Reading the Comics, January 13, 2019: January 13, 2019 Edition


I admit I’m including a fairly marginal strip in this, just so I can have the fun of another single-day edition. What can I say? I can be easily swayed by silly things. Also, somehow, all four strips today have circumstances where one might mistake them for reruns. Let’s watch.

Bill Amend’s FoxTrot for the 13th is wordplay, mashing up ‘cell division’ with ‘long division’. As you might expect from Bill Amend — who loves sneaking legitimate mathematics and physics in where it’s not needed — Paige’s long cell division is a legitimate one. If you’d like a bit of recreational mathematics fun, you can figure out which microscopic organisms correspond to which numerals. The answer is also the Featured Comment on the page, at least as I write this. So if you need an answer, or you want to avoid having the answer spoiled, know what’s there.

A long division problem, with microbes representing the digits. Science teacher: 'Paige, about your diagram of cell division ... ' Paige: 'Did I get the math wrong?'
Bill Amend’s FoxTrot for the 13th of January, 2019. Essays discussing topics raised by FoxTrot, whether new (Sunday strips) or rerun (the weekdays), should be at this link.

Greg Evans’s Luann Againn for the 13th is the strip of most marginal relevance here. Part of Luann’s awful ay is a mathematics test. The given problems are nothing particularly meaningful. There is the sequence ‘mc2’ in the problem, although written as m^c 2 . There’s also a mention of ‘googleplex’, which when the strip was first published in 1991 was nothing more than a misspelling of the quite large number. (‘Googol’ is the number; ‘Google’ a curious misspelling. Or perhaps a reversion. The name was coined in 1938 by Milton Sirotta. Sirotta was seven years old at the time. I accept that it is at least possible Sirotta was thinking of the then-very-popular serial-comic strip Barney Google, and that his uncle Edward Kasner, who brought the name to mathematics, wrote it down wrong.) And that carries with it the connotation that big numbers are harder than small numbers. This is … kind of true. At least, long numbers are more tedious than short numbers. But you don’t really do different work, dividing 1428 by 7, than you do dividing 147 by 7. It’s just longer. “Hard” is a flexible idea.

Panels showing a day in Luann's life: she gets dressed and made up. Then misses the bus and has to run to school, steps in gum, slides into base at gym class, sweats a mathematics test, gets food spilled on her at lunch, and walks in the rain back home. Brad looks over the mess: 'Jeez, Luann, no wonder you don't have any boyfriends. Lookit how you go to school!'
Greg Evans’s Luann Againn for the 13th of January, 2019. It originally ran the 13th of January, 1991. Essays discussing topics raised by Luann, whether new (current day) or rerun (1991 vintage), should be at this link.

Mac King and Bill King’s Magic in a Minute for the 13th felt like a rerun to me. It took a bit of work to find, but yeah, it was. The strip itself, as presented, is new. But the same neat little modular-arithmetic coincidence was used the 31st of July, 2016.

Hickory-Trickery-Clock. From a picture of a standard analog watch, here's what you do: think of any number, one through twelve. Place your fingertip on the number 12 of the clock. Spell the number you thought of, moving one number clockwise for each letter; eg, if you thought 'one', move three spaces, stopping at the 3. Now spell out the number you're touching, advancing the numbers by the same rule. And now do this one more time. You will have reached ... 1:00.
Mac King and Bill King’s Magic in a Minute for the 13th of January, 2019. Essays discussing topics raised by Magic In A Minute, whether new or re-drawn magic, should be at this link.

Mathematics on clock faces is often used as a way to introduce modular arithmetic, a variation on arithmetic with only finitely many integers. This can help, if you’re familiar with clock faces. Like regular arithmetic, modular arithmetic can form a group and a ring. Clock faces won’t give you a group or ring, not unless you replace the number before ‘1’ with a ‘0’. To be a group, you need a collection of items, and a binary operation on the items. This operation we often think of as either addition or multiplication, depending on what makes sense for the problem. To be a ring, you need two binary operations, which interact by a distributive law. So the operations are often matched to addition and multiplication. Modular arithmetic is fun, yes. It’s also useful, not just as a way to do something like arithmetic that’s different. Many schemes for setting up checksums, quick and easy tests against data entry errors, rely on modular arithmetic on the data. And many schemes for generating ‘random’ numbers are built on finding multiplicative inverses in modular arithmetic. This isn’t truly random, of course. But you can look at a string of digits and not see any clear patterns. This is often as close to random as you need.

Avis: 'My niece Jasmine is one of those Millennials.' Nick: 'Ah yes, Generation Y.' Avis: 'Y? Why? I'd like to know! Why can't they read cursive? Why can't they do simple multiplication? Why can't they parallel park? Why can't they talk to each other? Why are they always complaining?' Nick: 'Avis, complaining is hardly limited to millennials.' (Avis's questions are illustrated with young adults trying to read cursive or to multiply 3 x 6 or such.)
Rick DeTorie’s One Big Happy for the 13th of January, 2019. Essays discussing topics raised by One Big Happy, whether new (on Creators.com) or rerun (on GoComics.com), should be at this link.

Rick DeTorie’s One Big Happy for the 13th is mostly a bunch of complaints the old always have against the young. Well, the complaint about parallel parking I haven’t seen before. But the rest are common enough. Featured in it is a complaint that the young can’t do arithmetic. I’m not sure there was ever a time that the older generation thought the young were well-trained in arithmetic. Nor that there was ever a time that the current educational vogue wasn’t blamed for destroying a generation’s ability to calculate. I’m sure there are better and worse ways to teach calculation. But I suspect any teaching method will fall short of addressing a couple issues. One is that people over-rate their own competence and under-rate other’s competence. So the older generation will see itself as having got the best possible arithmetic education and anything that’s different is a falling away. And another is that people get worse at stuff they don’t think is enjoyable or don’t have to do a lot. If you haven’t got a use for the fact, or an appreciation for the beauty in it, three times six is a bit of trivia, and not one that inspires much conversation when shared.


There’s more comics with something of a mathematical theme that got published last week. When I get to them the essays should be at this link.

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How All Of 2018 Treated My Mathematics Blog


It’s looking as though WordPress has really and permanently discontinued its year-in-review posts. That’s a shame. They had this animation that presented your year as a set of fireworks, one for each post, paced the same way your posts for the year were. The size of the fireworks explosion corresponded to how much it was liked or drew comments or something. Great stuff. Haven’t seen it in a couple of years. The web washes away everything whimsical.

I can do it manually, at least, looking at the summaries for yearly readership and all that. It’s just a bit different from the monthly reviews. And then I can see what lessons I draw from that, and go on to ignore them all. My impression of 2018 had been that I’d had a mildly better-read year than I had in 2017, but that my comments and likes had cratered. That is, people might find something they wanted to read, but saw no reason to stick around and chat with me, which I understand. But here’s what the data says.

2018 statistics: 16,597 page views. 9,769 visitors. 1.70 views per visitor. 182 posts published.
Also I swear they’ve changed the color scheme on this statistics page. This is probably setting up to a complete redesign of the statistics page to make it into something that gives almost the same information but takes way more JavaScript to do. I’m being old again. Pay no mind.

And, for the sake of convenience, let me put things since 2012 — my first full year — in a coherent table.

Year Posts Published Page Views Unique Visitors Likes Comments
2012 6,094 180 275* 97 190
2013 106 5,729 2,905 262 161
2014 129 7,020 3,382 1,045 308
2015 188 11,241 5,159 3,273 822
2016 213 12,851 7,168 2,163 474
2017 164 12,214 7,602 1,094 301
2018 182 16,597 9,769 1,016 386

The 2012 visitors count doesn’t; they only started keeping track of those numbers (where they’d admit to us) partway through the year.

2015 you can see was a busy year. That’s the first year I did an A-To-Z sequence, and that got a fantastic response. In 2016 I tried two over the year and while neither was as well-received, it did turn out nicely. 2017 and 2018 had a single A-To-Z sequence each. I’m surprised how nearly I track to a post every other day over seven years straight. And I’m surprised that my page-view count grew by about one-third from 2017 to 2018. And that unique visitors grew by about the same amount, and has been except for 2016-to-2017. I’m certainly not doing much to be better about promoting myself, so something else is at work. The evaporating number of likes and comments I can’t explain. It’s looking like 2015 and 2016 were exceptional years, but what was the exception?

