Reading the Comics, April 10, 2019: Grand Avenue and Luann Want My Attention Edition


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

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

On the blackboard is a square-based pyramid with 'apex' labelled; also a circular cone with 'vertex' labelled. Wavehead: 'And if you put them together they're a duplex.'
Mark Anderson’s Andertoons for the 7th of March, 2019. I write about this strip a lot. Essays mentioning Andertoons are at this link.

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

Here's a tough little puzzle to get your brain firing on all four cylinders. See if you can connect the matching numbered boxes with three lines. The catch is that the liens cannot cross over each other. From left to right are disjoint boxes labelled 1, 2, 1, and 2. Above and below the center of the row are two boxes labelled 3.
Mac King and Bill King’s Magic in a Minute for the 7th of March, 2019. I should have the essays mentioning Magic In A Minute at this link.

One common problem, found early in studying graph theory, is about whether a graph is planar. That is, can you draw the whole graph, all its vertices and edges, without any lines cross each other? This graph, with six vertices and three edges, is planar. There are graphs that are not. If the challenge were to connect each number to a 1, a 2, and a 3, then it would be nonplanar. That’s a famous non-planar graph, given the obvious name K3, 3. A fun part of learning graph theory — at least fun for me — is looking through pictures of graphs. The goal is finding K3, 3 or another one called K5, inside a big messy graph.

Exam Question: Jack bought seven marbles and lost six. How many additional marbles must Jack buy to equal seven? Kid's answer: 'Jack wouldn't know. He's lost his marbles.'
Mike Thompson’s Grand Avenue for the 8th of March, 2019. I’m not always cranky about this comic strips. Examples of when I’m not are at this link, as are the times I’m annoyed with Grand Avenue.

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

Gunther: 'You put a question mark next to the part about using the slope-intercept form of a linear equation. What don't you understand?' Luann: 'Lemme see. Oh ... yeah. I don't understand why on earth I need to know this.'
Greg Evans’s Luann Againn for the 10th of March, 2019. This strip originally ran the 10th of March, 1991. Essays which include some mention of Luann, either current or 1990s reprints, are at this link.

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

How To Survive a Shark Attack (illustrated with a chicken surviving a shark.0 Keep your eye on the shark and move slowly toward safety. Don't make any sudden movements such as splashing or jazz hands. If the shark comes at you, punch it in the gills, snout, or eyes. You won't hurt the shark, but it will be surprised by your audacity. If all else fails, try to confuse it with logical paradoxes. Chicken: 'This statement is false.' Shark, wide-eyed and confused: '?'
Doug Savage’s Savage Chickens for the 10th of March, 2019. And when I think of something to write about Savage Chickens the results are at this link.

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


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

Myself, I keep all my Reading the Comics essays at this link, and those mentioning some aspect of Rhymes With Orange at this link.

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Is 1/x a Continuous Function?


So this is a question I got by way of a friend. It’s got me thinking because there is an obviously right answer. And there’s an answer that you get to if you think about it longer. And then longer still and realize there are several answers you could give. So I wanted to put it out to my audience. Figuring out your answer and why you stand on that is the interesting bit.

The question is as asked in the subject line: is \frac{1}{x} a continuous function?

Mathematics majors, or related people like physics majors, already understand the question. Other people will want to know what the question means. This includes people who took a class calculus class, who remember three awful weeks where they had to write ε and δ a lot. The era passed, even if they did not. And people who never took a mathematics class, but like their odds at solving a reasoning problem, can get up to speed on this fast.

The colloquial idea of a “continuous function” is, well. Imagine drawing a curve that represents the function. Can you draw the whole thing without lifting your pencil off the page? That is, no gaps, no jumps? Then it’s continuous. That’s roughly the idea we want to capture by talking about a “continuous function”. It needs some logical rigor to pass as mathematics, though. So here we go.

A function is continuous if, and only if, it’s continuous at every point in the function’s domain. That I start out with that may inspire a particular feeling. That feeling is, “our Game Master grinned ear-to-ear and took out four more dice and a booklet when we said we were sure”.

The red-and-brown ground of a rocky outcropping far above the lush green tree-covered hills below.
A discontinuous ground level. I totally took a weeklong vacation to the Keweenaw Peninsula of upper Michigan in order to get this picture just for my readers. Fun fact: there was also a ham radio event happening on the mountain.

But our best definition of continuity builds on functions at particular points. Which is fair. We can imagine a function that’s continuous in some places but that’s not continuous somewhere else. The ground can be very level level and smooth right up to the cliff. And we have a nice, easy enough, idea of what it is to be continuous at a point.

I’ll get there in a moment. My life will be much easier if I can give you some more vocabulary. They’re all roughly what you might imagine the words meant if I didn’t tell you they were mathematics words.

The first is ‘map’. A function ‘maps’ something in its domain to something in its range. Like if ‘a’ is a point in the domain, ‘f’ maps that point to ‘f(a)’, in its range. Like, if your function is ‘f(x) = x2‘, then f maps 2 to 4. It maps 3 to 9. It maps -2 to 4 again, and that’s all right. There’s no reason you can’t map several things to one thing.

The next is ‘image’. Take something in the domain. It might be a single point. It might be a couple of points. It might be an interval. It might be several intervals. It’s a set, as big or as empty as you like. The `image’ of that set is all the points in the range that any point in the original set gets mapped to. So, again play with f(x) = x2. The image of the interval from 0 to 2 is the interval from 0 to 4. The image of the interval from 3 to 4 is the interval from 9 to 16. The image of the interval from -3 to 1 is the interval from 0 to 9.

That’s as much vocabulary as I need. Thank you for putting up with that. Now I can say what it means to be continuous at a point.

Is a function continuous at a point? Let me call that point ‘a’? It is continuous at ‘a’ we can do this. Take absolutely any open set in the range that contains ‘f(a)’. I’m going to call that open set ‘R’. Is there an open set, that I’ll call ‘D’, inside the domain, that contains ‘a’, and with an image that’s inside ‘R’? ‘D’ doesn’t have to be big. It can be ridiculously tiny; it just has to be an open set. If there always is a D like this, no matter how big or how small ‘R’ is, then ‘f’ is continuous at ‘a’. If there is not — if there’s even just the one exception — then ‘f’ is not continuous at ‘a’.

I realize that’s going back and forth a lot. It’s as good as we can hope for, though. It does really well at capturing things that seem like they should be continuous. And it never rules as not-continuous something that people agree should be continuous. It does label “continuous” some things that seem like they shouldn’t be. We accept this because not labelling continuous stuff as non-continuous is worse.

And all this talk about open sets and images gets a bit abstract. It’s written to cover all kinds of functions on all kinds of things. It’s hard to master, but, if you get it, you’ve got a lot of things. It works for functions on all kinds of domains and ranges. And it doesn’t need very much. You need to have an idea of what an ‘open set’ is, on the domain and range, and that’s all. This is what gives it universality.

But it does mean there’s the challenge figuring out how to start doing anything. If we promise that we’re talking about a function with domain and range of real numbers we can simplify things. This is where that ε and δ talk comes from. But here’s how we can define “continuous at a point” for a function in the special case that its domain and range are both real numbers.

Take any positive ε. Is there is some positive δ, so that, whenever ‘x’ is a number less than δ away from ‘a’, we know that f(x) is less than ε away from f(a)? If there always is, no matter how large or small ε is, then f is continuous at a. If there ever is not, even for a single exceptional ε, then f is not continuous at a.

That definition is tailored for real-valued functions. But that’s enough if you want to answer the original question. Which, you might remember, is, “is 1/x a continuous function”?

That I ask the question, for a function simple and familiar enough a lot of people don’t even need to draw it, may give away what I think the answer is. But what’s interesting is, of course, why the answer. So I’ll leave that for an essay next week.

Reading the Comics, April 5, 2019: The Slow Week Edition


People reading my Reading the Comics post Sunday maybe noticed something. I mean besides my correct, reasonable complaining about the Comics Kingdom redesign. That is that all the comics were from before the 30th of March. That is, none were from the week before the 7th of April. The last full week of March had a lot of comic strips. The first week of April didn’t. So things got bumped a little. Here’s the results. It wasn’t a busy week, not when I filter out the strips that don’t offer much to write about. So now I’m stuck for what to post Thursday.

Jason Poland’s Robbie and Bobby for the 3rd is a Library of Babel comic strip. This is mathematical enough for me. Jorge Luis Borges’s Library is a magnificent representation of some ideas about infinity and probability. I’m surprised to realize I haven’t written an essay specifically about it. I have touched on it, in writing about normal numbers, and about the infinite monkey theorem.

At a tower. Bobby: 'The library of Babel!' Robbie: 'Inside is every book that will ever be written! It may take the rest of our lives to search, but it'll be worth it!' Bobby: 'What? No index?' Robbie: 'The search for meaning has no index.' Bobby (on the phone): 'I just downloaded one.' Robbie: 'It can't have everything. ... Mark Twain vs Frankenstein? Dante in Space? Harry Potter Infinity?' Bobby: 'Yep. All available as e-books too! Wow, Jeff Goldblum does the audio books.' Robbie: 'pfff. Well, forget this place!' (They leave a 'BORING' sign across the library's door.)
Jason Poland’s Robbie and Bobby for the 3rd of April, 2019. I would have sworn that I write more about this strip. But this seems to be the first time I’ve mentioned it since 2017. Well, that and other Robbie and Bobby-based essays are at this link.

The strip explains things well enough. The Library holds every book that will ever be written. In the original story there are some constraints. Particularly, all the books are 410 pages. If you wanted, say, a 600-page book, though, you could find one book with the first 410 pages and another book with the remaining 190 pages and then some filler. The catch, as explained in the story and in the comic strip, is finding them. And there is the problem of finding a ‘correct’ text. Every possible text of the correct length should be in there. So every possible book that might be titled Mark Twain vs Frankenstein, including ones that include neither Mark Twain nor Frankenstein, is there. Which is the one you want to read?

Over a pizza. Reggie: 'Don't let Jughead near the pizza! He always ends up eating half of it!' Jughead, with the cutter: 'Relax! I've divided it into four equal slices! Check it yourself!' Reggie: 'OK, I guess they do look equal.' Archie: 'Except for one thing! There are only three of us!' (Reggie and Archie each have one slice; Jughead has two.)
Henry Scarpelli and Craig Boldman’s Archie for the 4th of April, 2019. Now this strip I’ve written about as recently as October. That appearance, and other Archie strips, are discussed at this link.

Henry Scarpelli and Craig Boldman’s Archie for the 4th features an equal-divisions problem. In principle, it’s easy to divide a pizza (or anything else) equally; that’s what we have fractions for. Making them practical is a bit harder. I do like Jughead’s quick work, though. It’s got the slight-of-hand you expect from stage magic.

Caterpillars in an algebra classroom. On the back of one caterpillar student is a sign, 'Kick^{10} me'.
Scott Hilburn’s The Argyle Sweater for the 4th of April, 2019. And this strip I’ve written about … wait, can I really have gone since early March without mentioning? Huh. Well, so it appears. Essays discussing The Argyle Sweater appear at this link.

Scott Hilburn’s The Argyle Sweater for the 4th takes place in an algebra class. I’m not sure what algebraic principle 7^4 \times 13^6 demonstrates, but it probably came from somewhere. It’s 4,829,210. The exponentials on the blackboard do cue the reader to the real joke, of the sign reading “kick10 me”. I question whether this is really an exponential kicking situation. It seems more like a simple multiplication to me. But it would be harder to make that joke read clearly.

Tony Cochran’s Agnes for the 5th is part of a sequence investigating how magnets work. Agnes and Trout find just … magnet parts inside. This is fair. It’s even mathematics.

Looking over a pile of debris and a hammer on the table. Agnes: 'OK, we smashed a magnet. What do we see?' Trout: 'Uh. Magnet crumbs.' Agnes: 'Me too. I see magnet crumbs.' Trout: 'No gizmos, no gears, no wires. Just dirty black magnet crumbs.' Agnes: 'So what does this tell us about magnet function?' Trout: 'That it's one of God's many mysteries. Let's go eat.'
Tony Cochran’s Agnes for the 5th of April, 2019. And this strip I quite like, but don’t get to discuss enough. My essays featuring Agnes appears at this link.

Thermodynamics classes teach one of the great mathematical physics models. This is about what makes magnets. Magnets are made of … smaller magnets. This seems like question-begging. Ultimately you get down to individual molecules, each of which is very slightly magnetic. When small magnets are lined up in the right way, they can become a strong magnet. When they’re lined up in another way, they can be a weak magnet. Or no magnet at all.

How do they line up? It depends on things, including how the big magnet is made, and how it’s treated. A bit of energy can free molecules to line up, making a stronger magnet out of a weak one. Or it can break up the alignments, turning a strong magnet into a weak one. I’ve had physics instructors explain that you could, in principle, take an iron rod and magnetize it just by hitting it hard enough on the desk. And then demagnetize it by hitting it again. I have never seen one do this, though.

This is more than just a physics model. The mathematics of it is … well, it can be easy enough. A one-dimensional, nearest-neighbor model, lets us describe how materials might turn into magnets or break apart, depending on their temperature. Two- or three-dimensional models, or models that have each small magnet affected by distant neighbors, are harder.


And then there’s the comic strips that didn’t offer much to write about.
Brian Basset’s Red and Rover for the 3rd,
Liniers’s Macanudo for the 5th, Stephen Bentley’s Herb and Jamaal rerun for the 5th, and Gordon Bess’s Redeye rerun for the 5th all idly mention mathematics class, or things brought up in class.

Doug Savage’s Savage Chickens for the 2nd is another more-than-100-percent strip. Richard Thompson’s Richard’s Poor Almanac for the 3rd is a reprint of his Christmas Tree guide including a fir that “no longer inhabits Euclidean space”.

