even if we just consider “dribbling during games” to be what’s paid for. (And any reasonable person would have to count all the dribbling done during warm-up and practice as part of what’s being paid for.) But someone could come up with an estimate of, for example, about how long a typical player has the ball for a game, and how much of that time is spent moving the ball or preparing for a free throw or other move that calls for dribbling. How long a dribble typically takes. How many games a player typically plays over the year. The estimate you get from this will never, ever, be exactly right. But it should be close enough to give an idea how much money a player earns in the time it takes to dribble the ball once. So occasionally the comics put forth a good story problem after all.

Quincy on the 7th is again worrying about his mathematics and spelling tests. It’s a cute coincidence that these are the subjects worried about in Wee Pals too.

Paul Gilligan’s Pooch Cafe for the 7th is part of a string of jokes about famous dogs. This one’s a riff on Albert Einstein, mentioned here because Albert Einstein has such strong mathematical associations.


And that’s all the week there was. I’ll be Reading the Comics for their mathematics content next week, too, and be glad to see you then. My guess is: some jokes about π.

Paul Dirac discussed on the In Our Time podcast


It’s a touch off my professed mathematics focus. Also off my comic strips focus. But Paul Dirac was one of the 20th century’s greatest physicists, this in a century rich in great physicists. Part of his genius was in innovative mathematics, and in trusting strange implications of his mathematics.

This week the BBC podcast In Our Time, a not-quite-hourlong panel show discussing varied topics, came to Paul Dirac. It can be heard here, or from other podcast sources. I get it off iTunes myself. The discussion is partly about his career and about the magnitude of his work. It’s not going to make anyone suddenly understand how to do any of his groundbreaking work in quantum mechanics. But it is, after all, an hourlong podcast for the general audience about, in this case, a physicist. It couldn’t explain spinors.

And even if you know a fair bit about Dirac and his work you might pick up something new. This might be slight: one of the panelists mentioned Dirac, in retirement, getting to know Sting. This is not something impossible, but it’s also not a meeting I would have ever imagined happening. So my week has been broadened a bit.

The web site for In Our Time doesn’t have a useful archive category for mathematics, at least that I could find. But many mathematical topics are included in the archive of science subjects, including important topics like the kinetic theory of gases and the work of Emmy Noether.

How To Multiply Numbers By Multiplying Other Numbers Instead


I do read other people’s mathematics writing, even if I don’t do it enough. A couple days ago RJ Lipton and KW Regan’s Reductions And Jokes discussed how one can take a problem and rewrite it as a different problem. This is one of the standard mathematician’s tricks. The point to doing this is that you might have a better handle on the new problem.

“Better” is an aesthetic judgement. It reflects whether the new problem is easier to work with. Along the way, they offer an example that surprised and delighted me, and that I wanted to share. It’s about multiplying whole numbers. Multiplication can take a fair while, as anyone who’s tried to do 38 times 23 by hand has found out. But we can speed that up. A multiplication table is a special case of a lookup table, a chunk of stored memory which has computed ahead of time all the multiplications someone is likely to do. Then instead of doing them, you just look them up.

The catch is that a multiplication table takes memory. To do all the multiplications for whole numbers 1 through 10 you need … well, not 100 memory cells. But 55. To have 1 through 20 worked out ahead of time you need 210 memory cells. Can we do better?

If addition and subtraction are easy enough to do? And if dividing by two is easy enough? Then, yes. Instead of working out every pair multiplication, work out the squares of the whole numbers. And then make use of this identity:

a \times b = \frac{1}{2}\left( \left(a + b\right)^2 - a^2 - b^2\right)

And that delights me. It’s one of those relationships that’s sitting there, waiting for anyone who’s ever squared a binomial to notice. I don’t know that anyone actually uses this. But it’s fun to see multiplication worked out by a different yet practical way.

Reading the Comics, February 29, 2020: Leap Day Quiet Edition


I can clear out all last week’s mathematically-themed comic strips in one move, it looks like. There were a fair number of strips; it’s just they mostly mention mathematics in passing.

Bill Amend’s FoxTrot for the 23rd — a new strip; it’s still in original production for Sundays — has Jason asking his older sister to double-check a mathematics problem. Double-checking work is reliably useful, as proof against mistakes both stupid and subtle. But that’s true of any field.

Mark Tatulli’s Heart of the City for the 23rd has Heart preparing for an algebra test.

Jim Unger’s Herman for the 23rd has a parent complaining about the weird New Math. The strip is a rerun and I don’t know from when; it hardly matters. The New Math has been a whipping boy for mathematics education since about ten minutes after its creation. And the complaint attaches to every bit of mathematics education reform ever. I am sympathetic to parents, who don’t see why their children should be the test subjects for a new pedagogy. And who don’t want to re-learn mathematics in order to understand what their children are doing. But, still, let someone know you were a mathematics major and they will tell you how much they didn’t understand or like mathematics in school. It’s hard to see why not try teaching it differently.

(If you do go out pretending to be a mathematics major, don’t worry. If someone challenges you on a thing, cite “Euler’s Theorem”, and you’ll have said something on point. And I’ll cover for you.)

Phil Dunlap’s Ink Pen rerun for the 24th has Bixby Rat complain about his mathematics skills.

Father and child duck sitting on the starry sky. Father: 'Hey, Champ, I know you're only 5, but I think it's time I introduce you to the wonders of the universe! See those stars? How many do you think there are?' Child: 'Um ... 12?' Father: 'Actually, there's over 300 sextillion stars! That's a 3 with 23 zeroes after it.' Child: 'And that's more than 12?' Father: 'Maybe I should introduce you to the wonders of math, first.'
Brian Gordon’s Fowl Language for the 25th of February, 2020. This strip previously ran the 5th of February, 2016, which happens to be the only other time I have an essay mentioning this comic. That’s from before I tagged comic strips by title, though. So this essay and any future repetitions that happen to mention Fowl Language should be at this link, although the previous one probably won’t be.

Brian Gordon’s Fowl Language for the 25th has a father trying to explain the vastness of Big Numbers to their kid. Past a certain point none of us really know how big a thing is. We can talk about 300 sextillion stars, or anything else, and reason can tell us things about that number. But do we understand it? Like, can we visualize that many stars the way we can imagine twelve stars? This gets us into the philosophy of mathematics pretty soundly. 300 sextillion is no more imaginary than four is, but I know I feel more confident in my understanding of four. How does that make sense? And can you explain that to your kid?

Vic Lee’s Pardon my Planet for the 28th has an appearance by Albert Einstein. And a blackboard full of symbols. The symbols I can make out are more chemistry than mathematics, but they do exist just to serve as decoration.

Bud Blake’s Tiger rerun for the 28th has Hugo mourning his performance on a mathematics test.

Ruben Bolling’s Super-Fun-Pak Comix for the 28th is an installment of The Uncertainty Principal. This is a repeat, even allowing that Super-Fun-Pak Comix are extracted reruns from Tom The Dancing Bug. As I mention in the essay linked there, the uncertainty principle being referred to here is a famous quantum mechanics result. It tells us there are sets of quantities whose values we can’t, even in principle, measure simultaneously to unlimited precision. A precise measurement of, for example, momentum destroys our ability to be precise about position. This is what makes the joke here. The mathematics of this reflects non-commutative sets of operators.

Dave Blazek’s Loose Parts for the 29th is another with a blackboard full of symbols used to express deep thought on a subject.


And that takes care of last week. I’ll be Reading the Comics for their mathematics content next week, too, although the start of the week has been a slow affair so far. We’ll see if that changes any.