I can say what’s popular: posts that tell you how to do something. And, of course, my participation in the Playful Mathematics Education Blog Carnival. I hope to do that again this year. The ten most popular things from 2018 were:

Fascinating, to me, is that only one piece (the Playful Mathematics Education Blog Carnival) was posted in 2018. But overall it suggests I should start more pieces with the tag “How to … ”.

122 of the world’s countries sent me any readers at all in 2018. Here they are, and how many came from each, as WordPress organizes them and thinks dubious things like the “European Union” or the “United Kingdom” are countries:

Country Readers
United States 10,545
Philippines 803
United Kingdom 737
Canada 641
India 635
Australia 285
Singapore 246
Denmark 199
Turkey 148
Germany 122
South Africa 114
Sweden 106
Brazil 105
Slovenia 105
France 85
Italy 83
Netherlands 72
Spain 71
Hong Kong SAR China 70
Puerto Rico 67
European Union 66
Switzerland 63
Poland 62
Austria 53
Indonesia 53
New Zealand 50
Mexico 45
Ireland 44
Pakistan 43
Belgium 41
Norway 39
Malaysia 37
Greece 36
South Korea 35
Russia 29
Algeria 28
Romania 27
Israel 25
Argentina 24
Kenya 22
Japan 21
Czech Republic 20
Finland 20
United Arab Emirates 20
Thailand 19
Egypt 18
Vietnam 16
Ghana 15
Peru 15
Portugal 14
Bangladesh 13
Nigeria 13
Croatia 12
Lithuania 12
Ukraine 12
Taiwan 11
Bulgaria 10
Bhutan 9
Brunei 9
Chile 9
Serbia 9
Hungary 8
Nepal 8
Saudi Arabia 8
Slovakia 8
Belize 7
China 7
Kazakhstan 7
Venezuela 7
Afghanistan 6
Morocco 6
Qatar 6
Sri Lanka 6
American Samoa 5
Colombia 5
Iraq 5
Kuwait 5
Lebanon 5
Macau SAR China 5
Mongolia 5
Albania 4
Estonia 4
Georgia 4
Jamaica 4
Jordan 4
Uruguay 4
Barbados 3
Costa Rica 3
Guernsey 3
Iceland 3
Latvia 3
Mauritius 3
Palestinian Territories 3
Panama 3
Cambodia 2
Cyprus 2
Ecuador 2
Laos 2
Libya 2
Luxembourg 2
Namibia 2
St. Kitts & Nevis 2
Tanzania 2
Trinidad & Tobago 2
Armenia 1
Bahamas 1
Bahrain 1
Botswana 1
El Salvador 1
Ethiopia 1
Fiji 1
Gibraltar 1
Guam 1
Kyrgyzstan 1
Macedonia 1
Malta 1
Mozambique 1
Myanmar (Burma) 1
Oman 1
Senegal 1
Sint Maarten 1
Tunisia 1

I’m quite surprised to have so many readers from the Philippines and wonder if some peculiar event happened, like a teacher told the school to look at my piece about the number of grooves on a record. I figured to appeal more to countries where English is a primary language, and know I have a strong United States cultural bias. (Quick, name a non-American comic strip that’s ever got into a Reading The Comics post. Time’s up! You were trying to think of Sandra Bell-Lundy’s Between Friends.) But the gap in readers per capita between, say, the United States and Canada seems more than I should have expected.

In all, in 2018, I posted 182 things. They came out to 186,612 words overall, for an average of 1,025 words per post. On average posts attracted 5.3 likes, and 2.8 comments. Seems as though I could do more. I don’t really know what.

Reading the Comics, January 12, 2019: A Edition


As I said Sunday, last week was a slow one for mathematically-themed comic strips. Here’s the second half of them. They’re not tightly on point. But that’s all right. They all have titles starting with ‘A’. I mean if you ignore the article ‘the’, the way we usually do when alphabetizing titles.

Tony Cochran’s Agnes for the 11th is basically a name-drop of mathematics. The joke would be unchanged if the teacher asked Agnes to circle all the adjectives in a sentence, or something like that. But there are historically links between religious thinking and mathematics. The Pythagoreans, for example, always a great and incredible starting point for any mathematical topic or just some preposterous jokes that might have nothing to do with their reality, were at least as much a religious and philosophical cult. For a long while in the Western tradition, the people with the time and training to do advanced mathematics work were often working for the church. Even as people were more able to specialize, a mystic streak remained. It’s easy to understand why. Mathematics promises to speak about things that are universally true. It encourages thinking about the infinite. It encourages thinking about the infinitely tiny. It courts paradoxes as difficult as any religious Mystery. It’s easy to snark at someone who takes numerology seriously. But I’m not sure the impulse that sees magic in arithmetic is different to the one that sees something supernatural in a “transfinite” item.

Teacher: 'Agnes, what is 18 divided by 7?' Agnes: 'It matters not. I am being called to a religious life of heavy piety, general holiness, and things of that nature. I want none of math's Satanity to taint my walk.' Later, Agnes, to the principal: 'I'm toying with starting a compound in New Mexico, if you want to blow this cheap pop stand and find the light.'
Tony Cochran’s Agnes for the 11th of January, 2019. I’m always glad to have the chance to write about Agnes, and you can see things I’ve written about it at this link.

Scott Hilburn’s The Argyle Sweater for the 11th is another mistimed Pi Day joke. π is, famously, an irrational number. But so is every number, except for a handful of strange ones that we’ve happened to find interesting. That π should go on and on follows from what an irrational number means. It’s a bit surprising the 4 didn’t know all this before they married.

Cartoon Labelled 'Wife of Pi'. At a therapist's office. The therapist is a division sign. A pi with a face rolls its eyes. A 4 with a face complains, 'He's irrational and he goes on and on.'
Scott Hilburn’s The Argyle Sweater for the 11th of January, 2019. It’s not quite the most common strip mentioned here. But you can find other topics raised by The Argyle Sweater at this link.

I appreciate the secondary joke that the marriage counselor is a “Hugh Jripov”, and the counselor’s being a ripoff is signaled by being a ÷ sign. It suggests that maybe successful reconciliation isn’t an option. I’m curious why the letters ‘POV’ are doubled, in the diploma there. In a strip with tighter drafting I’d think it was suggesting the way a glass frame will distort an image. But Hilburn draws much more loosely than that. I don’t know if it means anything.

Wavehead working on the problem 17 + 7 at the blackboard. He's circled the 17, put a ? under the equals sign, added an ! before the +, and a * after it. Teacher: 'This is math; you don't need to annotate.'
Mark Anderson’s Andertoons for the 12th of January, 2019. There’s more essays with Andertoons at this link.

Mark Anderson’s Andertoons for the 12th is the Mark Anderson’s Andertoons for the essay. I’m so relieved to have a regular stream of these again. The teacher thinks Wavehead doesn’t need to annotate his work. And maybe so. But writing down thoughts about a problem is often good practice. If you don’t know what to do, or you aren’t sure how to do what you want? Absolutely write down notes. List the things you’d want to do. Or things you’d want to know. Ways you could check your answer. Ways that you might work similar problems. Easier problems that resemble the one you want to do. You find answers by thinking about what you know, and the implications of what you know. Writing these thoughts out encourages you to find interesting true things.

And this was too marginal a mention of mathematics even for me, even on a slow week. But Georgia Dunn’s Breaking Cat News for the 12th has a cat having a nightmare about mathematics class. And it’s a fun comic strip that I’d like people to notice more.


And that’s as many comics as I have to talk about from last week. Sunday, I should have another Reading the Comics post and it’ll be at this link.

Reading the Comics, January 9, 2018: I Go On About Johnny Appleseed Edition


This was a slow week for mathematically-themed comic strips. Such things happen. I put together a half-dozen that see on-topic enough to talk about, but I stretched to do it. You’ll see.