Mike Baldwin’s Cornered for the 31st depicts a common idiom about numbers. Eric the Circle for the 5th, by Rafoliveira, plays on the ∞ symbol.


And that covers the mathematically-themed comic strips from last week. There are more coming, though. I’ll show them on Sunday. Thanks for reading.

Reading the Comics, March 30, 2019: Comics Kingdom is Screwed Up Edition


It doesn’t affect much this batch of comics, as they’re a bunch that all came from GoComics.com. But Comics Kingdom suffered a major redesign of the web site this week, and so it’s lost a lot of functionality. The ability to load your whole comics page at once, for example. Or the ability of archives to work. I’d had the URL for one strip copied down because it mentioned mathematics, albeit in so casual a manner I didn’t mean to write a paragraph about it. Good luck that I didn’t, as that URL now directs to a Spanish translation of a Katzenjammer Kids strip. Why? That’s a good question, and one that deserves an answer.

Anyway, I’m hoping that Comics Kingdom is able to get over their redesign soon. But I know they won’t. There’s never been a web site redesign that lowered functionality and made the page more infuriating to work with that was ever abandoned for the older, working version instead.

Enough about Comics Kingdom. Let me share a couple comic strips from a web site that works, although not as well as it did before its 2018 redesign.

Jim Meddick’s Monty for the 27th is part of a fun storyline. In it Monty and Moondog’s cell phones start texting on their own. It’s presented as the start of an Artificial Intelligence-based singularity, computers transcending human thought and going into business for themselves. This is shown by their working out mathematical truths, starting with arithmetic and going into Boolean algebra. Humans learn arithmetic first and Boolean algebra — logical statements and their combinations — later on, if ever.

Monty: 'Doc! Glad you're here! Our phones started texting without us!' Moondog: 'Now they're doing math!' (The phones text '2 x 2 = 4' and '4^4 = 256'.) Monty: 'They started with 2x2 = 4'. Professor Xemit: 'And now they're swapping advanced Boolean algebraic operations!' Monty: 'Doc ... what's going on?' Xemit: 'A dangerous nexus of AI has formed in your devices! And it must be stopped before it surpasses the combined intellect of all humanity!' Moondog: 'Not sure what that means, but it sounds like some serious data charges.'
Jim Meddick’s Monty for the 27th of March, 2019. Essays discussing anything from Monty should appear at this link.

Computers are certainly able to discover mathematics on their own. Or at least without close guidance; someone still has to write a program to do it. Automated proof finders are a well-established thing, though. They have not, so far as I’ve heard, discovered anything likely to threaten humanity.

Prisoner number 81861^3, talking to prisoner 3757^5: 'Man, this is one *big* prison!'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th of March, 2019. Appearances of The Chuckle Brothers in this line of essays should be gathered here.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th is built on representing huge numbers. 818613 is a big number: 548,568,842,280,381. Even bigger is 37575: it’s 748,524,423,279,410,560. It’s silly to imagine needing an identification number that large. But it’s also a remarkable coincidence that both prisoners here have numbers that can be represented with no more than six digits. There aren’t so many 15-digit numbers that could be represented with as few as six digits. But then it would be an absurdly large prison if it “only” had 818,613 prisoners in it. That seems like the joke would have been harder to recognize, though.

Anthropomorphic numeral 9, looking at temperatures on the Weather Channel. Caption: 'Nate wanted to work at the Weather Channel but didn't have a degree.'
Mark Parisi’s Off The Mark for the 28th of March, 2019. Essays inspired by something mentioned in Off The Mark should be gathered here.

Mark Parisi’s Off The Mark for the 28th is sort of the anthropomorphic numerals joke for the week. It’s also a joke for my friend with the meteorology degree, who I think doesn’t actually read these posts. Well, he probably got the comic forwarded to him anyway.

Daniel Beyer’s Long Story Short for the 29th is another prison joke. I’m not sure if someone at Comic Strip Master Command was worried about something. But a scrawl of mathematics is used as icon of skills learned in prison.

Prisoner, to the other in his cell: 'You learn new skills after being here awhile.' On his wall are several lines of mathematical scrawls.
Daniel Beyer’s Long Story Short for the 29th of March, 2019. Essays discussing topics raised by Long Story Short should be here. There aren’t many, yet.

Mathematics has the reputation of being a subject someone can still do useful work in while in prison. Maybe even do more work, as it seems to offer the prospect of undistracted time to think. And there are examples of mathematicians doing noteworthy work while imprisoned. Bertrand Russell wrote the Introduction To Mathematical Philosophy while jailed for protesting the First World War. André Weil advanced his work in arithmetic geometry while in prison for resisting service in the Second World War. Évariste Galois spent six months in prison shortly before the end of his life, and used some of the time to work on the theory of equations for which we still remember him. I would not recommend prison as a way to advance one’s mathematical research. But it’s something which could happen.

Colin, showing a card: 'Check it out, Dad! This is one of the most powerful Spen-Yo-Do Cards, the Crystal Conquerer! It's worth 2500 power units, has a multiplier ratio of 10, and a destruction ratio of 2 over 15! It can only be defeated by a mage card of equal value or greater!' Dad: 'That's fine, Colin, but I came to check your homework. C'mon, what's 9 divided by 3?' Colin: 'How am I supposed to know?!'
Terry LaBan and Patty LaBan’s Edge City rerun for the 30th of March, 2019. It originally ran in 2004; I can’t say whether it ran the 30th of March then. And topics raised by Edge City should be discussed at this link.

Terry LaBan and Patty LaBan’s Edge City for the 30th showcases the motivation problem. Colin, like many people, is easily able to do complicated algorithms to do something he likes doing. Arithmetic drills, though, not so much. This is why we end up writing story problems with dubious amounts of story in them.


And I don’t want to devote too much space to this. But Brian Fies’s The Last Mechanical Monster for the 29th included the lead character, the Mad Scientist, working out the numbers of the Fibonacci sequence as a way to keep his mind going. The strip is a rerun and I discussed it when it first ran on GoComics.


There were quite a lot of mathematically-themed comic strips the week of the 24th of March. I’ll get to the actual strips of the past week soon, at this link. Also if anyone knows a way to get the old Comics Kingdom back please let me know.

Reading the Comics, March 26, 2019: March 26, 2019 Edition


And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.

Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.

Caption: Eric on an inclined plane. It shows a circle on a right triangle, with the incline of the angle labelled 'x'. The force of gravity is pointing vertically down, labelled (Eric)(g). The force parallel to the incline is labelled (Eric)(g)sin(x); the force perpendicular to the incline is labelled (Eric)(g)cos(x).
Eric the Circle, by Kyle, for the 26th of March, 2019. Essays inspired at all by Eric the Circle are at this link.

Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.

The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.

The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.

Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.

Ruthie, on the phone: 'Homework hot line? On the Same/Different page of our workbook there are two circles like this. They're called Venn diagrams and I wanna know who this Venn person is. And if I put two squares together, can we call it the Ruthie diagram, and how much money do I get for that? ... Huh? Well, I'll wait here 'til you find somebody who DOES know!'
Rick Detorie’s One Big Happy for the 26th of March, 2019. This is a rerun from … 2007, I want to say? There are two separate feeds, one of current and one of several-years-old, strips on the web. Essays including One Big Happy, current or years-old reruns, should be at this link.

John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.

Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.

Although we always start Venn diagrams off with circles, they don’t have to be. Circles are good shapes if you have two or three sets. It gets hard to represent all the possible intersections with four circles, though. This is when you start seeing weirder shapes. Wikipedia offers some pictures of Venn diagrams for four, five, and six sets. Meanwhile Mathworld has illustrations for seven- and eleven-set Venn diagrams. At this point, the diagrams are more for aesthetic value than to clarify anything, though. You could draw them with squares. Some people already do. Euler diagrams, particularly, are often squares, sometimes with rounded corners.

Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.

An illustration of an abacus. Caption: 'No matter what the category, you'll usually find me in the upper 99%.'
Ashleigh Brilliant’s Pot-Shots for the 26th of March, 2019. The strip originally appeared sometime in 1979. Essays discussing anything from Pot-Shots should appear at this link.

Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.

It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.

Snow, cat, to a kitten: '1 + 1 = 2 ... unless it's spring.' (Looking at a bird's nest with five eggs.) 'Then 1 + 1 = 5.'
T Shepherd’s Snow Sez for the 26th of March, 2019. Essays including an appearance of Essays inspired at all by Snow Sez should be gathered at this link. They will be, anyway; this is a new tag.

T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.


There were a lot of mathematically-themed comic strips last week. There’ll be another essay soon, and it should appear at this link. And then there’s always Sunday, as long as I stay ahead of deadline. I am never ahead of deadline.

How March 2019 Treated My Mathematics Blog


So, I did something dangerous in March. I try not paying attention to the day-to-day statistics. But there’s a little graphic that shows the last several hours of views. And it’s easy to see while doing administrative stuff. And I happened to see a surge in readers. I couldn’t find an obvious cause for it. There’s some data available about where readers are coming from, but not much. I never did figure out why several hundred people wanted to read my mathematics blog all at once. But it did make me go back and check and re-check what my readership was like. And that’s dangerous stuff, especially since I had a quite variable month. Like, the day before a 113-views day there were 19 views. And that wasn’t the least-read day of the month. Watching the readership statistics, day-by-day, is a terrible habit. It’s even worse for a blog like this with relatively low, irregular readership volume.

So that’s what I did to drive myself mad this past month. And how well did that work?

A box chart showing, for the last several months, fairly uniform numbers of views and visitors. For march, 1,391 views from 954 visitors, 1.46 views per visitor, and 14 posts published.
Yes but I’ve learned how to fiddle with the date range that these figures show so next month I should have an amazing box plot to show you.

For all those slow days I had an uptick in pages viewed: 1,391 in March, up from February’s 1,275 and January’s 1,375. But is that significant? Not really; there were 45 views per day on average in March, 46 in February, and 44 in January. So this is all keeping to the level I’ve been at since about October 2018. There were 14 posts published in March, up from February’s 11 and January’s 12.

The number of unique visitors was up, noticeably: 954 in March. So I’m still holding at only one thousand-visitor month so far. (March 2018 saw 999 visitors, though. It almost makes me think there’s some event or other in the middle of March which attracts people to pop mathematics blogs.) Well, February 2019 had 835 unique visitors, and January 856, and I’d been around 850 per month going back through November 2018. There’s a level there.

Reader engagement is a more erratic thing. One measure was positive, as I see things: there were 97 likes given to my writing in March. That’s the greatest number in twelve months. February only saw 44 likes; January, 63. But that’s a surprisingly variable measure. But the other side of things? Comments? There were four in all March. Comments are always erratic, yes. February had 10 comments, and January 22, and there’ve been as many as 60 in the past year. But four comments? If I haven’t missed anything I haven’t had a month that sparse since November 2012, which, just … wow.

I can explain some of this. I’ve been doing a lot of Reading the Comics posts, which are fun to write but have almost nothing to respond to. I’ve gotten some comments on Twitter. This has to be the first month I’ve seen more comments on Twitter than on WordPress. And I haven’t been in the midst of an A-To-Z or similar themed event that’s really open to comments. Still, mm. I should do more things that are open to comments, but how would I learn what those are?

For all that people read without commenting, they did still read things. The most popular posts in March were:

So, two perennials, and a bunch of comics. I’m curious why the 2016 Pi Day comics was so much more popular than the 2019. There were more strips for the 2016 version, but the 2015 Pi Day comics were even more robust than that. Also now that I’m reminded I’d had a Barely Mathematics Edition I realize I should have named Sunday’s Reading the Comics, with all those Bear With Me strips, the Bearly Mathematics Edition. Maybe I’ll be lucky enough to get to use that one sometime.

59 countries sent me readers in March. That’s down from 73 in February and equal to the 59 in January. There were 17 single-reader countries, down from February’s 20 and from January’s 19. Here’s where readers were:

It's just the one view from China for the month, which isn't much per capita, I admit. But it's still, like, fourteen times more readers than I usually see from China. The real curious thing is that it shows 51 views from Sweden, while six from Norway and from Denmark, and four from Finland. I'd have expected those countries to be a bit closer together.
A Mecator-style map of the world, showing the United States in dark red; Canada, Mexico, Australia, and western and central Europe, plus Russia, in light red, and a surprising number of readers from Pakistan through China. Also several in Colombia, Ecuador, Peru, and Brazil.
Country Readers
United States 902
United Kingdom 65
Sweden 51
Canada 45
Philippines 45
India 40
France 21
Germany 17
Brazil 15
Singapore 14
South Africa 11
Nepal 10
Pakistan 10
Spain 10
Australia 9
Slovenia 9
Netherlands 8
Turkey 8
Mexico 7
Denmark 6
Italy 6
Norway 6
Ireland 5
Thailand 5
Austria 4
Finland 4
Malaysia 4
South Korea 4
American Samoa 3
Belgium 3
Hungary 3
Sri Lanka 3
Switzerland 3
Croatia 2
Ecuador 2
Hong Kong SAR China 2
Israel 2
Latvia 2
Poland 2
Portugal 2
United Arab Emirates 2
Vietnam 2
Bahrain 1
Bangladesh 1
China 1
Colombia 1
Czech Republic 1 (*)
Ethiopia 1
Indonesia 1
Jamaica 1
Japan 1 (***)
Jordan 1 (***)
Lithuania 1 (**)
Myanmar (Burma) 1
Peru 1 (*)
Puerto Rico 1 (*)
Russia 1
Saudi Arabia 1
Slovakia 1

Czech, Peru, and Puerto Rico have sent a single page view two months running now. Lithuania’s been a single view a month for three months. Japan and Jordan have four-month streaks going.

In 37 posts through the start of April I’ve put up 36,734 in 2019. This is 15,984 words in March. My average post length this year has been 993 words, up from the 902 at the end of February and even the 966 at the end of January. Hm. Well, that’s what fourteen posts at 1,142 words per post will do. I’ve reached 52 comments on the whole year, an average of 1.4 comments per posting. That’s down from the start of March’s 1.5 comments per post. There’ve been 182 total likes this year to date, for an average of 4.9 likes per post. That’s an increase, at least. At the start of March there had been an average 4.3 likes per post.

The month started with my having made 1,239 posts in total. They’ve attracted in total 76,956 page views from an acknowledged 38,905 unique visitors.

If you’d like to not miss any posts, you can add my work to your RSS reader, using this link. Or you can use the “Follow Nebusresearch” button in the upper right corner of the page. And I am on Twitter as @nebusj, so it should be easy enough to spot me somewhere. Thank you for being around.

Reading the Comics, March 25, 2019: Bear Edition


I’m again stepping slightly outside the normal chronological progression of these posts. This is to let me share several days’ worth of Bob Scott’s Bear With Me. It’ll make for cleaner thematic breaks in the week.

Wayno and Piraro’s Bizarro for the 25th is a precision joke. That a proposal might be more than half-baked is reasonable enough. Pinning down its baked-ness to one part in a thousand? Nice gentle absurdity. The panel does showcase two things that connote accuracy, though. Percentages read as confident knowledge: to say something is half-done seems somehow a more uncertain thing than to say something is 50 percent done. And decimal places suggest precision also.

Boss at the Precision Calibration Corporation, chewing out a subordinate: 'Another one of your 64.3 percent-baked proposals, Wilkins?'
Wayno and Piraro’s Bizarro for the 25th of March, 2019. Essays that discuss Bizarro are gathered at this link.

There are different, but not wholly separate, things to value in a measurement. Precision seems like the desirable one. It looks like superior knowledge. But there are other and more important things. One is repeatability: if you measure the same thing again, do you get approximately the same number? If the boss re-read the proposal and judged it to be 24.7 percent baked, would we feel confident in the numbers? And another is whether the measurement corresponds to what we would like to know. The diameter of a person’s head can be measured precisely. And repeatably; the number won’t change very much day to day. But suppose what we really care to know is the person’s intelligence. Does this precision and repeatability matter, given how much intelligence varies for even people of about the same head size?

Amanda, on the phone: 'Hey, Gramps, what's up?' Grandpa: 'I'm watching Wheel of Fortune.' 'How's that going?' 'There's a group of purses on here totalling $2900!! I buy a seven-dollar wallet, and it lasts me ten years!' Amanda ;'So these purses will last them ... [ math ] over four thousand years [*] by your math. That's value.' Grandpa: 'That's almost one thousand dollars per bag, Amanda!' [*]: I think My degree is not in math.
Amanda El-Dweek’s Amanda the Great for the 25th of March, 2019. I had expected this to be a new tag, but no. I’ve mentioned Amanda the Great before, in an essay at this link. Still, it has been a while.

Amanda El-Dweek’s Amanda the Great for the 25th starts from someone watching a game show. That’s a great way to find casual mathematics problems. Often these involve probability questions, and expectation values. That is, what would be the wisest course if you could play this game thousands or millions or billions of times?

This one dodges that, though, as the strip gets to Gramps shocked by the high price of designer women’s purses. And it features a great bit of mental arithmetic on Amanda’s part. A $2900 purse is more than four hundred times the cost of a $7 wallet. The way I spot that is noticing that 29 is awfully close to 28, but more than it. And 2800 divided by 7 is easy: it’s a hundred times 28 divided by 7. Grant the supposition that cost scales with the wallet or purse’s lifespan. Amanda nails it. If we pretend that more precision would help, she’d be forecasting a nearly 4,143-year lifespan for the purses. I admit that seems to me like an over-engineered purse.

Molly, reading homework: 'Farmer Smith has 49 acres of blueberries, and each acre has 179 pounds of blueberries, so how many pounds of blueberries will Farmer Smith harvest?' Bear: '8,771 pounds. That would be 1,710,355 blueberries.' Molly: 'Wow, Bear! How do you know that?' Bear: 'How can you NOT? Blueberries are amazing!'
Bob Scott’s Bear With Me for the 25th of March, 2019. This is also not a new tag, although I thought it might be. Bear With Me essays should appear at this link.

Bob Scott’s Bear With Me for the 25th starts a string of word problem jokes. I like them, not just for liking Bear. I also like the comic motif of the character who’s ordinarily a buffoon but has narrow areas of extreme competence. There was a fun bit on one episode of The Mary Tyler Moore Show in which Ted Baxter was able to do some complex arithmetic in his head just by imagining there was a dollar sign in front of it, for an example close to this one.

Molly: 'Farmer Brown has 3 crates of carrots and each crate holds 12 pounds of carrots. How many carrots does Farmer Brown have?' Bear: 'I have no idea.' Molly: 'But you solved yesterday's math problem easily.' Bear: 'That problem was about blueberries and this one is about carrots.' Molly: 'So?' Bear: 'I don't like carrots.'
Bob Scott’s Bear With Me for the 26th of March, 2019. The comic strip Bear With Me started its run as Molly And The Bear. The title shifted as, I think, Scott realized the most reliably interesting interactions were between Bear and Molly’s Dad. But Molly isn’t out of the strip either.

Bob Scott’s Bear With Me for the 26th Bear’s arithmetic skills vary with his interest in solving the problem. This is comically exaggerated, yes. It’s something I think is basically true though. I’ve noticed I have an easier time solving problems I’m curious about, for example. I suspect most of us think the sae way, or at least expect people to do so. If we din’t, we wouldn’t worry so about motivating the solving of problems. Molly only has story problems about farmers gathering things because it’s supposed a person would want to know, given this setup, what they might expect to gather.

Molly: 'Bear, what's 4,400 blueberries divided by 2?' Bear: 'What?' Molly: 'Divide 4,400 blueberries by 2.' Bear: 'Why would I want to divide them?!' Molly: 'So you can share them with me!' Bear: 'But you can just go to the store and buy your own. And wow, what a great idea!! Let's go to the store now!!' Molly: 'I hate math.'
Bob Scott’s Bear With Me for the 27th of March, 2019. Peculiar thing is I don’t have any strips tagged as Molly And The Bear, even though the strip runs a lot of reruns and some of those have mathematical content. If I’m searching my own archives correctly that’s just because I didn’t find any old comics with enough mathematical content to discuss here. Curious, that. You’d think Molly would do more word problems, especially given the retro aesthetic Bob Scott’s chosen for the comic strip.

Bob Scott’s Bear With Me for the 27th shows a hazard in making a story too real-world: someone might want to bring in solutions that fall outside the course material. I don’t think that happens much in mathematics. My love teaches philosophy, though, and there is a streak of students who will not accept the premises of a thought experiment. They’ll insist on disproving that the experiment could happen, or stand on solutions that involve breaking the selection of options.


Last week was busy for mathematically-themed comic strips. I’ll have more Reading the Comics posts, at this link, in a couple days. Thanks as always for reading any of these.

How To Find Logarithms Without Using Powerful Computers


I got to remembering an old sequence of mine, and wanted to share it for my current audience. A couple years ago I read a 1949-published book about numerical computing. And it addressed a problem I knew existed but hadn’t put much thought into. That is, how to calculate the logarithm of a number? Logarithms … well, we maybe don’t need them so much now. But they were indispensable for computing for a very long time. They turn the difficult work of multiplication and division into the easier work of addition and subtraction. They turn the really hard work of exponentiation into the easier work of multiplication. So they’re great to have. But how to get them? And, particularly, how to get them if you have a computing device that’s able to do work, but not very much work?

Machines That Think About Logarithms sets out the question, including mentioning Edmund Callis Berkeley’s book that got me started on this. And some talk about the kinds of logarithms and why we use each of them.

Machines That Do Something About Logarithms sets out some principles. These are all things that are generically true about logarithms, including about calculating logarithms. They’re just the principles that were put into clever play by Harvard’s IBM Automatic Sequence-Controlled Calculator in the 1940s.

Machines That Give You Logarithms explains how to use those tools. And lays out how to get the base-ten logarithm for most numbers that you would like with a tiny bit of computing work. I showed off an example of getting the logarithm of 47.2286 using only three divisions, four additions, and a little bit of looking up stuff.

Without Machines That Think About Logarithms closes out the cycle. One catch with the algorithm described is that you need to work out some logarithms ahead of time and have them on hand, ready to look up. They’re not ones that you care about particularly for any problem, but they make it easier to find the logarithm you do want. This essay talks about which logarithms to calculate, in order to get the most accurate results for the logarithm you want, using the least custom work possible.

And there we go. Logarithms are still indispensable for mathematical work, although I realize not so much because we ever care what the logarithm of 47.2286 or any other arbitrary number is. Logarithms have some nice analytic properties, though, and they make other work easier to do. So they’re still in use, but for different problems.

Reading the Comics, March 23, 2019: March 23, 2019 Edition


I didn’t cover quite all of last week’s mathematics comics with Sunday’s essay. There were a handful that all ran on Saturday. And, as has become tradition, I’ll also list a couple that didn’t rate a couple paragraphs.

Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd has a neat variation on story problems. Zoe’s given the assignment to make her own. I don’t remember getting this as homework, in elementary school, but it’s hard to see why I wouldn’t. It’s a great exercise: not just set up an arithmetic problem to solve, but a reason one would want to solve it.

Composing problems is a challenge. It’s a skill, and you might be surprised that when I was in grad school we didn’t get much training in it. We were just taken to be naturally aware of how to identify a skill one wanted to test, and to design a question that would mostly test that skill, and to write it out in a question that challenged students to identify what they were to do and how to do it, and why they might want to do it. But as a grad student I wasn’t being prepared to teach elementary school students, just undergraduates.

Dad: 'Homework?' Zoe: 'Yeah, math. Our teacher is having us write our own story problem.' Dad: 'What have you got?' Zoe: 'If Hammie picks his nose at the rate of five boogers an hour ... ' Hammie: 'Ooh! Put me on a jet ski!'
Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd of March, 2019. Essays inspired by some Baby Blues strip appear at this link.

Mastroianni and Hart’s B.C. for the 23rd is a joke in the funny-definition category, this for “chaos theory”. Chaos theory formed as a mathematical field in the 60s and 70s, and it got popular alongside the fractal boom in the 80s. The field can be traced back to the 1890s, though, which is astounding. There was no way in the 1890s to do the millions of calculations needed to visualize any good chaos-theory problem. They had to develop results entirely by thinking.

Wiley’s definition is fine enough about certain systems being unpredictable. Wiley calls them “advanced”, although they don’t need to be that advanced. A compound pendulum — a solid rod that swings on the end of another swinging rod — can be chaotic. You can call that “advanced” if you want but then people are going to ask if you’ve had your mind blown by this post-singularity invention, the “screw”.

Cute Chick, reading Wiley's Dictionary: 'Chaos Theory. Mathematical principle that advanced systems are wholly unpredictable due to the introduction of random tweets.'
Mastroianni and Hart’s B.C. for the 23rd of March, 2019. Appearances here inspired by B.C., current syndication or 1960s reprints on GoComics, are at this link. Yeah, the character here is named ‘Cute Chick’ because that was funny when the comic started in 1958 and it can’t be updated for some reason?

What makes for chaos is not randomness. Anyone knows the random is unpredictable in detail. That’s no insight. What’s exciting is when something’s unpredictable but deterministic. Here it’s useful to think of continental divides. These are the imaginary curves which mark the difference in where water runs. Pour a cup of water on one side of the line, and if it doesn’t evaporate, it eventually flows to the Pacific Ocean. Pour the cup of water on the other side, it eventually flows to the Atlantic Ocean. These divides are often wriggly things. Water may mostly flow downhill, but it has to go around a lot of hills.

So pour the water on that line. Where does it go? There’s no unpredictability in it. The water on one side of the line goes to one ocean, the water on the other side, to the other ocean. But where is the boundary? And that can be so wriggly, so crumpled up on itself, so twisted, that there’s no meaningfully saying. There’s just this zone where the Pacific Basin and the Atlantic Basin merge into one another. Any drop of water, however tiny, dropped in this zone lands on both sides. And that is chaos.

Neatly for my purposes there’s even a mountain at a great example of this boundary. Triple Divide Peak, in Montana, rests on the divides between the Atlantic and the Pacific basins, and also on the divide between the Atlantic and the Arctic oceans. (If one interprets the Hudson Bay as connecting to the Arctic rather than the Atlantic Ocean, anyway. If one takes Hudson Bay to be on the Atlantic Ocean, then Snow Dome, Alberta/British Columbia, is the triple point.) There’s a spot on this mountain (or the other one) where a spilled cup of water could go to any of three oceans.

There's at least a 99.9 percent chance that in a group of 70 people at least two will share a birthday. The Pentagon had to ban staff from playing Pokemon Go in the building. Picasso created more than 13,500 paintings and designs, 10,000 prints and engravings, 34,000 book illustrations, and 300 sculptures and ceramics --- making him one of the world's most prolific artists.
John Graziano’s Ripley’s Believe It Or Not for the 23rd of March, 2019. The various pieces of mathematics trivia featured in Ripley’s Believe It Or Not get shown off at this link. I still think it’s weird to write Graziano’s Ripley’s. Anyway, with 57,800 listed pieces of art here Picasso is only credited as “one of” the world’s most prolific artists? Who’s out there with 57,802 pieces?