Mark Anderson’s Andertoons for the 6th mentions addition as one of the things you learn in an average day of elementary school. I can’t help noticing also the mention of Johnny Appleseed, who’s got a weird place in my heart as he and I share a birthday. He got to it first. Although Johnny Appleseed — John Champan — is legendary for scattering apple seeds, that’s not what he mostly did. He would more often grow apple-tree nurseries, from which settlers could buy plants and demonstrate they were “improving” their plots. He was also committed to spreading the word of Emanuel Swedenborg’s New Church, one of those religious movements that you somehow don’t hear about. But there was this like 200-year-long stretch where a particular kind of idiosyncratic thinker was Swedenborgian, or at least influenced by that. I don’t know offhand of any important Swedenborgian mathematicians, I admit, but I’m glad to hear if someone has news.

Wavehead, walking home, talking to another kid: 'Today we learned about Columbus planting apple seeds using two-digit addition. I also daydreamed a lot.'
Mark Anderson’s Andertoons for the 6th of January, 2019. Andertoons often appears in these essays. You can see the proof of that Andertoons claim at this link.

Justin Thompson’s MythTickle rerun for the 9th mentions “algebra” as something so dreadful that even being middle-aged is preferable. Everyone has their own tastes, yes, although it would be the same joke if it were “gym class” or something. (I suppose that’s not one word. “Dodgeball” would do, but I never remember playing it. It exists just as a legendarily feared activity, to me.) Granting, though, that I had a terrible time with the introduction to algebra class I had in middle school.

Karma: 'What's wrong, Dziva?' Dziva: 'Watching Boody act so young and carefree makes me long for my own youth. I could run faster then, eat more and care less. I'm getting sad about it. Karma, isn't there some magical word that could make me quit wanting to be young again? Some profound reminder that being a kid wasn't so --- ' Karma: 'Algebra.' (Both rest, happy.)
Justin Thompson’s MythTickle rerun for the 9th of January, 2019. MythTickle has only barely appeared before in these essays. You can see the proof of that MythTickle claim at this link.

Tom Wilson’s Ziggy for the 9th is a very early Pi Day joke, so, there’s that. There’s not much reason a take-a-number dispenser couldn’t give out π, or other non-integer numbers. What the numbers are doesn’t matter. It’s just that the dispensed numbers need to be in order. It should be helpful if there’s a clear idea how uniformly spaced the numbers are, so there’s some idea how long a wait to expect between the currently-serving number and whatever number you’ve got. But that only helps if you have a fair idea of how long an order should on average take.

Ziggy, at a pie counter, takes a number. It's pi.
Tom Wilson’s Ziggy for the 9th of January, 2019. Ziggy has turned up once or twice in these essays. You can see the proof of that Ziggy claim at this link.

I’ll close out last week’s comics soon. The next Reading the Comics post, like all the earlier ones, should be at this link.

Reading the Comics, January 5, 2019: Start of the Year Edition


With me wrapping up the mathematically-themed comic strips that ran the first of the year, you can see how far behind I’m falling keeping everything current. In my defense, Monday was busier than I hoped it would be, so everything ran late. Next week is looking quite slow for comics, so maybe I can catch up then. I will never catch up on anything the rest of my life, ever.

Scott Hilburn’s The Argyle Sweater for the 2nd is a bit of wordplay about regular and irregular polygons. Many mathematical constructs, in geometry and elsewhere, come in “regular” and “irregular” forms. The regular form usually has symmetries that make it stand out. For polygons, this is each side having the same length, and each interior angle being congruent. Irregular is everything else. The symmetries which constrain the regular version of anything often mean we can prove things we otherwise can’t. But most of anything is the irregular. We might know fewer interesting things about them, or have a harder time proving them.

Teacher: 'Well, class, who'd like to show Mr Hoffmeyer how to correctly make an irregular polygon regular?' On the blackboard is an irregular pentagon and, drawn by Mr Hoffmeyer, a box of Ex-Lax.
Scott Hilburn’s The Argyle Sweater for the 2nd of January, 2019. The many appearances of Argyle Sweater in these pages are at this link.

I’m not sure what the teacher would be asking for in how to “make an irregular polygon regular”. I mean if we pretend that it’s not setting up the laxative joke. I can think of two alternatives that would make sense. One is to draw a polygon with the same number of sides and the same perimeter as the original. The other is to draw a polygon with the same number of sides and the same area as the original. I’m not sure of the point of either. I suppose polygons of the same area have some connection to quadrature, that is, integration. But that seems like it’s higher-level stuff than this class should be doing. I hate to question the reality of a comic strip but that’s what I’m forced to do.

Mutt, to Jeff in the hospital bed: 'Don't be afraid! Surgery on the tonsils is very simple!' Doctor: 'Don't you worry about the results!' Jeff: 'How do you know I'll be all right?' Doctor: 'Well, I lost my last eleven patients! So if the law of probabilities doesn't lie, you'll be all right! May I do something for you before I begin?' Jeff: 'Oh, yes, Doc! Help me put on my trousers and my jacket!'
Bud Fisher’s Mutt and Jeff rerun for the 4th of January, 2019. The several appearances of Mutt and Jeff in these pages are at this link.

Bud Fisher’s Mutt and Jeff rerun for the 4th is a gambler’s fallacy joke. Superficially the gambler’s fallacy seems to make perfect sense: the chance of twelve bad things in a row has to be less than the chance of eleven bad things in a row. So after eleven bad things, the twelfth has to come up good, right? But there’s two ways this can go wrong.

Suppose each attempted thing is independent. In this case, what if each patient is equally likely to live or die, regardless of what’s come before? And in that case, the eleven deaths don’t make it more likely that the next will live.

Suppose each attempted thing is not independent, though. This is easy to imagine. Each surgery, for example, is a chance for the surgeon to learn what to do, or not do. He could be getting better, that is, more likely to succeed, each operation. Or the failures could reflect the surgeon’s skills declining, perhaps from overwork or age or a loss of confidence. Impossible to say without more data. Eleven deaths on what context suggests are low-risk operations suggest a poor chances of surviving any given surgery, though. I’m on Jeff’s side here.

On the blackboard: 'Ratios: Apples 9, Oranges 6'. Wavehead, to teacher: 'Technically the ratio is 3:2, but as a practical matter we shouldn't even really be considering this.'
Mark Anderson’s Andertoons for the 5th of January, 2019. The amazingly many appearances of Andertoons in these pages are at this link.

Mark Anderson’s Andertoons for the 5th is a welcome return of Wavehead. It’s about ratios. My impression is that ratios don’t get much attention in themselves anymore, except to dunk on stupid Twitter comments. It’s too easy to jump right into fractions, and division. Ratios underlie this, at least historically. It’s even in the name, ‘rational numbers’.

Wavehead’s got a point in literally comparing apples and oranges. It’s at least weird to compare directly different kinds of things. This is one of those conceptual gaps between ancient mathematics and modern mathematics. We’re comfortable stripping the units off of numbers, and working with them as abstract entities. But that does mean we can calculate things that don’t make sense. This produces the occasional bit of fun on social media where we see something like Google trying to estimate a movie’s box office per square inch of land in Australia. Just because numbers can be combined doesn’t mean they should be.

Kid: 'Dad, I need help with a math problem. If striking NFL players who get $35,000 a game are replaced by scab players who get $1,000 a game ... what will be the point spread in a game between the Lions and the Packers?'
Larry Wright’s Motley rerun for the 5th of January, 2019. The occasional appearances of Motley in these pages are at this link.

Larry Wright’s Motley rerun for the 5th has the form of a story problem. And one timely to the strip’s original appearance in 1987, during the National Football League players strike. The setup, talking about the difference in weekly pay between the real players and the scabs, seems like it’s about the payroll difference. The punchline jumps to another bit of mathematics, the point spread. Which is an estimate of the expected difference in scoring between teams. I don’t know for a fact, but would imagine the scab teams had nearly meaningless point spreads. The teams were thrown together extremely quickly, without much training time. The tools to forecast what a team might do wouldn’t have the data to rely on.