John Graziano’s Ripley’s Believe It Or Not for the 23rd mentions one of those beloved bits of mathematics trivia, the birthday problem. That’s finding the probability that no two people in a group of some particular size will share a birthday. Or, equivalently, the probability that at least two people share some birthday. That’s not a specific day, mind you, just that some two people share a birthday. The version that usually draws attention is the relatively low number of people needed to get a 50% chance there’s some birthday pair. I haven’t seen the probability of 70 people having at least one birthday pair before. 99.9 percent seems plausible enough.

The birthday problem usually gets calculated something like this: Grant that one person has a birthday. That’s one day out of either 365 or 366, depending on whether we consider leap days. Consider a second person. There are 364 out of 365 chances that this person’s birthday is not the same as the first person’s. (Or 365 out of 366 chances. Doesn’t make a real difference.) Consider a third person. There are 363 out of 365 chances that this person’s birthday is going to be neither the first nor the second person’s. So the chance that all three have different birthdays is \frac{364}{365} \cdot \frac{363}{365} . Consider the fourth person. That person has 362 out of 365 chances to have a birthday none of the first three have claimed. So the chance that all four have different birthdays is \frac{364}{365} \cdot \frac{363}{365} \cdot \frac{362}{365} . And so on. The chance that at least two people share a birthday is 1 minus the chance that no two people share a birthday.

As always happens there are some things being assumed here. Whether these probability calculations are right depends on those assumptions. The first assumption being made is independence: that no one person’s birthday affects when another person’s is likely to be. Obvious, you say? What if we have twins in the room? What if we’re talking about the birthday problem at a convention of twins and triplets? Or people who enjoyed the minor renown of being their city’s First Babies of the Year? (If you ever don’t like the result of a probability question, ask about the independence of events. Mathematicians like to assume independence, because it makes a lot of work easier. But assuming isn’t the same thing as having it.)

The second assumption is that birthdates are uniformly distributed. That is, that a person picked from a room is no more likely to be born the 13th of February than they are the 24th of September. And that is not quite so. September births are (in the United States) slightly more likely than other months, for example, which suggests certain activities going on around New Year’s. Across all months (again in the United States) birthdates of the 13th are slightly less likely than other days of the month. I imagine this has to be accounted for by people who are able to select a due date by inducing delivery. (Again if you need to attack a probability question you don’t like, ask about the uniformity of whatever random thing is in place. Mathematicians like to assume uniform randomness, because it akes a lot of work easier. But assuming it isn’t the same as proving it.)

Do these differences mess up the birthday problem results? Probably not that much. We are talking about slight variations from uniform distribution. But I’ll be watching Ripley’s to see if it says anything about births being more common in September, or less common on 13ths.


And now the comics I didn’t find worth discussing. They’re all reruns, it happens. Morrie Turner’s Wee Pals rerun for the 20th just mentions mathematics class. That could be any class that has tests coming up, though. Percy Crosby’s Skippy for the 21st is not quite the anthropomorphic numerals jokes for the week. It’s getting around that territory, though, as Skippy claims to have the manifestation of a zero. Bill Rechin’s Crock for the 22nd is a “pick any number” joke. I discussed as much as I could think of about this when it last appeared, in May of 2018. Also I’m surprised that Crock is rerunning strips that quickly now. It has, in principle, decades of strips to draw from.


And that finishes my mathematical comics review for last week. I’ll start posting essays about next week’s comics here, most likely on Sunday, when I’m ready.

Reading the Comics, March 19, 2019: Average Edition


This time around, averages seem important.

Mark Anderson’s Andertoons for the 18th is the Mark Anderson’s Andertoons for the week. This features the kids learning some of the commonest terms in descriptive statistics. And, as Wavehead says, the similarity of names doesn’t help sorting them out. Each is a kind of average. “Mean” usually is the arithmetic mean, or the thing everyone including statisticians calls “average”. “Median” is the middle-most value, the one that half the data is less than and half the data is greater than. “Mode” is the most common value. In “normally distributed” data, these three quantities are all the same. In data gathered from real-world measurements, these are typically pretty close to one another. It’s very easy for real-world quantities to be normally distributed. The exceptions are usually when there are some weird disparities, like a cluster of abnormally high-valued (or low-valued) results. Or if there are very few data points.

On the blackboard the teacher's written Median, Mode, and Mean, with a bunch of numbers from 3 through 15. Wavehead: 'I know they're all subtly different, but I have to say, the alliteration doesn't help.'
Mark Anderson’s Andertoons for the 18th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

The word “mean” derives from the Old French “meien”, that is, “middle, means”. And that itself traces to the Late Latin “medianus”, and the Latin “medius”. That traces back to the Proto-Indo-European “medhyo”, meaning “middle”. That’s probably what you might expect, especially considering that the mean of a set of data is, if the data is not doing anything weird, likely close to the middle of the set. The term appeared in English in the middle 15th century.

The word “median”, meanwhile, follows a completely different path. That one traces to the Middle French “médian”, which traces to the Late Latin “medianus” and Latin “medius” and Proto-Indo-European “medhyo”. This appeared as a mathematical term in the late 19th century; Etymology Online claims 1883, but doesn’t give a manuscript citation.

The word “mode”, meanwhile, follows a completely different path. This one traces to the Old French “mode”, itself from the Latin “modus”, meaning the measure or melody or style. We get from music to common values by way of the “style” meaning. Think of something being done “á la mode”, that is, “in the [ fashionable or popular ] style”. I haven’t dug up a citation about when this word entered the mathematical parlance.

So “mean” and “median” don’t have much chance to do anything but alliterate. “Mode” is coincidence here. I agree, it might be nice if we spread out the words a little more.

Edison, pointing to a checkerboard: 'So Grandpa if you put one cookie on the first square, two on the second, four on the next, then eight, and you keep doubling them until you fill all 64 squares do you know what you'll end up with?' Grandpa: 'A stomachache for a month?'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of March, 2019. I’ve been talking about this strip a lot lately, it seems to me. Essays where I do discuss Edison Lee are at this link.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th has Edison introduce a sequence to his grandfather. Doubling the number of things for each square of a checkerboard is an ancient thought experiment. The notion, with grains of wheat rather than cookies, seems to be first recorded in 1256 in a book by the scholar Ibn Khallikan. One story has it that the inventor of chess requested from the ruler that many grains of wheat as reward for inventing the game.

If we followed Edison Lee’s doubling through all 64 squares we’d have, in total, need for 263-1 or 18,446,744,073,709,551,615 cookies. You can see why the inventor of chess didn’t get that reward, however popular the game was. It stands as a good display of how exponential growth eventually gets to be just that intimidatingly big.

Edison, like many a young nerd, is trying to stagger his grandfather with the enormity of this. I don’t know that it would work. Grandpa ponders eating all that many cookies, since he’s a comical glutton. I’d estimate eating all that many cookies, at the rate of one a second, eight hours a day, to take something like eighteen billion centuries. If I’m wrong? It doesn’t matter. It’s a while. But is that any more staggering than imagining a task that takes a mere ten thousand centuries to finish?

Toby, looking at his homework, and a calculator, and a textbook: '... Wow. Crazy ... Huh. How about that? ... Am I stupid if math *always* has a surprise ending?'
Greg Cravens’s The Buckets for the 19th of March, 2019. It also seems like I discuss The Buckets more these days, as seen at this link.

Greg Cravens’s The Buckets for the 19th sees Toby surprised by his mathematics homework. He’s surprised by how it turned out. I know the feeling. Everyone who does mathematics enough finds that. Surprise is one of the delights of mathematics. I had a great surprise last month, with a triangle theorem. Thomas Hobbes, the philosopher/theologian, entered his frustrating sideline of mathematics when he found the Pythagorean Theorem surprising.

Mathematics is, to an extent, about finding interesting true statements. What makes something interesting? That depends on the person surprised, certainly. A good guideline is probably “something not obvious before you’ve heard it, thatlooks inevitable after you have”. That is, a surprise. Learning mathematics probably has to be steadily surprising, and that’s good, because this kind of surprise is fun.

If it’s always a surprise there might be trouble. If you’re doing similar kinds of problems you should start to see them as pretty similar, and have a fair idea what the answers should be. So, from what Toby has said so far … I wouldn’t call him stupid. At most, just inexperienced.

Caption: Eric had good feelings about his date. It turned out, contrary to what he first thought ... [ Venn Diagram picture of two circles with a good amount of overlap ] ... They actually had quite a lot in common.
Eric the Circle for the 19th of March, 2019, this one by Janka. This and other essays with Eric the Circle, by any artist, should be at this link.

Eric the Circle for the 19th, by Janka, is the Venn Diagram joke for the week. Properly any Venn Diagram with two properties has an overlap like this. We’re supposed to place items in both circles, and in the intersection, to reflect how much overlap there is. Using the sizes of each circle to reflect the sizes of both sets, and the size of the overlap to represent the size of the intersection, is probably inevitable. The shorthand calls on our geometric intuition to convey information, anyway.

She: 'What time is it?' He: 'The clock in the hall says 7:56, the oven clock says 8:02, the DVD clock says 8:07, and the bed table clock says 8:13.' She: 'Which one's right?' He: 'I dunno.' She: 'Let's ee. Four clocks ... carry the one ... divided by four ... ' He: 'Whenever we want to know what time it is we have to do algebra.' She: 'We better hurry, it's 8:05 and five-eighths!'
Tony Murphy’s It’s All About You for the 19th of March, 2019. And the occasional time I discuss something from It’s All About You are here.

Tony Murphy’s It’s All About You for the 19th has a bunch of things going on. The punch line calls “algebra” what’s really a statistics problem, calculating the arithmetic mean of four results. The work done is basic arithmetic. But making work seem like a more onerous task is a good bit of comic exaggeration, and algebra connotes something harder than arithmetic. But Murphy exaggerates with restraint: the characters don’t rate this as calculus.

Then there’s what they’re doing at all. Given four clocks, what’s the correct time? The couple tries averaging them. Why should anyone expect that to work?

There’s reason to suppose this might work. We can suppose all the clocks are close to the correct time. If they weren’t, they would get re-set, or not looked at anymore. A clock is probably more likely to be a little wrong than a lot wrong. You’d let a clock that was two minutes off go about its business, in a way you wouldn’t let a clock that was three hours and 42 minutes off. A clock is probably as likely to show a time two minutes too early as it is two minutes too late. This all suggests that the clock errors are normally distributed, or something like that. So the error of the arithmetic mean of a bunch of clock measurements we can expect to be zero. Or close to zero, anyway.

There’s reasons this might not work. For example, a clock might systematically run late. My mantle clock, for example, usually drifts about a minute slow over the course of the week it takes to wind. Or the clock might be deliberately set wrong: it’s not unusual to set an alarm clock to five or ten or fifteen minutes ahead of the true time, to encourage people to think it’s later than it really is and they should hurry up. Similarly with watches, if their times aren’t set by Internet-connected device. I don’t know whether it’s possible to set a smart watch to be deliberately five minutes fast, or something like that. I’d imagine it should be possible, but also that the people programming watches don’t see why someone might want to set their clock to the wrong time. From January to March 2018, famously, an electrical grid conflict caused certain European clocks to lose around six minutes. The reasons for this are complicated and technical, and anyway The Doctor sorted it out. But that sort of systematic problem, causing all the clocks to be wrong in the same way, will foil this take-the-average scheme.

Murphy’s not thinking of that, not least because this comic’s a rerun from 2009. He was making a joke, going for the funnier-sounding “it’s 8:03 and five-eights” instead of the time implied by the average, 8:04 and a half. That’s all right. It’s a comic strip. Being amusing is what counts.


There were just enough mathematically-themed comic strips this past week for one more post. When that is ready, it should be at this link. I’ll likely post it Tuesday.

Reading the Comics, March 13, 2019: Ziggy Rerun Scandal Edition


I do not know that the Ziggy printed here is a rerun. I don’t seem to have mentioned it in previous Reading the Comics posts, but that isn’t definite. How much mathematical content a comic strip needs to rate a mention depends on many things, and a strip that seems too slight one week might inspire me another. I’ll explain why I’ve started to get suspicious of the quite humanoid figure.

Tom II Wilson’s Ziggy for the 12th is framed around weather forecasts. It’s the probability question people encounter most often, unless they’re trying to outsmart the contestants on Let’s Make A Deal. (And many games on The Price Is Right, too.) Many people have complained about not knowing the meaning of a “50% chance of rain” for a day. If I understand it rightly, it means, when conditions have been like this in the recorded past, it’s rained about 50% of the time. I’m open to correction from meteorologists and it just occurred to me I know one. Mm.

Few people ask about the probability a forecast is correct. In some ways it’s an unanswerable question. To say there is a one-in-six chance a fairly thrown die will turn up a ‘1’ is not wrong just because it’s rolled a ‘1’ eight times out of the last ten. But it does seem like a forecast such as this should include a sense of confidence, how sure the forecaster is that the current weather is all that much like earlier times.

Weather forecaster on the TV Ziggy watches: 'Tomorrow's weather, there's a 50% chance of rain, and a 50% chance I'm even right about the 50%!!'
Tom II Wilson’s Ziggy for the 12th of March, 2019. When I do find a mathematical context to discuss Ziggy the results should appear at this link. Speculating about the comic’s rerun schedule isn’t really my business.

I’m not sure how much of the joke is meant to be the repetition of “50% chance”. The joke might be meant to say that if he’s got a 50% chance of being wrong, then, isn’t the 50% chance of rain “correctly” a 50% chance of not-rain … which is the same chance of rain? The logic doesn’t hold up, if you pay attention, but it sounds like it should make sense, and having the “wrong” version of something be the same as the original is a valid comic construction.

So now for the promised Ziggy rerun scandal. To the best of my knowledge Ziggy is presented as being in new run. It’s done by the son of the comic strip’s creator, but that’s common enough for long-running comic strips. This Monday, though, ran a Ziggy-at-the-psychiatrist joke that was, apart from coloring, exactly the comic run the 2nd of March, barely two weeks before. (Compare the scribbles in the psychiatrist’s diploma.) It wouldn’t be that weird if a comic were accidentally repeated; production mistakes happen, after all. It’s slightly weird that the daily, black-and-white, original got colored in two different ways, but I can imagine this happening by accident.

Still, that got me primed to look for Ziggy repeats. I couldn’t find this one having an earlier appearance. But I did find that the 9th of January this year was a reprint of the Ziggy from the 11th of January, 2017. I wrote about both appearances, without noticing they were reruns. Here’s the 2017 essay, and over here is the 2019 essay, from before I was very good at remembering what the year was. Mercifully I didn’t say anything contradictory on the two appearances. I’m more interested in how I said things differently in the two appearances. Anyway this earlier year seems to have been part of a week’s worth of reruns, noticeable by the copyright date. I can’t begrudge a cartoonist their vacation. The psychiatrist strip doesn’t seem to be part of that, though, and its repetition is some as-yet-unexplained event.

Pete: 'Have you seen my ... ' Peggy: 'Top drawer, dresser.' Pete: 'What day is the ... ' Peggy: 'Monday.' Pete: 'Do we have any ... ' Peggy: 'Middle cabinet, kitchen.' Pete: 'What's the square root of 532?' Peggy: '23.06512518.' (In the last panel Peggy looks smugly at the reader.)
Tony Rubino and Gary Markstein’s Daddy’s Home for the 13th of March, 2019. The steadily growing number of essays with a mention of Daddy’s Home are at this link.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 13th has a much more casual and non-controversial bit of mathematics. Pete tosses out a calculate-the-square-root problem as a test of Peggy’s omniscience. One of the commenters points out that the square root of 532 is closer to 23.06512519 than it is Peggy’s 23.06512818. It suggests the writers found the square root by something that gave plenty of digits. For example, the macOS Calculator program offers me “23.065 125 189 341 592”. But then they chopped off, rather than rounding off, digits when the panel space ran out.

Teacher: 'Nancy, Esther, I'm making you partners for classwork today.' Nancy, thinking: 'How are we supposed to work together? We're fighting!' Nancy, tearing a page of mathematics problems down the center: 'Here, you take the right side of the equals sign and I'll take the left.'
Olivia Jaimes’s Nancy for the 13th of March, 2019. Essays mentioning Nancy, either current-run or the “classic” vintage reprints, should appear here.

Olivia Jaimes’s Nancy for the 13th has Nancy dividing up mathematics problems along the equals sign. That’s cute and fanciful enough. One could imagine working out expressions on either side of the equals sign in the hopes of getting them to match. That wouldn’t work for these algebra problems, but, that’s something.

This isn’t what Nancy might do, unless she flashed forward to college and became a mathematics or physics major. But one great trick in differential equations is called the separation of variables. Differential equations describe how quantities change. They’re great. They’re hard. A lot of solving differential equations amounts to rewriting them as simpler differential equations.

Separation is a trick usable when there’s two quantities whose variation affect each other. If you can rewrite the differential equation so that one variable only appears on the left side, and the other variable only appears on the right? Then you can split this equation into two simpler equations. Both sides of the equation have to be some fixed number. So you can separate the differential equations of two variables into two differential equations, each with one variable. One with the first variable, one with the other. And, usually, a differential equation of one variable is easier than a differential equation with two variables. So Nancy and Esther could work each half by themselves. But the work would have to be put together at the end, too.


And for a truly marginal mathematics topic: Lincoln Pierce’s Big Nate: First Class for the 13th, reprinting the 2nd of March, 1994, mentions a mathematics test for Nate’s imminent doom.


And this wraps up the comic strips for the previous week. Come Sunday there should be a fresh new comic post. Yes, Andertoons is scheduled to be there.

Let Me Tell You How Interesting March Madness Could Possibly Be


I read something alarming in the daily “Best of GoComics” e-mail this morning. It was a panel of Dave Whamond’s Reality Check. It’s a panel comic, although it stands out from the pack by having a squirrel character in the margins. And here’s the panel.

Three mathematicians standing around chalkboards. One says: 'My pick is (9 +/- sqrt(5)). What's your bracket?' '(x + 3).' '(n - 1).' Caption: 'March Mathness'.
Dave Whamond’s Reality Check for the 2nd of March, 2019. Edge City inspires discussions in the essays at this link. I don’t know when or how Reality Check dropped off my comics page. It must have been after October 2018. Here’s essays including Reality Check, to serve as proof. I probably won’t go reading five months’ worth of the strip to get all the strips I’d missed. But the strip should return to my regular Reading the Comics posts now.

Certainly a solid enough pun to rate a mention. I don’t know of anyone actually doing a March Mathness bracket, but it’s not a bad idea. Rating mathematical terms for their importance or usefulness or just beauty might be fun. And might give a reason to talk about their meaning some. It’s a good angle to discuss what’s intersting about mathematical terms.

And that lets me segue into talking about a set of essays. The next few weeks see the NCAA college basketball tournament, March Madness. I’ve used that to write some stuff about information theory, as it applies to the question: is a basketball game interesting?

Along the way here I got to looking up actual scoring results from major sports. This let me estimate the information-theory content of the scores of soccer, (US) football, and baseball scores, to match my estimate of basketball scores’ information content.

  • How Interesting Is A Football Score? Football scoring is a complicated thing. But I was able to find a trove of historical data to give me an estimate of the information theory content of a score.
  • How Interesting Is A Baseball Score? Some Partial Results I found some summaries of actual historical baseball scores. Somehow I couldn’t find the detail I wanted for baseball, a sport that since 1845 has kept track of every possible bit of information, including how long the games ran, about every game ever. I made do, though.
  • How Interesting Is A Baseball Score? Some Further Results Since I found some more detailed summaries and refined the estimate a little.
  • How Interesting Is A Low-Scoring Game? And here, well, I start making up scores. It’s meant to represent low-scoring games such as soccer, hockey, or baseball to draw some conclusions. This includes the question: just because a distribution of small whole numbers is good for mathematicians, is that a good match for what sports scores are like?

Reading the Comics, March 12, 2019: Back To Sequential Time Edition


Since I took the Pi Day comics ahead of their normal sequence on Sunday, it’s time I got back to the rest of the week. There weren’t any mathematically-themed comics worth mentioning from last Friday or Saturday, so I’m spending the latter part of this week covering stuff published before Pi Day. It’s got me slightly out of joint. It’ll all be better soon.

Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for this week. That’s nice to have. It’s built on the concept of story problems. That there should be “stories” behind a problem makes sense. Most actual mathematics, even among mathematicians, is done because we want to know a thing. Acting on a want is a story. Wanting to know a thing justifies the work of doing this calculation. And real mathematics work involves looking at some thing, full of the messiness of the real world, and extracting from it mathematics. This would be the question to solve, the operations to do, the numbers (or shapes or connections or whatever) to use. We surely learn how to do that by doing simple examples. The kid — not Wavehead, for a change — points out a common problem here. There’s often not much of a story to a story problem. That is, where we don’t just want something, but someone else wants something too.

On the classroom chalkboard: 'Today's number story. 2 dogs are at the park. 3 more dogs arrive. How many dogs are there?' Non-Wavehead student: 'Really? That's it? I don't mean anything by it, but where the conflict? Where's the drama? OK, try this ... a cat shows up. Or a mailman! Ooh, a cat mailman! Now we're talking!'
Mark Anderson’s Andertoons for the 11th of March, 2019. Essays which discuss topics raised by Andertoons can be found at this link. Also at this link, nearly enough.

Parker and Hart’s The Wizard of Id for the 11th is a riff on the “when do you use algebra in real life” snark. Well, no one disputes that there are fields which depend on advanced mathematics. The snark comes in from supposing that a thing is worth learning only if it’s regularly “useful”.

King: 'Wiz! We need your help! There's a giant meteor heading towards earth!' Wizard: 'oh boy! Oh boy!' (He runs off and pulls out a chalkboard full of mathematics symbols.) King: 'What are you doing?' Wizard: 'Finally! This is where algebra saves lives!'
Parker and Hart’s The Wizard of Id for the 11th of March, 2019. This is part of the current run of The Wizard of Id, such as you might find in a newspaper. Both the current and 1960s-vintage reruns get discussion at this link.

Rick Detorie’s One Big Happy for the 12th has Joe stalling class to speak to “the guy who invented zero”. I really like this strip since it’s one of those cute little wordplay jokes that also raises a legitimate point. Zero is this fantastic idea and it’s hard to imagine mathematics as we know it without the concept. Of course, we could say the same thing about trying to do mathematics without the concept of, say, “twelve”.

Teacher: 'Are there any questions before the test? Yes, Joe?' Joe: 'I would like to say a few words to the guy who invented zero. Thanks for NOTHING, dude! ... I think we can all enjoy a little math humor at a time like this.'
Rick Detorie’s One Big Happy for the 12th of March, 2019. This is part of the current run of One Big Happy, such as you might find in a newspaper. At GoComics.com strips from several years ago are reprinted. Both runs of One Big Happy get their discussion at this link.

We don’t know who’s “the guy” who invented zero. It’s probably not all a single person, though, or even a single group of people. There are several threads of thought which merged together to zero. One is the notion of emptiness, the absense of a measurable thing. That probably occurred to whoever was the first person to notice a thing wasn’t where it was expected. Another part is the notion of zero as a number, something you could add to or subtract from a conventional number. That is, there’s this concept of “having nothing”, yes. But can you add “nothing” to a pile of things? And represent that using the addition we do with numbers? Sure, but that’s because we’re so comfortable with the idea of zero that we don’t ponder whether “2 + 1” and “2 + 0” are expressing similar ideas. You’ll occasionally see people asking web forums whether zero is really a number, often without getting much sympathy for their confusion. I admit I have to think hard to not let long reflex stop me wondering what I mean by a number and why zero should be one.

And then there’s zero, the symbol. As in having a representation, almost always a circle, to mean “there is a zero here”. We don’t know who wrote the first of that. The oldest instance of it that we know of dates to the year 683, and was written in what’s now Cambodia. It’s in a stone carving that seems to be some kind of bill of sale. I’m not aware whether there’s any indication from that who the zero was written for, or who wrote it, though. And there’s no reason to think that’s the first time zero was represented with a symbol. It’s the earliest we know about.

Lemont: 'My son asked what my favorite number is. I thought, who has a favorite number? Then remembered *I* did. When I was a kid I loved the number eight, because it reminded me of infinity.' C Dog: 'Nuh-uh, Big L. You tol' me you loved it 'cause it was the only number that reminded you of food.' Lemont: 'You sure? I remember being a much more profound kid.' C Dog: 'People always re-writing e'reything.'
Darrin Bell’s Candorville for the 12th of March, 2019. Essays featuring some topic raised by of Candorville should appear here.

Darrin Bell’s Candorville for the 12th has some talk about numbers, and favorite numbers. Lemont claims to have had 8 as his favorite number because its shape, rotated, is that of the infinity symbol. C-Dog disputes Lemont’s recollection of his motives. Which is fair enough; it’s hard to remember what motivated you that long ago. What people mostly do is think of a reason that they, today, would have done that, in the past.

The ∞ symbol as we know it is credited to John Wallis, one of that bunch of 17th-century English mathematicians. He did a good bit of substantial work, in fields like conic sections and physics and whatnot. But he was also one of those people good at coming up with notation. He developed what’s now the standard notation for raising a number to a power, that x^n stuff, and showed how to define raising a number to a rational-number power. Bunch of other things. He also seems to be the person who gave the name “continued fraction” to that concept.

Wallis never explained why he picked ∞ as a shape, of all the symbols one could draw, for this concept. There’s speculation he might have been varying the Roman numeral for 1,000, which we’ve simplified to M but which had been rendered as (|) or () and I can see that. (Well, really more of a C and a mirror-reflected C rather than parentheses, but I don’t have the typesetting skills to render that.) Conflating “a thousand” with “many” or “infinitely many” has a good heritage. We do the same thing when we talk about something having millions of parts or costing trillions of dollars or such. But, Wallis never explained (so far as we’re aware), so all this has to be considered speculation and maybe mnemonic helps to remembering the symbol.

Colin: 'These math problems are impossible!' Dad: 'Calm down, I'll help you think it through. [ Reading ] Dot has 20 gumballs. Jack has twice as many, and Joe has a third as many as Jack. If Dot gives Jack half her gumballs, and Jack chews half the new amount, how many will the teacher have to take away for all the kids' gumballs to be equal?' ... Now I see why people buy the answers to these things on the Internet!' Colin: 'I'm never going to harvard, am I?'
Terry Laban and Patty LaBan’s Edge City rerun for the 12th of March, 2019. (The strip has ended.) This comic originally ran in 2004. Edge City inspires discussions in the essays at this link.

Terry LaBan and Patty LaBan’s Edge City for the 12th is another story problem joke. Curiously the joke seems to be simply that the father gets confused following the convolutions of the story. The specific story problem circles around the “participation awards are the WORST” attitude that newspaper comics are surprisingly prone to. I think the LaBans just wanted the story problem to be long and seem tedious enough that our eyes glazed over. Anyway you could not pay me to read whatever the comments on this comic are. Sorry not sorry.


I figure to have one more Reading the Comics post this week. When that’s posted it should be available at this link. Thanks for being here.

Reading the Comics, March 14, 2019: Pi Day 2019 Edition


Some weeks there’s an obvious theme. Most weeks there’s not. But mid-March has formed a traditional theme for at least one day. I’m going to excerpt that from the rest of the week’s comics, because I’ve noticed what readership around here is like for stuff tagged “Pi Day” in mid-March. You all can do what you like with your pop-mathematics blogs.