The at-least-weekly appearances of Reading the Comics in these pages are at this link.

Reading the Comics, January 1, 2019: New Year’s Day Edition


It’s a new year. That doesn’t mean I’m not going to keep up some of my old habits. One of them is reading the comics for the mathematics bits. For example …

Johnny Hart’s Back To BC for the 30th presents some curious use of mathematics. At least the grammar of mathematics. It’s a bunch of statements that are supposed to, taken together, overload … I’m going to say BC’s … brain. (I’m shaky on which of the characters is Peter and which is BC. Their difference in hair isn’t much of a visual hook.) Certainly mathematics inspires that feeling that one’s overloaded one’s brain. The long strings of reasoning and (ideally) precise definitions are hard to consider. And the proofs mathematicians find the most fun are, often, built cleverly. That is, going about their business demonstrating things that don’t seem relevant, and at the end tying them together. It’s hard to think.

Peter: 'The sum of the four sides of an isosceles triangle is equal to the ... ' BC, thinking: 'Four?' Peter: 'Hypotenuse of a rectangular circle ... ' BC, thinking: 'A rectangular circle?' Peter: 'Having a mean radius of x divided by alpha centura ... ' BC, thinking: 'Having.' SNAP! (He rubs his head.) BC: 'I think that must have been my mind.'
Johnny Hart’s Back To BC for the 30th of December, 2018. The strip says it’s from the 18th of February, 1962. Essays inspired by B.C., both 1962 vintage and 2019 current, should be at this link.

But then … Peter … isn’t giving a real mathematical argument. He’s giving nonsense. And obvious nonsense, rather than nonsense because the writer wanted something that sounded complicated without caring what was said. Talking about a “four-sided triangle” or a “rectangular circle” has to be Peter trying to mess with BC’s head. Confidently-spoken nonsense can sound as if it’s deeper wisdom than the listener has. Which, fair enough: how can you tell whether an argument is nonsense or just cleverer than you are? Consider the kind of mathematics proof I mentioned above, where the structure might almost be a shaggy dog joke. If you can’t follow the logic, is it because the argument is wrong or because you haven’t worked out why it is right?

I believe that … Peter … is just giving nonsense and trusting that … BC … won’t know the difference, but will wear himself out trying to understand. Pranks.

Doctor: 'I'm not an accountant. I'm your doctor. However, by trying to do your taxes by yourself, I've calculated your brain has depreciated by nearly 68%.'
Tim Lachowski’s Get A Life for the 31st of December, 2018. Essays discussing things raised by Get A Life should be at this link.

Tim Lachowski’s Get A Life for the 31st just has some talk about percentages and depreciation and such. It’s meant to be funny that we might think of a brain depreciating, as if anatomy could use the same language as finance. Still, one of the virtues of statistics is the ability to understand a complicated reality with some manageable set of numbers. If we accept the convention that some number can represent the value of a business, why not the convention that some number could represent the health of a brain? So, it’s silly, but I can imagine a non-silly framing for it.

Trout: 'Two thousand and nineteen years, that's a long time.' Agnes: 'Yep! Earth has been around a while!' Trout: 'Were people even alive back then?' Agnes: 'Someone had to start counting, so I guess so, yeah.' Trout: 'Who taught them to count?' Agnes: 'Probably the people selling calendars.'
Tony Cochran’s Agnes for the 1st of January, 2019. This and other essays discussing Agnes should be at this link.

Tony Cochran’s Agnes for the 1st is about calendars. The history of calendars is tied up with mathematics in deep and sometimes peculiar ways. One might imagine that a simple ever-increasing index from some convenient reference starting time would do. And somehow that doesn’t. Also, the deeper you go into calendars the more you wonder if anyone involved in the project knew how to count. If you ever need to feel your head snapping, try following closely just how the ancient Roman calendar worked. Especially from the era when they would occasionally just drop an extra month in to the late-middle of February.

The Julian and Gregorian calendars have a year number that got assigned proleptically, that is, with the year 1 given to a set of dates that nobody present at the time called the year 1. Which seems fair enough; not many people in the year 1 had any idea that something noteworthy was under way. Calendar epochs dated to more clear events, like the reign of a new emperor or the revolution that took care of that whole emperor problem, will more reliably start with people aware of the new numbers. Proleptic dating has some neat side effects, though. If you ever need to not impress someone, you can point out that the dates from the 1st of March, 200 to the 28th of February, 300 both the Julian and the Gregorian calendar dates exactly matched.

Dad: 'OK, Suzie, you hate math. And actually, 1/37 of me understands exactly how you feel.' Suzie: 'Lay off, Dad.'
Niklas Eriksson’s Carpe Diem for the 2nd of January, 2019. And appearances by Carpe Diem in my essays should be at this link.

Niklas Eriksson’s Carpe Diem for the 2nd is a dad joke about mathematics. And uses fractions as emblematic of mathematics, fairly enough. Introducing them and working with them are the sorts of thing that frustrate and confuse. I notice also the appearance of “37” here. Christopher Miller’s fascinating American Cornball: A Laffopedic Guide to the Formerly Funny identifies 37 as the current “funniest number”, displacing the early 20th century’s preferred 23 (as in skidoo). Among other things, odd numbers have a connotation of seeming more random than even numbers; ask someone to pick a whole number from 1 to 50 and you’ll see 37’s and 33’s more than you’ll see, oh, 48’s. Why? Good question. It’s among the mysteries of psychology. There’s likely no really deep reason. Maybe a sense that odd numbers are, well, odd as in peculiar, and that a bunch of peculiarities will be funny. Now let’s watch the next decade make a food of me and decide the funniest number is 64.


I’m glad to be back on schedule publishing Reading the Comics posts. I should have another one this week. It’ll be at this link when it’s ready. Thanks for reading.

How December 2018 Treated My Mathematics Blog


With the end of December it’s the time to see what was popular around here, and just how popular it was. I keep figuring I’ll learn something useful from these explorations. Now and then I come to conclusions and one of these months I’ll even act on them.

December was an exhausting month. The last couple weeks of any A To Z sequence always are. These sequences are great fun, of course, or I wouldn’t keep doing them. But fatigue sets in, especially as I discover I’m not getting as far ahead of deadline as I imagined I would be and I get to the difficult letters of the alphabet’s end. And the Fall 2018 A To Z made up for being less frequent than past glossaries — two, rather than three, essays a week — with being crazily longer. So I was exhausted by that. And then the mathematically-themed comics for my Reading the Comics posts completely dried up. Add to that real-life obligations that I would not skip — being with family, and going to pinball events — and I ended up posting 17 things in December. Which is more than usual, yes. A typical month is 12 to 14 posts. But it’s down from the 23 of October and November, and I’m as convinced as I can be without evidence that the number of posts determines how many page views I get.

So October 2018, aided in part by my hosting the Playful Mathematics Education Blog Carnival, had 2,010 page views from 1,063 unique visitors. November I got in 1,611 page views from 847 visitors. December, well, that saw 1,409 page views from 875 visitors.

December 2018: 1,409 views; 875 visitors; 1.61 views per visitor; 17 posts published.
I’d have liked to have taken this screenshot at just past midnight, Universal Time, on the 1st of January, but I was busy with something or other. Who can really say what, or why, though?

There were 82 things liked in December. It’s a slight drop from November’s 85, and October’s 94. I suppose it’s a rise in likes per page view, at least. The number of comments utterly collapsed, which probably reflects the end of the A To Z project. In October and November I had appeals for suggested topics; December didn’t have time for them. So what was 60 comments in October and 36 in November dropped to 17 for December. It’s not my least talkative period of the last year, but it’s up there. I need, seriously, to work on opening my posts to more comments. I’d ask people for suggestions how to do that, but who would answer?