Pi Day seems to have brought out fewer comics than in years past. The ones that were made, among the set I read, were also less on point. There was a lot of actual physical pie involved, too, suggesting the day might be escaping the realm of pop-mathematics silliness straight into pun nobody thinks about. Or maybe cartoonists just didn’t have a fresh angle this year.

John Hambrock’s The Brilliant Mind of Edison Lee shows off a nerd kind of mistake. At least one I think of as particularly nerdy. Wanting to calculate is a natural urge, especially for those who do it well. But to calculate the circumference of a pie from its diameter? What is exciting about that? More, does Grandpa recognize what a circumference is? It’s relatively easy to see the diameter of a pie. Area, also. But circumference? I’m not sure people are good at estimating the circumference of things, not by sight. You’d need a tape measure, or a similar flexible ruler, to start with and we don’t see that. Without the chance to measure it himself, Grandpa has to take the circumference (and, for that matter, diameter) at Edison Lee’s word. What would convince Grandpa of anything?

Edison: 'Happy Pi Day, Grandpa.' Grandpa: 'Is that today?' Edison: 'I'll demonstrate Pi by using it to calculate the circumference of this pie. [ He sets a pie on the table and calculates. ] If the diameter is 12 inches and we multiply by pi, which is 3.14, we'll end up with ... [ he looks up ] nothing.' Grandpa, who's already eaten the whole pie: 'Sorry, were you saying something?'
John Hambrock’s The Brilliant Mind of Edison Lee for the 14th of March, 2019. This and other essays inspired by Edison Lee can be found at this link.

For example, even if Grandpa accepted that Edison Lee had multiplied one number by 3.14 and gotten another number he might ask: how do we know pi is the same for pies of all sizes? Could a small pie’s circumference be only three times the diameter’s length, while a large pie’s is four times that? Could Edison offer an answer for why 3.14, or some nearby number, is all that interesting?

Hamster, holding up a pie: 'Guess what? It's national pie day!' Capybara: 'It's also my birthday.' Hamster: 'uh ... aand I got you this pie!'
Liz Climo’s Cartoons for the 14th of March, 2019. I haven’t had reason to discuss this comic here before. This and any future essays discussing Liz Climo Cartoons should appear at this new tag.

Liz Climo’s Cartoons is an example of the second kind of strip I mentioned during my introductory paragraphs. While it’s nominally built on Pi Day, any mathematics is gone. It’s just about the pun. And, well, the fun of having a capybara around.

Mark Parisi’s Off The Mark is the most on-topic strip for the day. And the anthropomorphic numerals joke for the day, too. It’s built on there being infinitely many digits to π, which, true enough. There are also infinitely many digits to \frac{1}{3} , mind; they’re just not so interesting a set. π being irrational gives us a never-ending variety of digits. It’s almost certainly normal, too. Any finite string of digits most likely appears infinitely often in this string.

Anthropomorphic 3, holding up a selfie stick; a decimal and the digits 1, 4, 1, 5, 9, 2, etc, all waving hands. 3: 'I don't think I can fit everyone in ... '
Mark Parisi’s Off The Mark for the 14th of March, 2019. The essays inspired by Off The Mark should appear at this link.

We won’t ever know enough digits of π to depict all of them. But we can depict the digits we know, and many different ways. Here’s a 2015 Washington Post article with several pictures representing the digits, including some neat “random walk” ones. In those the digits are used to represent directions and distances for a thing to move, and it represents the number as this curious wispy structure. There’s amazing pictures to be made of this.

Roy, who has a pie tin and mess on his face: 'It's OK, Norm. Kath and I agreed we both deserve to wear gag pies for forgetting what yesterday was.' Norm: 'My gosh, Roy --- you mean you both forgot your anniversary?' Roy: 'Oh, that's not yet. No, we forgot it was Pi day!' Norm: 'I'm officially in over my head ... '
John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2019. And this comic appears often enough. Working Days strips should appear in discussions at this link.

John Zakour and Scott Roberts’s Working Daze for the 15th is built more around the pie pun. I was relieved to see this. The kind of nerd jokes routinely made in Working Daze made me think it was bizarre the comic strip didn’t do a Pi Day joke. They were saving the setup.

Pierpoint, porcupine, to Gunther, bear: 'Heh! Heh! If I baked 13 apple pies and gave you half of them, how many would you have?' Gunther: 'Obviously I'd have all of them.' Pierpoint, dejected: 'Obviously.'
Bill Schorr’s The Grizzwells for the 13th of March, 2019. I’ve had a few chances to mention The Grizzwells and those essays are at this link.

And last, a comic strip that I don’t think was trying to set up a Pi Day joke. But Bill Schorr’s The Grizzwells for the 13th is a routine story problem joke. But that the setup mentions pies? If this ran on the 14th I would feel confident Schorr was going for a Pi Day comic. But it didn’t, so I don’t know if Schorr was going for that or not.


And those are the surprisingly few Pi Day 2019 comic strips. Later this week I should post, at this link, other recent mathematically-themed comic strips. Thanks for reading.

Six Or Arguably Four Things For Pi Day


I hope you’ll pardon me for being busy. I haven’t had the chance to read all the Pi Day comic strips yet today. But I’d be a fool to let the day pass without something around here. I confess I’m still not sure that Pi Day does anything lasting to encourage people to think more warmly of mathematics. But there is probably some benefit if people temporarily think more fondly of the subject. Certainly I’ll do more foolish things than to point at things and say, “pi, cool, huh?” this week alone.

I’ve got a couple of essays that discuss π some. The first noteworthy one is Calculating Pi Terribly, discussing a way to calculate the value of π using nothing but a needle, a tile floor, and a hilariously excessive amount of time. Or you can use an HTML5-and-JavaScript applet and slightly less time, and maybe even experimentally calculate the digits of π to two decimal places, if you get lucky.

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

In Calculating Pi Less Terribly I showed a way to calculate π that’s … well, you see where that sentence was going. This is a method that uses an alternating series. To get π exactly correct you have to do an infinite amount of work. But if you just want π to a certain precision, all right. This will even tell you how much work you have to do. There are other formulas that will get you digits of π with less work, though, and maybe I’ll write up one of those sometime.

Jack-o-lantern standing on a scale: 'Hey! I weigh exactly 3.14 pounds!' Caption: 'Pumpkin Pi'.
Dave Whamond’s Reality Check for the 27th of October, 2018. Does the weight count if the jack-o-lantern is wearing sneakers?

And the last of the relevant essays I’ve already written is an A To Z essay about normal numbers. I don’t know whether π is a normal number. No human, to the best of my knowledge, does. Well, anyone with an opinion on the matter would likely say, of course it’s normal. There’s fantastic reasons to think it is. But none of those amount to a proof it is.

[PI sces ] Guy at bar talking to Pi: 'Wow, so you were born on March 14th at 1:59, 26 seconds? What're the odds?'
Scott Hilburn’s The Argyle Sweater for the 14th of March, 2018. Also a free probability question, if you’re going to assume that every second of the year is equally likely to be the time of birth.

That’s my three items. After that I’d like to share … I don’t know whether to classify this as one or three pieces. They’re YouTube videos which a couple months ago everybody in the world was asking me if I’d seen. Now it’s your turn. I apologize if you too got this, a couple months ago, but don’t worry. You can tell people you watched and not actually do it. I’ll alibi you.

Pi figure, wearing glasses, reading The Neverending Story.
Mark Parisi’s Off The Mark for the 14th of March, 2018. Really the book seems a little short for that.

It’s a string of videos posted on youTube by 3Blue1Brown. The first lays out the matter with a neat physics problem. Imagine you have an impenetrable wall, a frictionless floor, and two blocks. One starts at rest. The other is sliding towards the first block and the wall. How many times will one thing collide with another? That is, will one block collide with another block, or will one block collide with a wall?

[ How ancient mathematicians amused themselves, AKA how to celebrate Pi Day today; third annual Pi-Easting Contest. Emcee: 'And HERE he is, our defending champ, that father of conic sections --- ARCHIMEDES!' They're all eating cakes shaped like pi.
Michael Cavna’s Warped for the 14th of March, 2018. Yes, but have you seen Pythagoras and his golden thigh?

The answer seems like it should depend on many things. What it actually depends on is the ratio of the masses of the two blocks. If they’re the same mass, then there are three collisions. You can probably work that sequence out in your head and convince yourself it’s right. If the outer block has ten times the mass of the inner block? There’ll be 31 collisions before all the hits are done. You might work that out by hand. I did not. You will not work out what happens if the outer block has 100 times the mass of the inner block. That’ll be 314 collisions. If the outer block has 1,000 times the mass of the inner block? 3,141 collisions. You see where this is going.

[ To Stephen Hawking, Thanks for making the Universe a little easier for the rest of us to understand ] Jay: 'I suppose it's only appropriate that he'd go on Pi Day.' Roy: 'Not to mention, Einstein's birthday.' Katherine: 'I'll bet they're off in some far reach of the universe right now playing backgammon.'
John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2018. No, you should never read the comments, but here, really, don’t read the comments.

The second video in the sequence explains why the digits of π turn up in this. And shows how to calculate this. You could, in principle, do this all using Newtonian mechanics. You will not live long enough to finish that, though.

Pie chart. Most of the chart: 'likes pie'. Small wedge of the chart: 'likes charts'.
Daniel Beyer’s Long Story Short for the 14th of March, 2015.

The video shows a way that saves an incredible load of work. But you save on that tedious labor by having to think harder. Part of it is making use of conservation laws, that energy and linear momentum are conserved in collisions. But part is by recasting the problem. Recast it into “phase space”. This uses points in an abstract space to represent different configurations of a system. Like, how fast blocks are moving, and in what direction. The recasting of the problem turns something that’s impossibly tedious into something that’s merely … well, it’s still a bit tedious. But it’s much less hard work. And it’s a good chance to show off you remember the Inscribed Angle Theorem. You do remember the Inscribed Angle Theorem, don’t you? The video will catch you up. It’s a good show of how phase spaces can make physics problems so much more manageable.

'Happy Pi Day.' 'Mmm. I love apple pie.' 'Pi day, not Pie Day. Pi ... you know ... 3.14 ... March 14th. Get it?' 'Today is a pie-eating holiday?' 'Sort of. They do celebrate it with pie, but it's mostly about pi.' 'I don't understand what that kid says half the time.'
John Hambrock’s The Brilliant Mind of Edison Lee for the 14th of March, 2016. The strip is like this a lot.

The third video recasts the problem yet again. In this form, it’s about rays of light reflecting between mirrors. And this is a great recasting. That blocks bouncing off each other and walls should have anything to do with light hitting mirrors seems ridiculous. But set out your phase space, and look hard at what collisions and reflections are like, and you see the resemblance. The sort of trick used to make counting reflections easy turns up often in phase spaces. It also turns up in physics problems on toruses, doughnut shapes. You might ask when do we ever do anything on a doughnut shape. Well, real physical doughnuts, not so much. But problems where there are two independent quantities, and both quantities are periodic? There’s a torus lurking in there. There might be a phase space using that shape, and making your life easier by doing so.

Anthropomorphic numerals at a cocktail party. 2: 'You're greater than me. I could listen to you forever.' Pi: 'Aw, shucks. I'm blushing.' (It is.) Caption: 'Humble Pi.'
Scott Hilburn’s The Argyle Sweater for the 14th of March, 2017. And while the strip is true, arguably, 2 goes on forever also; it’s just not very interesting how it does.

That’s my promised four or maybe six items. Pardon, please, now, as I do need to get back to reading the comics.

Reading the Comics, March 9, 2019: In Which I Explain Eleven Edition


I thought I had a flood of mathematically-themed comic strips last week. On reflection, many of them were slight enough not to need further context. You’ll see in the paragraph of not-discussed strips at the end of this. What did rate discussion turned out to get more interesting to me the more I wrote about them.

Stephen Beals’s Adult Children for the 6th uses mathematics as icon of things that are indisputably true. Two plus two equals four is a good example of such. If we take the ordinary meanings of ‘two’ and ‘plus’ and ‘equals’ and ‘four’ there’s no disputing it. The result follows from some uncontroversial-seeming axioms and a lot of deduction. By the rules of logic, the conclusion has to be true, whoever makes it. Even, for that matter, if nobody makes it. It’s difficult to imagine a universe in which nobody ever notices two plus two equals four. But we can imagine that there are mathematical truths that will never be noticed by anyone. (Here’s one. There is some largest finite whole number that any human-created project will ever use in any context. Consider the equation represented by “that number plus two equals (even bigger number)”.)

Harvey: 'Everyone ignores facts! Two plus two equals four, you know what I mean?' Friend: 'Yes. In your opinion, two plus two equals four.' Harvey: 'Noooo! Facts aren't opinions! There are no true facts, fake facts, iffy facts ... just facts! Let's judge things based on the facts!' Friend: 'And how do these facts make you feel?' Harvey, clutching his chest. 'Like you're giving me a fact attack.'
Stephen Beals’s Adult Children for the 6th of March, 2019. Essays inspired by something mentioned in Adult Children appear at this link.

But you see cards palmed there. What do we mean by ‘two’? Have we got a good definition? Might there be a different definition that’s more useful? Probably not, for ‘two’ anyway. But a part of mathematics, especially as a field develops, is working out what are the important concepts, and what their definitions should be. What a ‘function’ is, for example, went through a lot of debate and change over the 19th century. There is an elusiveness to facts, even in mathematics, where you’d think epistemology would be simpler.