The most popular essays around here in December were two perennials, two A To Z pieces, and some comics:

What countries sent me page views, and in what quantity? These, and these:

Country Readers
United States 819
Canada 77
United Kingdom 68
India 46
Turkey 39
Philippines 31
Singapore 31
Australia 27
Denmark 23
Indonesia 15
Slovenia 14
South Africa 13
Italy 11
New Zealand 11
Sweden 10
Greece 9
Netherlands 9
South Korea 9
Bangladesh 8
Ireland 8
Lithuania 8
Germany 7
Brazil 6
Hong Kong SAR China 6
Poland 6
American Samoa 5
European Union 5
Mexico 5
Russia 5
Switzerland 5
Finland 4
France 4
Norway 4
Belgium 3
Czech Republic 3
Ghana 3
Hungary 3
Israel 3
Kenya 3
Pakistan 3
Serbia 3
Austria 2
Chile 2
Egypt 2
Libya 2
Romania 2
Spain 2
Taiwan 2
Ukraine 2
Vietnam 2
Albania 1
Algeria 1
Argentina 1
Botswana 1
China 1
Colombia 1 (****)
Iceland 1
Iraq 1
Japan 1
Jordan 1
Kuwait 1
Latvia 1
Palestinian Territories 1 (*)
Portugal 1
Puerto Rico 1
Saudi Arabia 1 (*)
Sint Maarten 1
Slovakia 1

That was 68 different countries sending readers, down from November’s 70 and October’s 74. There were 17 single-reader countries, up from November’s 13 and down from October’s 23. The Palestinian Territories and Saudi Arabia were single-reader countries in November too. Colombia’s been a single-reader country five months now. I don’t see how China can be a single-reader country, even given that English isn’t a primary language there. More than one person has to stumble across here just by accident. There’s something going on there.

According to Insights, I start the month and year with 72,915 total page views, from an admitted 36,260 unique visitors.

I published something like 18,587 words here in December, which is a drop from the 26,644 of November. I write “something like” because I don’t know how WordPress tallies stuff like words in captions, and I don’t think it counts words in comments. And the idea of a “word” in a count like this is difficult to make precise and indisputable. So don’t be fooled by the digits into thinking there’s any precision there. Also it’s still 1,093 words per post, which is a bit down from the 1,158 in an average November essay but still.

For the year-to-date, by the end of December, I was writing an average of 1,025 words per post. That is, posts for the whole of the year, rather than just in December. That’s down from the end-of-November average of 1,108 words per post. I averaged 5.2 likes per post, down from the end-of-November average of 5.3. And 2.8 comments per post, up from the end-of-November average of 2.7. That’s certainly not a significant change.

I’m glad to have you as a reader, even if it’s just for this sort of self-preening post. You can put my posts in your RSS reader and enjoy them at your convenience. If you’re using WordPress regularly you can also add me to your WordPress Reader. That’s from the button at the upper-right corner of the page. On Twitter I’m @Nebusj. And just about every Sunday plus, usually, some other day of the week I’ll be Reading the Comics for the mathematics stuff. Thanks for being here.

Reading the Comics, December 28, 2018: More Christmas Break Edition


I apologize for running quite so late. Comic Strip Master Command tried to make it easy for me, by issuing few comic strips that had any mathematical content to speak of. I was just busier than all that, and even now, I can’t say quite how. Well, living, I suppose. But I’ve done plenty of things now and can settle back to the usual, if anyone knows just what that was.

Also I am drawing down on the number of cancelled, in-eternal-reruns comic strips on my daily feed. So that should reduce the number of times I feature a comic strip and realize I’ve described it four times already and haven’t got anything new to say. It’s hard for me, since most of these comics have some charms, or at least pleasant weirdness. But clearly just making a note to myself that I’ve said everything there is to say about Randolph Itch, 2 am, isn’t enough. I’m sorry, Randolph.

Bill Holbrook’s On The Fastrack for the 28th is an example of the cartoonist’s habit of drawing metaphors literally. Dethany does ask the auditor Fi about “accepting his numbers”. In this context the numbers aren’t intersting as numbers. They’re interesting as representations for a narrative. If the numbers are consistent with a believable story? If it’s more believable that they represent a truth than that they’re a hoax? We call that “accepting the numbers”, but what we’re accepting is the story they’re given as evidence for.

Man giving a presentation; it's a string of digits on the whiteboard. A giant thumb swipes across it, and the numbers all fall off, leaving the man disheartened. Last panel, the man walks out, dejected. Dethany: 'Did you accept his numbers?' Fi: 'I swiped left.'
Bill Holbrook’s On The Fastrack for the 28th of December, 2018. Essays discussing topics raised by On The Fastrack should all appear at this link.

Auditing, and any critical thinking about numbers, involves some subtle uses of Bayesian probability. We’re working out the probability that this story is something we should believe. Each piece of evidence makes us think this probability is greater or lesser. With experience and skill one learns of patterns which suggest the story is false. Benford’s Law, for example, is often useful. Honestly-taken samples show tendencies, for example, in what leading digits appear. A discrepancy between what’s expected and what appears, if it can’t be explained, can be a sign of forgery.

B.C.: 'How many grains of sand are on this beach?' Peter: 'Five hundred and sixty gillion.' B.C.: 'You're a genius, Peter.'
Johnny Hart’s Back To BC rerun for the 27th of December, 2018. From the dating it apparently originally ran the 1st of July, 1961. Essays mentioning B.C., both 1961 vintage and 2019 current, should be at this link.

Johnny Hart’s Back To BC rerun for the 27th is built on estimating the grains of sand on a beach. This is, as fits the setting, a very old query. Archimedes wrote The Sand Reckoner which estimated how many grains of sand could fit in the universe. Estimating the number of grains of sand on a beach, or in a universe, is a fun mathematical problem. Perhaps not a practical one, not directly. The answer is after all “lots”, and there is no way to verify the number.

But it can still be indirectly practical. To work with enormous but finite numbers of things is hard. We do well working with small numbers like ‘six’ and ‘fourteen’ and some of us are even good at around ‘thirty’. We don’t have a good intuition for how a number like 480,000,000,000,000,000 should work. And that’s important; if we try adding six and fourteen and get thirty, we realize there’s something not quite right before we’ve done too much more work. With enormous numbers we can go on not noticing the mistake’s there. We need to find ways to understand these inconvenient numbers using the skills and intuitions we already have. Aristotle had to develop new terminology for numbers to get the Ancient Greek numerals system to handle the problem coherently. Peter’s invention of a gillion is — I’ll go ahead and say — a sly reenactment of that.

Neil: 'Hey, a Rubik's Cube! I used to be really good at those! But you can only solve them five times. Then you have to buy another one.' Leticia: 'Five times? Er ... why's that, Neil?' Neil: 'After that, the little square stickers aren't sticky anymore.'
Mark Pett’s Lucky Cow rerun for the 27th of December, 2018. So I am dropping the strip from my routine reading. But the various appearances of Lucky Cow should remain at this link.

Mark Pett’s Lucky Cow rerun for the 27th I do intend to make this enjoyable but cancelled strip’s last appearance here. It’s a Rubik’s Cube joke. It’s one about using a solution outside the rules of the problem. And as marginal as this one, I couldn’t quite bring myself to write a paragraph about the Todd the Dinosaur strip of the 29th, which also features the Rubik’s Cube.

An anthropomorphic numeral 8 talks to the doctor about its weight and its eating. The doctor performs surgery, cutting off a loop and turning the figure into a 9. It smiles and waves, and the new 9 goes off to join its friends 2, 0, and 1.
Ryan Pagelow’s Buni for the 28th of December, 2018. Essays which mention Buni should be gathered at this link.

Ryan Pagelow’s Buni for the 28th I’ll list as the anthropomorphic-numerals joke for the week, since it did turn out to be that slow a week here. I’m a bit curious what the now-9 is figuring to do next year. I suppose that one’s easy; it’s going to be going from 3 to 4 in a couple years that’s a real problem.


The various Reading the Comics posts should all be at this link. I like to think I’ll be back to having a post this coming Sunday, and maybe a second one next week if there are enough comic strips near enough to on-topic. Thanks for reading.

Yes, I Am Late With The Comics Posts Today


I apologize that, even though the past week was light on mathematically-themed comic strips, I didn’t have them written up by my usual Sunday posting time. It was just too busy a week, and I am still decompressing from the A to Z sequence. I’ll have them as soon as I’m able.

In the meanwhile may I share a couple of things I thought worth reading, and that have been waiting in my notes folder for the chance to highlight?

This Fermat’s Library tweet is one of those entertaining consequences of probability, multiplied by the large number of people in the world. If you flip twenty coins in a row there’s a one in 1,048,576 chance that all twenty will come up heads, or all twenty will come up tails. So about one in every million times you flip twenty coins, they all come up the same way. If the seven billion people in the world have flipped at least twenty coins in their lives, then something like seven thousand of them had the coins turn up heads every single one of those twenty times. That all seven billion people have tossed a coin seems like the biggest point to attack this trivia on. A lot of people are too young, or don’t have access to, coins. But there’s still going to be thousands who did start their coin-flipping lives with a remarkable streak.

Also back in October, so you see how long things have been circulating around here, John D Cook published an article about the World Series. Or any series contest. At least ones where the chance of each side winning don’t depend on the previous games in the series. If one side has a probability ‘p’ of winning any particular game, what’s the chance they’ll win a best-four-of-seven? What makes this a more challenging mathematics problem is that a best-of-seven series stops after one side’s won four games. So you can’t simply say it’s the chance of four wins. You need to account for four wins out of five games, out of six games, and out of seven games. Fortunately there’s a lot of old mathematics that explores just this.

The economist Brandford DeLong noticed the first write-up of the Prisoners Dilemma. This is one of the first bits of game theory that anyone learns, and it’s an important bit. It establishes that the logic of cooperatives games — any project where people have to work together — can have a terrible outcome. What makes the most sense for the individuals makes the least sense for the group. That a good outcome for everyone depends on trust, whether established through history or through constraints everyone’s agreed to respect.

And finally here’s part of a series about quick little divisibility tests. This is that trick where you tell what a number’s divisible by through adding or subtracting its (base ten) digits. Everyone who’d be reading this post knows about testing for divisibility by three or nine. Here’s some rules for also testing divisibility by eleven (which you might know), by seven (less likely), and thirteen. With a bit of practice, and awareness of some exceptional numbers, you can tell by sight whether a number smaller than a thousand is prime. Add a bit of flourish to your doing this and you can establish a reputation as a magical mathematician.

Reading the Comics, December 22, 2018: Christmas Break Edition


There were just enough mathematically-themed comic strips last week for me to make two posts out of it. This current week? Is looking much slower, at least as of Wednesday night. But that’s a problem for me to worry about on Sunday.

Eric the Circle for the 20th, this one by Griffinetsabine, mentions a couple of shapes. That’s enough for me, at least on a slow comics week. There is a fictional tradition of X marking the spot. It can be particularly credited to Robert Louis Stevenson’s Treasure Island. Any symbol could be used to note a special place on maps, certainly. Many maps are loaded with a host of different symbols to convey different information. Circles and crosses have the advantage of being easy to draw and difficult to confuse for one another. Squares, triangles, and stars are good too.

Eric on a treasure hunt: Eric asking, 'Wait ... triangle marks the spot? No ... rhombus marks the spot? Dodecahedron marks the spot?' A square sighs; an X coughs, 'ahem!'
Eric the Circle for the 20th of December, 2018, this one by Griffinetsabine. This and other essays featuring Eric the Circle are at this link.

Bill Whitehead’s Free Range for the 22nd spoofs Wheel of Fortune with “theoretical mathematics”. Making a game out of filling in parts of a mathematical expression isn’t ridiculous, although it is rather niche. I don’t see how the revealed string of mathematical expressions build to a coherent piece, but perhaps a few further pieces would help.

Wheel Of Theoretical Mathematics. Contestant: 'I'd like to buy a sqrt(x).' On the board are several mathematical expressions, including 'dx = sqrt{pi}', 'a^2 + b^2 = (a + b)', and 'dy/dx x^4 - (1 - x^2)^4$.
Bill Whitehead’s Free Range for the 22nd of December, 2018. Appearances by Free Range should be at this link.

The parts shown are all legitimate enough expressions. Well, like a^2 + b^2 = (a + b) is only true for some specific numbers ‘a’ and ‘b’, but you can find solutions. -b \pm \sqrt{b^2 - x^2y^2} is just an expression, not picking out any particular values of ‘b’ or ‘x’ or ‘y’ as interesting. But in conjunction with a^2 + b^2 = (a + b) or other expressions there might be something useful. On the second row is a graph, highlighting a region underneath a curve (and above the x-axis) between two vertical lines. This is often the sort of thing looked at in calculus. It also turns up in probability, as the area under a curve like this can show the chance that an experiment will turn up something in a range of values. And \frac{dy}{dx} = x^4 - \left(1 - x^2\right)^4 is a straightforward differential equation. Its solution is a family of similar-looking polynomials.

Scientist guy runs in to the Lucky Cow restaurant. The scientist begs of cashier Neil, the left half of Schrodinger's equation equals ?!? Neil thinks about it some and then provides the answer, earning the scientist's gratitude and the admiration of his coworkers. Later, out in back, Neil pays off the scientist.
Mark Pett’s Lucky Cow for the 22nd of December, 2018. This and other discussions inspired by Lucky Cow should be at this link.

Mark Pett’s Lucky Cow for the 22nd has run before. I’ve even made it the title strip for a Reading the Comics post back in 2014. So it’s probably time to drop this from my regular Reading the Comics reporting. The physicists comes running in with the left half of the time-dependent Schrödinger Equation. This is all over quantum mechanics. In this form, quantum mechanics contains information about how a system behaves by putting it into a function named \psi . Its value depends on space (‘x’). It can also depend on time (‘t’). The physicists pretends to not be able to complete this. Neil arranges to give the answer.

Schrödinger’s Equation looks very much like a diffusion problem. Normal diffusion problems don’t have that \imath which appears in the part of Neil’s answer. But this form of equation turns up a lot. If you have something that acts like a fluid — and heat counts — then a diffusion problem is likely important in understanding it.

And, yes, the setup reminds me of a mathematical joke that I only encounter in lists of mathematics jokes. That one I told the last time this strip came up in the rotation. You might chuckle, or at least be convinced that it is a correctly formed joke.


Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

What I Wrote About in My 2018 Mathematics A To Z


I have reached the end! Thirteen weeks at two essays per week to describe a neat sampling of mathematics. I hope to write a few words about what I learned by doing all this. In the meanwhile, though, I want to gather together the list of all the essays I did put into this project.

Reading the Comics, December 19, 2018: Andertoons Is Back Edition


I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.

Mrs Olsen: 'How do you know I haven't got my flu shot?' Caulfield: 'Just playing the odds.' Mrs Olsen: 'Maybe I was playing some odds myself. Maybe I got to the pharmacy and remembered that this year's vaccine is 30-40% effective.' Caulfield: 'I'd take those odds.' Mrs Olsen: 'They're not my kind of odds.' Caulfield: 'And what are the odds you bought a lottery ticket on your way out?' (Pause.) Mrs Olsen: 'You are getting under my skin.' Caulfield: 'That's good news. Now there's a 30-40% chance you'll develop a resistance.'
Jef Mallett’s Frazz for the 16th of December, 2018. Other essays discussing topics raised by Frazz are at this link.

Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.

Guard, to new prisoner: 'Never mind Professor Phillip. He's always preoccupied with some theory of escape probability.' The cell walls are covered with mathematical scrawls.
Tak Bui’s PC and Pixel for the 16th of December, 2018. This and other essays, when they’re written, inspired by PC and Pixel should be at this link. It’s a new tag, which surprises me.

Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.

Amazing Yet Tautological strip: 'Each year America consumes enough EGG SALAD ... ' (Picture of a woman holding up a lumpy pile that context indicates is egg salad.) ' ... to give EACH AMERICAN an annualized national-average serving of the tasty concoction!'
Ruben Bolling’s Super-Fun-Pak Comix for the 18th of December, 2018. Essays based on Super-Fun-Pak Comix are at this link. (Amazing Yet Tautological is one of the features that turns up in Super-Fun-Pak Comix, which is why it doesn’t rate a tag on its own)

Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.