Lauren's problem: '(x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2). Which of the following is equivalent to the expression above? a. 4x^2 y^2. b. 8xy^2 - 6y^2. c. 2x^2 + 2xy^2. d. 2x^2 y + 8xy^2 - 6y^2.' Next problem: 'If a/b = 2 what's the value of 4b/a? a. 0. b. 1. c. 2. d. 4.' Bob, holding up empty ice trays: 'If a and b are empty because Lauren is selfish and not thinking of Bob, what are the chances he gets to have an iced drink? a. slim, b. none, c. all of the above?'
Frank Page’s Bob the Squirrel for the 6th of March, 2019. When I’m moved to write something based on Bob the Squirrel the essays should be tagged to appear at this link.

Frank Page’s Bob the Squirrel for the 6th continues the SAT prep questions from earlier in the week. There’s two more problems in shuffling around algebraic expressions here. The first one, problem 5, is probably easiest to do by eliminating wrong answers. (x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2) is a tedious mess. But look at just the x^2 y terms: they have to add up to 2x^2 y , so, the answer has to be either c or d. So next look at the 3y^2 terms and oh, that’s nice. They add up to zero. The answer has to be c. If you feel like checking the 5xy^2 terms, go ahead; that’ll offer some reassurance, if you do the addition correctly.

The second one, problem 8, is probably easier to just think out. If \frac{a}{b} = 2 then there’s a lot of places to go. What stands out to me is that 4\frac{b}{a} has the reciprocal of \frac{a}{b} in it. So, the reciprocal of \frac{a}{b} has to equal the reciprocal of 2 . So \frac{a}{b} = \frac{1}{2} . And 4\frac{b}{a} is, well, four times \frac{b}{a} , so, four times one-half, or two. There’s other ways to go about this. In honestly, what I did when I looked at the problem was multiply both sides of \frac{a}{b} = 2 by \frac{b}{a} . But it’s harder to explain why that struck me as an obviously right thing to do. It’s got shortcuts I grew into from being comfortable with the more methodical approach. Someone who does a lot of problems like these will discover shortcuts.

Ruthie on the phone: 'Hello, homework hotline? I have an arithmetic question. Why isn't eleven called oneteen, and twelve called twoteen? ... You don't know? ... May I speak to your supervisor, please?'
Rick Detorie’s One Big Happy for the 6th of March, 2019. This particular strip is several years old, but I can’t pin down its original run more precisely than that. Essays featuring One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 6th asks one of those questions you need to be a genius or a child to ponder. Why don’t the numbers eleven and twelve follow the pattern of the other teens, or for that matter of twenty-one and thirty-two, and the like? And the short answer is that they kind of do. At least, “eleven” and “twelve”, etymologists agree, derive from the Proto-Germanic “ainlif” and “twalif”. If you squint your mouth you can get from “ain” to “one” (it’s probably easier if you go through the German “ein” along the way). Getting from “twa” to “two” is less hard. If my understanding is correct, etymologists aren’t fully agreed on the “lif” part. But they are settled on it means the part above ten. Like, “ainlif” would be “one left above ten”. So it parses as one-and-ten, putting it in form with the old London-English preference for one-and-twenty or two-and-thirty as word constructions.

It’s not hard to figure how “twalif” might over centuries mutate to “twelve”. We could ask why “thirteen” didn’t stay something more Old Germanic. My suspicion is that it amounts to just, well, it worked out like that. It worked out the same way in German, which switches to “-zehn” endings from 13 on. Lithuanian has all the teens end with “-lika”; Polish, similarly, but with “-ście”. Spanish — not a Germanic language — has “custom” words for the numbers up to 15, and then switches to “diecis-” as a prefix to the numbers 6 through 9. French doesn’t switch to a systematic pattern until 17. (And no I am not going to talk about France’s 80s and 90s.) My supposition is that different peoples came to different conclusions about whether they needed ten, or twelve, or fifteen, or sixteen, unique names for numbers before they had to resort to systemic names.

Here’s some more discussion of the teens, though, including some exploration of the controversy and links to other explanations.

Caption: '4 out of 5 Doctors agree ... ' Four, of five, chickens dressed as doctors: 'We are 80% of the doctors!'
Doug Savage’s Savage Chickens for the 6th of March, 2019. And the occasional essay based on Savage Chickens should be gathered at this link.

Doug Savage’s Savage Chickens for the 6th is a percentages comic. It makes reference to an old series of (American, at least) advertisements in which four out of five dentists would agree that chewing sugarless gum is a good thing. Shifting the four-out-of-five into 80% riffs is not just fun with tautologies. Percentages have this connotation of technical precision; 80% sounds like a more rigorously known number than “four out of five”. It doesn’t sound as scientific as “0.80”, quite. But when applied to populations a percentage seems less bizarre than a decimal.


Oh, now, and what about comic strips I can’t think of anything much to write about?
Ruben Bolling’s Super-Fun-Pak Comix for the 4th featured divisibility, in a panel titled “Fun Facts for the Obsessive-Compulsive”. Olivia James’s Nancy on the 6th was avoiding mathematics homework. Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 7th has Skip avoiding studying for his mathematics test. Bob Scott’s Bear With Me for the 7th has Molly mourning a bad result on her mathematics test. (The comic strip was formerly known as Molly And The Bear, if this seems familiar but the name seems wrong.) These are all different comic strips, I swear. Bill Holbrook’s Kevin and Kell for the 8th has Rudy and Fiona in mathematics class. (The strip originally ran in 2013; Comics Kingdom has started running Holbrook’s web comic, but at several years’ remove.) And, finally, Alex Hallatt’s Human Cull for the 8th talks about “110%” as a phrase. I don’t mind the phrase, but the comic strip has a harder premise.


And that finishes the comic strips from last week. But Pi Day is coming. I’ll be ready for it. Shall see you there.

Reading the Comics, March 6, 2019: Fix This Joke Edition


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

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

Peter: 'Whatcha doing?' Jason: 'Teaching Quincy algebra.' Peter: 'Isn't that a little advanced for an iguana?' Jason: 'I tried teaching him simpler math like addition and subtraction, but he wouldn't stop yawning. I'm taking that as a sign he needed something more challenging to engage in. 'Chapter seven: Quadratic Equations'.' (Quincy falls asleep.) Peter: 'Well, he's not yawning.' Jason: 'Maybe I should just jump right to calculus.'
Bill Amend’s FoxTrot for the 3rd of March, 2019. Essays that discuss FoxTrot, both old and current vintage, are at this link.

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

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

'Disc-o-Magic'. It's a ring of ten magician names, linked clockwise, and an inner ring of five magician names. Starting from any of the outer ring and going clockwise a number of times equal to the number of letters in the magician's name (eg, so, 'Houdini' would move clockwise seven spaces), then insite and repeating this counterclockwise the number of letters in *that* magician's name lands you to 'a new name that is (arguably) the name of the world's greatest magician!'
Mac King and Bill King’s Magic in a Minute for the 3rd of March, 2019. Arithmetic-based tricks from Magic in a Minute get listed at this link.

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

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

Lauren's SAT prep question: if f(x) = 2x^2 + 4 for all real numbers x which of the following is equal to f(3) + f(5)? a. f(4). b. f(6). c. f(10). d. f(15). Bob comes up. Lauren: 'I'm *studying*, Bob. Don't bother me.' Bob: 'The answer is B.' Lauren: 'Wow ... that's the correct ... answer?' Bob: 'WHY do you gotta say it with all the dots and pauses like that?'
Frank Page’s Bob the Squirrel for the 4th of March, 2019. The occasional essay inspired by Bob The Squirrel is at this link.

The first thing I notice in this problem is f(x). For positive values of x this is an “increasing” function. That is, if you have two positive numbers x and y, and x is less than y, then f(x) is less than f(y). You can see that from how x^2 is an increasing function. Multiply an increasing function by a positive number and it stays increasing. Add a constant to an increasing function and it stays increasing. So this right away rules out f(4) as a possible answer. If Lauren guessed wildly at this point, she’d have a one-in-three chance of getting it right. If the SAT still scores by the rules in place when I took it, that’s a chance worth taking.

That x^2 is another tip. This value grows, and pretty fast. It grows even faster the bigger x gets. The difference between f(10) and f(11) is 42. The difference between f(11) and f(12) is 46. The difference between f(12) and f(13) is 50. So just from that alone it’s hard to imagine f(15) being the right answer. Easier to imagine f(10) being right. Less hard to imagine f(6) being right. If I had to guess, f(6) would be it. If I must know which is right? I’d start by calculating f(5) and f(6). Then check their difference. If that seems close to what f(3) must be, good, call it done. If that didn’t work I’d move reluctantly on to calculating f(10). But, bleah. Seems tedious. I’m glad to be past having to work that out.

Woman, to the man with her, as they see someone approaching the corner of the city street: 'It's that Fibonacci dude. His conversations are never-ending.'
S Camilleri Konar’s Six Chix for the 6th of March, 2019. Essays inspired by something mentioned in Six Chix, whichever cartoonist created it, are at this link.

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

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

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

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

That’s … really probably all you can expect to work with. There are fascinating other properties but you have to be really into number theory to know them. A positive number x is a Fibonacci number if and only if either 5x^2 + 4 or 5x^2 - 4 , or both, are perfect squares, for example. 1, 8, and 144 are the only Fibonacci numbers that are perfect powers of a whole number. Any Fibonacci number besides 1, 2, and 3 is the largest number of a Pythagorean triplet. Building a joke on any of these facts aims it at a particularly narrow audience.

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

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


That’s far from the full set of mathematics comics this past week. I’ll have another posting about them here soon enough. And yes, I know what Thursday is, too.

Reading the Comics, March 2, 2019: Process Edition


There were a handful of comic strips from last week which I didn’t already discuss. Two of them inspire me to write about how we know how to do things. That makes a good theme.

Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th gets into deep territory. How does we could count to a million? Maybe some determined soul has actually done it. But it would take the better part of a month. Things improve some if we allow that anything a computing machine can do, a person could do. This seems reasonable enough. It’s heady to imagine that all the computing done to support, say, a game of Roller Coaster Tycoon could be done by one person working alone with a sheet of paper. Anyway, a computer could show counting up to a million, a billion, a trillion, although then we start asking whether anyone’s checked that it hasn’t skipped some numbers. (Don’t laugh. The New York Times print edition includes an issue number, today at 58,258, at the top of the front page. It’s meant to list the number of published daily editions since the paper started. They mis-counted once, in 1898, and nobody noticed until 1999.)

Dennis, to Margaret: 'How do you know you can count to a million if you've never done it?'
Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 27th of February, 2019. I’m not quite confident that I have the credits right here, but if I am parsing Wikipedia’s entry correctly Hamilton and Ketcham work on the daily comics and Ron Ferdinand and Ketcham work on the Sunday strips. And I would have thought this was a new tag but it turns out I have several Dennis the Menace-based essays at this link.

Anyway, allow that. Nobody doubts that, if we put enough time and effort into it, we could count up to any positive whole number, or as they say in the trade, “counting number”. But … there is some largest number that we could possibly count to, even if we put every possible resource and all the time left in the universe to that counting. So how do we know we “could” count to a number bigger than that? What does it mean to say we “could” if the circumstances of the universe are such that we literally could not?

Counting up to a number seems uncontroversial enough. If I wanted to prove it I’d say something like “if we can count to the whole number with value N, then we can count to the whole number with value N + 1 by … going one higher.” And “We can count to the whole number 1”, proving that by enunciating as clearly as I can. The induction follows. Fine enough. That’s a nice little induction proof.

But … what if we needed to do more work? What if we needed to do a lot of work? There is a corner of logic which considers infinitely long proofs, or infinitely long statements. They’re not part of the usual deductive logic that any mathematician knows and relies on. We’re used to, at least in principle, being able to go through and check every step of a proof. If that becomes impossible is that still a proof? It’s not my field, so I feel comfortable not saying what’s right and what’s wrong. But it is one of those lectures in your Mathematical Logic course that leaves you hanging your jaw open.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th is a joke about algorithms. These are the processes by which we know how to do a thing. Here, Hansel and Gretel are shown using what’s termed a “greedy algorithm” to follow pebbles back home. This kind of thing reflects trying to find an acceptable solution, in this case, finding a path somewhere. What makes it “greedy” is each step. You’re at a pebble. You can see other pebbles nearby. Which one do you go to? Go to some extreme one; in this case, the nearest. It could instead have been the biggest, or the shiniest, the one at the greatest altitude, the one nearest a water source. Doesn’t matter. You choose your summum bonum and, at each step, take the move that maximizes that.

During the great famine, Hansel and Gretel's mother decided to leave them in the woods. Overhearing the conversation, Hansel had an idea. `I will take these bright pebbles and leave them along our path, then we can follow them home.` Little did they know, their mother overheard *their* conversation. That night she created loops of shiny pebbles at various points in the woods. The following evening she left them in the forest. Gretel: `Just always go to the nearest pebble, keep doing that until you are home.` On the path they encountered a loop which caused them to go in an endless cycle until they passed out from exhaustion. The moral of this story? There are arts far darker than witchcraft. (Shows the wicked stepmother reading Introduction to Algorithms.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of February, 2019. There’s no mistaking this for a new tag. Saturday Morning Breakfast Cereal inspires many discussions at this link.

The wicked mother knows something about this sort of algorithm, one that promises merely a solution and not the best solution. And that is that all these solutions can be broken. You can set up a problem that the algorithm can’t solve. Greedy algorithms are particularly vulnerable to this. They’re called “local maximums”. You find the best answer of the ones nearby, but not the best one you possibly could locate.

Why use an algorithm like this, that can be broken so? That’s because we often want to do problems like finding a path through the woods. There are so many possible paths that it’s hard to find one of the acceptable ones. But there are processes that will, typically, find an acceptable answer. Maybe processes that will let us take an acceptable answer and improve it to a good answer. And this is getting into my field.