Teacher showing solid geometry to the class. Wavehead: 'I saw a movie where the robot monster came right at me. If you want me to get excited about 3D shapes, you're going to have to do better than that.'
Mark Anderson’s Andertoons for the 19th of December, 2018. The many essays which discuss Andertoons are at this link.

Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.


Each of the Reading the Comics posts should all be at this link. And I have finished the alphabet in my Fall 2018 Mathematics A To Z glossary. There should be a few postscript thoughts to come this week, though.

My 2018 Mathematics A To Z: Zugzwang


My final glossary term for this year’s A To Z sequence was suggested by aajohannas, who’d also suggested “randomness” and “tiling”. I don’t know of any blogs or other projects they’re behind, but if I do hear, I’ll pass them on.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Zugzwang.

Some areas of mathematics struggle against the question, “So what is this useful for?” As though usefulness were a particular merit — or demerit — for a field of human study. Most mathematics fields discover some use, though, even if it takes centuries. Others are born useful. Probability, for example. Statistics. Know what the fields are and you know why they’re valuable.

Game theory is another of these. The subject, as often happens, we can trace back centuries. Usually as the study of some particular game. Occasionally in the study of some political science problem. But game theory developed a particular identity in the early 20th century. Some of this from set theory experts. Some from probability experts. Some from John von Neumann, because it was the 20th century and all that. Calling it “game theory” explains why anyone might like to study it. Who doesn’t like playing games? Who, studying a game, doesn’t want to play it better?

But why it might be interesting is different from why it might be important. Think of what a game is. It is a string of choices made by one or more parties. The point of the choices is to achieve some goal. Put that way you realize: this is everything. All life is making choices, all in the pursuit of some goal, even if that goal is just “not end up any worse off”. I don’t know that the earliest researchers in game theory as a field realized what a powerful subject they had touched on. But by the 1950s they were doing serious work in strategic planning, and by 1964 were even giving us Stanley Kubrick movies.

This is taking me away from my glossary term. The field of games is enormous. If we narrow the field some we can discuss specific kinds of games. And say more involved things about these games. So first we’ll limit things by thinking only of sequential games. These are ones where there are a set number of players, and they take turns making choices. I’m not sure whether the field expects the order of play to be the same every time. My understanding is that much of the focus is on two-player games. What’s important is that at any one step there’s only one party making a choice.

The other thing narrowing the field is to think of information. There are many things that can affect the state of the game. Some of them might be obvious, like where the pieces are on the game board. Or how much money a player has. We’re used to that. But there can be hidden information. A player might conceal some game money so as to make other players underestimate her resources. Many card games have one or more cards concealed from the other players. There can be information unknown to any party. No one can make a useful prediction what the next throw of the game dice will be. Or what the next event card will be.

But there are games where there’s none of this ambiguity. These are called games with “perfect information”. In them all the players know the past moves every player has made. Or at least should know them. Players are allowed to forget what they ought to know.

There’s a separate but similar-sounding idea called “complete information”. In a game with complete information, players know everything that affects the gameplay. At least, probably, apart from what their opponents intend to do. This might sound like an impossibly high standard, at first. All games with shuffled decks of cards and with dice to roll are out. There’s no concealing or lying about the state of affairs.

Set complete-information aside; we don’t need it here. Think only of perfect-information games. What are they? Some ancient games, certainly. Tic-tac-toe, for example. Some more modern versions, like Connect Four and its variations. Some that are actually deep, like checkers and chess and go. Some that are, arguably, more puzzles than games, as in sudoku. Some that hardly seem like games, like several people agreeing how to cut a cake fairly. Some that seem like tests to prove people are fundamentally stupid, like when you auction off a dollar. (The rules are set so players can easily end up paying more then a dollar.) But that’s enough for me, at least. You can see there are games of clear, tangible interest here.

The last restriction: think only of two-player games. Or at least two parties. Any of these two-party sequential games with perfect information are a part of “combinatorial game theory”. It doesn’t usually allow for incomplete-information games. But at least the MathWorld glossary doesn’t demand they be ruled out. So I will defer to this authority. I’m not sure how the name “combinatorial” got attached to this kind of game. My guess is that it seems like you should be able to list all the possible combinations of legal moves. That number may be enormous, as chess and go players are always going on about. But you could imagine a vast book which lists every possible game. If your friend ever challenged you to a game of chess the two of you could simply agree, oh, you’ll play game number 2,038,940,949,172 and then look up to see who won. Quite the time-saver.

Most games don’t have such a book, though. Players have to act on what they understand of the current state, and what they think the other player will do. This is where we get strategies from. Not just what we plan to do, but what we imagine the other party plans to do. When working out a strategy we often expect the other party to play perfectly. That is, to make no mistakes, to not do anything that worsens their position. Or that reduces their chance of winning.

… And yes, arguably, the word “chance” doesn’t belong there. These are games where the rules are known, every past move is known, every future move is in principle computable. And if we suppose everyone is making the best possible move then we can imagine forecasting the whole future of the game. One player has a “chance” of winning in the same way Christmas day of the year 2038 has a “chance” of being on a Tuesday. That is, the probability is just an expression of our ignorance, that we don’t happen to be able to look it up.

But what choice do we have? I’ve never seen a reference that lists all the possible games of tic-tac-toe. And that’s about the simplest combinatorial-game-theory game anyone might actually play. What’s possible is to look at the current state of the game. And evaluate which player seems to be closer to her goal. And then look at all the possible moves.

There are three things a move can do. It can put the party closer to the goal. It can put the party farther from the goal. Or it can do neither. On her turn the other party might do something that moves you farther from your goal, moves you closer to your goal, or doesn’t affect your status at all. It seems like this makes strategy obvious. On every step take the available move that takes one closest to the goal. This is known as a “greedy” strategy. As the name suggests it isn’t automatically bad. If you expect the game to be a short one, greed might be the best approach. The catch is that moves that seem less good — even ones that seem to hurt you initially — might set up other, even better moves. So strategy requires some thinking beyond the current step. Properly, it requires thinking through to the end of the game. Or at least until the end of the game seems obvious.

We should like a strategy that leaves us no choice but to win. Next-best would be one that leaves the game undecided, since something might happen like the other player needing to catch a bus and so resigning. This is how I got my solitary win in the two months I spent in the college chess club. Worst would be the games that leave us no choice but to lose.

It can be that there are no good moves. That is, that every move available makes it a little less likely that we win. Sometimes a game offers the chance to pass, preserving the state of the game but giving the other party the turn. Then maybe the other party will do something that creates a better opportunity for us. But if we are allowed to pass, there’s a good chance the game lets the other party pass, too, and we end up in the same fix. And it may be the rules of the game don’t allow passing anyway. One must move.

The phenomenon of having to make a move when it’s impossible to make a good move has prominence in chess. I don’t have the chess knowledge to say how common the situation is. But it seems to be a situation people who study chess problems love. I suppose it appeals to a love of lost causes and the hope that you can be brilliant enough to see what everyone else has overlooked. German chess literate gave it a name 160 years ago, “zugzwang”, “compulsion to move”. Somehow I never encountered the term when I was briefly a college chess player. Perhaps because I was never in zugzwang and was just too incompetent a player to find my good moves. I first encountered the term in Michael Chabon’s The Yiddish Policeman’s Union. The protagonist picked up on the term as he investigated the murder of a chess player and then felt himself in one.

Combinatorial game theorists have picked up the word, and sharpened its meaning. If I understand correctly chess players allow the term to be used for any case where a player hurts her position by moving at all. Game theorists make it more dire. This may reflect their knowledge that an optimal strategy might require taking some dismal steps along the way. The game theorist formally grants the term only to the situation where the compulsion to move changes what should be a win into a loss. This seems terrible, but then, we’ve all done this in play. We all feel terrible about it.

I’d like here to give examples. But in searching the web I can find only either courses in game theory. These are a bit too much for even me to sumarize. Or chess problems, which I’m not up to understanding. It seems hard to set out an example: I need to not just set out the game, but show that what had been a win is now, by any available move, turned into a loss. Chess is looser. It even allows, I discover, a double zugzwang, where both players are at a disadvantage if they have to move.