Actual persons encountering one of these pebble rings would (probably) notice they were caught in a loop. And what they’d do, then, is suspend the greedy rule: instead of going to the nearest pebble they could find, they’d pick something else. Maybe simply the nearest pebble they hadn’t recently visited. Maybe the second-nearest pebble. Maybe they’d give up and strike out in a random direction, trusting they’ll find some more pebbles. This can lead them out of the local maximum they don’t want toward the “global maximum”, the path home, that they do. There’s no reason they can’t get trapped again — this is why the wicked mother made many loops — and no reason they might not get caught in a loop of loops again. Every algorithm like this can get broken by some problem, after all. But sometimes taking the not-the-best steps can lead you to a better solution. That’s the insight at the heart of “Metropolis-Hastings” algorithms, which was my field before I just read comic strips all the time.

Father Figure Eight. A big 8, wearing ice skates and holding a tiny 8's hand, says, 'Son, I'll show you how to skate in the shape of a right-side-up infinity symbol!'
Dan Thompson’s Brevity for the 28th of February, 2019. This is another strip that’s inspired a host of essays. Brevity panels get shown off at this link.

Dan Thompson’s Brevity for the 28th is a nice simple anthropomorphic figures joke. It would’ve been a good match for the strips I talked about Sunday. I’m just normally reluctant to sort these comic strips other than by publication date.


And there were some comic strips I didn’t think worth making paragraphs about. Chris Giarrusso’s G-Man Webcomics for the 25th of February mentioned negative numbers and built a joke on the … negative … connotations of that word. (And inaugurates a tag for that comic strip. This fact will certainly come back to baffle me some later day.) Art Sansom and Chip Sansom’s The Born Loser for the 2nd of March has a bad mathematics report card. Tony Rubino and Gary Markstein’s Daddy’s Home for the 2nd has geometry be the subject parents don’t understand. Bill Amend’s FoxTrot Classics for the 2nd has a mathematics-anxiety dream.


And this closes out my mathematics comics for the week. Come Sunday I should have a fresh post with more comics, and I thank you for considering reading that.

Reading the Comics, February 25, 2019: Barely Mathematics Edition


These days I’ve been preparing these comics posts by making a note of every comic that seems like it might have a mathematical topic. Then at the end of the week I go back and re-read them all and think what I could write something about. This past week’s had two that seemed like nice juicy topics. And then I was busy all day Saturday so didn’t have time to put the thought into them that they needed. So instead I offer some comic strips with at least mentions of mathematical subjects. If they’re not tightly on point, well, I need to post something, don’t I?

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 24th is the anthropomorphic numerals joke for the week. It did get me thinking about the numbers which (in English) are homophones to other words. There don’t seem to be many, though: one, two, four, six, and eight seem to be about all I could really justify. There’s probably dialects where “ten” and “tin” blend together. There’s probably a good Internet Argument to be had about whether “couple” should be considered the name of a number. That there aren’t more is probably that there, in a sense, only a couple of names for numbers, with a scheme to compound names for a particular number of interest.

Anthropomorphized numerals 3 and 5 are at the golf course. 3 asks: 'Now where did four go?' 5: 'I don't know.' 3: 'Four? FOUR!!?' Caption: 'A tradition begins.'
Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 24th of February, 2019. I had thought this would be a new comics tag, but no. There’s already been another appearance here by Yaffle, which you can find at this link.

Scott Hilburn’s The Argyle Sweater for the 25th mentions algebra, but is mostly aimed at the Reading the Comics for some historian blogger. I kind of admire Hilburn’s willingness to go for the 70-year-old scandal for a day’s strip. But a daily strip demands a lot of content, especially when it doesn’t have recurring characters. The quiz answers as given are correct, and that’s easy to check. But it is typically easy to check whether a putative answer is correct. Finding an answer is the hard part.

A spy passes a sheet of quiz answers (4x + 3 = 7, x = 1. 18 - 4x = 5x, x = 2) to another spy. Caption: Algebra Hiss.
Scott Hilburn’s The Argyle Sweater for the 25th of February, 2019. There was never a moment I’d think this was a new tag. The Argyle Sweater gets discussed often and essays including it are at this link.

I’m not aware of any etymological link between the term algebra and the name Alger. The word “algebra” derivate from the Arabic “al-jabr”, which the Oxford English Dictionary tells me literally derives from a term for “the surgical treatment of fractures”. Less literally, it would mean putting things back together, restoring the missing parts. We get it from a textbook by the 9th century Persian mathematician Muhammad ibn Musa al-Khwarizmi, whose last name Europeans mutated into “algorithm”, as in, the way to solve a problem. That’s thanks to his book again. “Alger” as in a name seems to trace to Old English, although exactly where is debatable, as it usually is. (I’m assuming ‘Alger’ as a first name derives from its uses as a family name, and will angrily accept correction from people who know better.)

8-year-old Nicholas is doing addition problems. 4-year-old Alec asks 'Whatcha doing.' Nicholas: 'Math. And it's really hard.' Alec: 'Maybe I can help.' Nicholas: 'You're four years old. How can YOU help?' Alec: 'You can use my fingers too! Then you can count to twenty!'
Daniel Shelton’s Ben for the 25th of February, 2019. I had also thought this might be a new tag, but again no. Ben has appeared at least twice before, in essays at this link.

Daniel Shelton’s Ben for the 25th has a four-year-old offering his fingers as a way to help his older brother with mathematics work. Counting on fingers can be a fine way to get the hang of arithmetic and at least I won’t fault someone for starting there. Eventually, do enough arithmetic, and you stop matching numbers with fingers because that adds an extra layer of work that doesn’t do anything but slow you down.

Catching my interest though is that Nicholas (the eight-year-old, and I had to look that up on the Ben comic strip web site; GoComics doesn’t have a cast list) had worked out 8 + 6, but was struggling with 7 + 8. He might at some point get experienced enough to realize that 7 + 8 has to be the same thing as 8 + 7, which has to be the same thing as 8 + 6 + 1. And if he’s already got 8 + 6 nailed down, then 7 + 8 is easy. But that takes using a couple of mathematical principles — that addition commutes, that you can substitute one quantity with something equal to it, that you addition associates — and he might not see where those principles get him any advantage over some other process.

Caption: What does it mean when you see repeating numbers? A set of people say things: 'That's the third 8 I've seen this week.' 'Everywhere I go ... a 12 is following me.' 'When I turn on the TV ... there's that 5 again.' 'Is the DEEP STATE trying to tell us something?' 'Have THEY concealed the existence of 'certain numbers'?' 'If you see something stupid ... ' '... Say something STUPIDER!'
Ed Allison’s Unstrange Phenomena for the 25th of February, 2019. And I never seriously suspected this was a new tag. Unstrange Phenomena gets discussed in essays at this link.

Ed Allison’s Unstrange Phenomena for the 25th builds its Dadaist nonsense for the week around repeating numbers. I learn from trying to pin down just what Allison means by “repeating numbers” that there are people who ascribe mystical significance to, say, “444”. Well, if that helps you take care of the things you need to do, all right. Repeating decimals are a common enough thing. They appear in the decimal expressions for rational numbers. These expressions either terminate — they have finitely many digits and then go to an infinitely long sequence of 0’s — or they repeat. (We rule out “repeating nothing but zeroes” because … I don’t know. I would guess it makes the proofs in some corner of number theory less bothersome.)

You could also find interesting properties about numbers made up of repeating strings of numerals. For example, write down any number of 9’s you like, followed by a 6. The number that creates is divisible by 6. I grant this might not be the most important theorem you’ll ever encounter, but it’s a neat one. Like, a strong of 4’s followed by a 9 is not necessarily divisible by 4 or 9. There are bunches of cute little theorem like this, mostly good for making one admit that huh, there’s some neat coincidences(?) about numbers.

Although … Allison’s strip does seem to get at seeing particular numbers over and over. This does happen; it’s probably a cultural thing. One of the uses we put numbers to is indexing things. So, for example, a TV channel gets a number and while the station may have a name, it makes for an easier control to set the TV to channel numbered 5 or whatnot. We also use numbers to measure things. When we do, we get to pick the size of our units. We typically pick them so our measurements don’t have to be numbers too big or too tiny. There’s no reason we couldn’t measure the distance between cities in millimeters, or the length of toes in light-years. But to try is to look like you’re telling a joke. So we get see some ranges — 1 to 5, 1 to 10 — used a lot when we don’t need fine precision. We see, like, 1 to 100 for cases where we need more precision than that but don’t have to pin a thing down to, like, a quarter of a percent. Numbers will spill past these bounds, naturally. But we are more likely to encounter a 20 than a 15,642. We set up how we think about numbers so we are. So maybe it would look like some numbers just follow you.


Over the next few days I should have more chance to think. I’ll finish Reading the Comics from the past week and put an essay up at this link.

How February 2019 Treated My Mathematics Blog


February offered an interesting casual experiment for my mathematics blog. I didn’t actually leave it completely fallow. But I also didn’t do very much with it. I’d had an idea for a nice little project for it, but kept finding other things consuming the time.

So the short month ended up having a mere 11 posts. That’s on the low end of what I usually post around here. I’ve done as few as this several times in the roughly two years that WordPress makes it easy to find statistics for. But it hasn’t been common.

What did this do to my readership?

February 2019: 11 posts. 1,275 views. 835 visitors. 1.53 views per visitor.
1.53 views per visitor is the lowest that ratio’s been since May 2018 (1.52).

So I had a mere 1,275 views over the month, down from January’s 1,375 and December’s 1,409. What fascinates me is that this is an average of 46 views per day. In January there were an average 44 views per day; in December, 45. There were 835 unique visitors in February, down a touch from January’s 856 and December’s 875. That’s an average of 30 per day in February, 28 per day in January, and 28 per day in December. This suggests my blog may have reached the point that I don’t actually need to have stuff on it anymore. This would be quite the load off my schedule. It certainly suggests I’m improving my views-per-things-posted ratio.

My ‘likes’ continue to fall from the October 2018 local peak. There were 44 in February, my lowest total since July. Down from 63 in January and 82 in December. That’s rather more than can be accounted for by the shortness of February. Even per-post it’s still a drop, but not from much of a height. Comments plummeted even farther; there were ten in February, and one of those was about how there aren’t a lot of comments around here. There’d been 22 in January and 17 in December, numbers that seem more robust now. February was my lowest-comment month going back to May 2017, when there were eight comments.

The most popular posts this past month include a couple old reliables, and then one that I expect to be a steadily read one. The top five were:

There were 73 countries sending me readers in February. That’s well up from January’s 59, and even higher than December’s 68. Twenty of these were single-reader countries. That’s up from January’s 19 and December’s 17. I seem to have Europe pretty well-covered, apart from the Balkan, the Baltics, Bulgaria, and Belarus. I’m glad I have readers in Belgium at least. And how many?

A Mercator-style map of the world. The United States is the darkest pink on it; most of the Americas and Eurasia are a fairly uniform pink. Only a few African countries have any pink to them.
And ooh, hey! A reader from China. That like never happens.
Country Readers
United States 729
United Kingdom 67
Canada 50
Russia 42
Philippines 41
Denmark 39
India 38
Australia 28
Indonesia 13
Italy 13
Netherlands 13
Singapore 13
South Africa 12
Hong Kong SAR China 11
American Samoa 10
Germany 8
France 7
Poland 7
Austria 6
Belgium 6
Switzerland 6
China 5
Nepal 5
New Zealand 5
Pakistan 5
Sweden 5
Turkey 5
European Union 4
Slovenia 4
Spain 4
Thailand 4
Algeria 3
Iraq 3
Macedonia 3
Romania 3
Slovakia 3
United Arab Emirates 3
Brazil 2
Colombia 2
Finland 2
Greece 2
Guatemala 2
Ireland 2
Lebanon 2
Mexico 2
Nigeria 2
Norway 2
Panama 2
Saudi Arabia 2
Serbia 2
Taiwan 2
Uganda 2
Ukraine 2
Argentina 1 (**)
Cambodia 1
Cyprus 1
Czech Republic 1
Egypt 1
Hungary 1
Israel 1
Japan 1 (**)
Jordan 1 (**)
Kenya 1
Lithuania 1 (*)
Malaysia 1
Martinique 1
Mauritius 1
Papua New Guinea 1
Peru 1
Portugal 1
Puerto Rico 1
South Korea 1
Vietnam 1

Lithuania has been a single-reader country two months running now. Argentina, Japan, and Jordan have been single-reader countries three months now. Colombia ends its single-reader streak at six months as someone else came in to see what all the fuss was about. This spoils their chance to beat the European Union’s seven-month single reader streak, from December 2015 through June 2016. Sorry. Colombia still has the single-country streak record, though.

If I learn anything from the Insights panel, it’s that I write very long articles. They’re growing less so! According to Insights this year, to date, I’ve posted 20,750 words over 23 posts. This is an average 902 words per post. At the end of January I averaged 966 words per post. I posted a total of 9,162 words over February, or a mere 833 words each of those. I’m imposing less of a crushing workload on myself! Anyway, there were a total of 35 comments so far this year, an average of 1.5 comments per post, down from 1.9 at the start of February. There were 100 total likes, for an average of 4.3 likes per post, down from 4.8. Hm.

I start March with having made 1,225 total posts. They’ve attracted 75,565 views, from an acknowledged 37,951 unique visitors. So far.

I’d be glad to have you as a regular reader. You can have this blog added to your WordPress reader by using the “Follow nebusresearch” button in the upper right of the page. All these posts are also available by RSS, if you’d like to be read without being tracked. On Twitter I’m @Nebusj. And on Sundays and often another day of the week I go Reading the Comics for their mathematics topics. Thanks for reading this. I’m glad you’re there, trusting that you are there. And that it is you. You looked like you from a distance, anyway.