It’s a quite relatable problem. You see why game theory has this reputation as mathematics that touches all life.


And with that … I am done! All of the Fall 2018 Mathematics A To Z posts should be at this link. Next week I’ll post my big list of all the letters, though. And, as has become tradition, a post about what I learned by doing this project. And sometime before then I should have at least one more Reading the Comics post. Thanks kindly for reading and we’ll see when in 2019 I feel up to doing another of these.

My 2018 Mathematics A To Z: Yamada Polynomial


I had another free choice. I thought I’d go back to one of the topics I knew and loved in grad school even though I didn’t have the time to properly study it then. It turned out I had forgotten some important points and spent a night crash-relearning knot theory. This isn’t a bad thing necessarily.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Yamada Polynomial.

This is a thing which comes from graphs. Not the graphs you ever drew in algebra class. Graphs as in graph theory. These figures made of spots called vertices. Pairs of vertices are connected by edges. There’s many interesting things to study about these.

One path to take in understanding graphs is polynomials. Of course I would bring things back to polynomials. But there’s good reasons. These reasons come to graph theory by way of knot theory. That’s an interesting development since we usually learn graph theory before knot theory. But knot theory has the idea of representing these complicated shapes as polynomials.

There are a bunch of different polynomials for any given graph. The oldest kind, the Alexander Polynomial, J W Alexander developed in the 1920s. And that was about it until the 1980s when suddenly everybody was coming up with good new polynomials. The definitions are different. They give polynomials that look different. Some are able to distinguish between a knot and the knot that’s its reflection across a mirror. Some, like the Alexander aren’t. But they’re common in some important ways. One is that they might not actually be, you know, polynomials. I mean, they’ll be the sum of numbers — whole numbers, even — times a variable raised to a power. The variable might be t, might be x. Might be something else, but it doesn’t matter. It’s a pure dummy variable. But the variable might be raised to a negative power, which isn’t really a polynomial. It might even be raised to, oh, one-half or three-halves, or minus nine-halves, or something like that. We can try saying this is “a polynomial in t-to-the-halves”. Mostly it’s because we don’t have a better name for it.

And going from a particular knot to a polynomial follows a pretty common procedure. At least it can, when you’re learning knot theory and feel a bit overwhelmed trying to prove stuff about “knot invariants” and “homologies” and all. Having a specific example can be such a comfort. You can work this out by an iterative process. Take a specific drawing of your knot. There’s places where the strands of the knot cross over one another. For each of those crossings you ponder some alternate cases where the strands cross over in a different way. And then you add together some coefficient times the polynomial of this new, different knot. The coefficient you get by the rules of whatever polynomial you’re making. The new, different knots are, usually, no more complicated than what you started with. They’re often simpler knots. This is what saves you from an eternity of work. You’re breaking the knot down into more but simpler knots. Just the fact of doing that can be satisfying enough. Eventually you get to something really simple, like a circle, and declare that’s some basic polynomial. Then there’s a lot bit of adding up coefficients and powers and all that. Tedious but not hard.

Knots are made from a continuous loop of … we’ll just call it thread. It can fold over itself many times. It has to, really, or it hasn’t got a chance of being more interesting than a circle. A graph is different. That there are vertices seems to change things. Less than you’d think, though. The thread of a knot can cross over and under itself. Edges of a graph can cross over and under other edges. This isn’t too different. We can also imagine replacing a spot where two edges cross over and under the other with an intersection and new vertex.

So we get to the Yamada polynomial by treating a graph an awful lot like we might treat a knot. Take the graph and split it up at each overlap. At each overlap we have something that looks, at least locally, kind of like an X. An upper left, upper right, lower left, and lower right intersection. The lower left connects to the upper right, and the upper left connects to the lower right. But these two edges don’t actually touch; one passes over the other. (By convention, the lower left going to the upper right is on top.)

There’s three alternate graphs. One has the upper left connected to the lower left, and the upper right connected to the lower right. This looks like replacing the X with a )( loop. The second alternate has the upper left connected to the upper right, and the lower left connected to the lower right. This looks like … well, that )( but rotated ninety degrees. I can’t do that without actually including a picture. The third alternate puts a vertex in the X. So now the upper left, upper right, lower left, and lower right all connect to the new vertex in the center.

Probably you’d agree that replacing the original X with a )( pattern, or its rotation, probably doesn’t make the graph any more complicated. And it might make the graph simpler. But adding that new vertex looks like trouble. It looks like it’s getting more complicated. We might get stuck in an infinite regression of more-complicated polynomials.

What saves us is the coefficient we’re multiplying the polynomials for these new graphs by. It’s called the “chromatic coefficient” and it reflects how many different colors you need to color in this graph. An edge needs to connect two different colors. And — what happens if an edge connects a vertex to itself? That is, the edge loops around back to where it started? That’s got a chromatic number of zero and the moment we get a single one of these loops anywhere in our graph we can stop calculating. We’re done with that branch of the calculations. This is what saves us.

There’s a catch. It’s a catch that knot polynomials have, too. This scheme writes a polynomial not just for a particular graph but a particular way of rendering this graph. There’s always other ways to draw it. If nothing else you can always twirl a edge over itself, into a loop like you get when Christmas tree lights start tangling themselves up. But you can move the vertices to different places. You can have an edge go outside the rest of the figure instead of inside, that sort of thing. Starting from a different rendition of the shape gets you to a different polynomial.

Superficially different, anyway. What you get from two different renditions of the same graph are polynomials different by your dummy variable raised to a whole number. Also maybe a plus-or-minus sign. You can see a difference between, say, t^{-1} - 2 + 3t (to make up an example) and t - 2t^2 + 3t^3 . But you can see that second polynomial is just t^2\left(t^{-1} - 2 + 3t\right) . It’s some confounding factor times something that is distinctive to the graph.

And that distinctive part, the thing that doesn’t change if you draw the graph differently? That’s the Yamada polynomial, at last. It’s a way to represent this collection of vertices and edges using only coefficients and exponents.

I would like to give an impressive roster of uses for these polynomials here. I’m afraid I have to let you down. There is the obvious use: if you suspect two graphs are really the same, despite how different they look, here’s a test. Calculate their Yamada polynomials and if they’re different, you know the graphs were different. It can be hard to tell. Get anything with more than, say, eight vertices and 24 edges in it and you’re not going to figure that out by sight.

I encountered the Yamada polynomial specifically as part of a textbook chapter about chemistry. It’s easy to imagine there should be great links between knots and graphs and the way that atoms bundle together into molecules. The shape of their structures describes what they will do. But I am not enough of a chemist to say how this description helps chemists understand molecules. It’s possible that it doesn’t: Yamada’s paper introducing the polynomial was published in 1989. My knot theory textbook might have brought it up because it looked exciting. There are trends and fashions in mathematical thought too. I don’t know what several more decades of work have done to the polynomial’s reputation. I’m glad to hear from people who know better.


There’s one more term in the Fall 2018 Mathematics A To Z to come. Will I get the article about it written before Friday? We’ll know on Saturday! At least I don’t have more Reading the Comics posts to write before Sunday.

Reading the Comics, December 15, 2018: Early Holiday Edition


So then this happened: Comic Strip Master Command didn’t have much they wanted me to write about this week. I made out three strips as being relevant enough to discuss at all. And even they don’t have topics that I felt I could really dig into. Coincidence, surely, although I like to think they were trying to help me get ahead of deadline on my A To Z essays for this last week of the run. It’s a noble thought, but doomed. I haven’t been more than one essay ahead of deadline the last three months. I know in past years I’ve gotten three or even four essays ahead of time and I don’t know why it hasn’t worked this time. I am going ahead and blaming that this these essays have been way longer than previous years’. So anyway, I thank Comic Strip Master Command for trying to make my Monday and my Thursday this week be less packed. It won’t help.

Darrin Bell and Theron Heir’s Rudy Park for the 10th uses mathematics as shorthand for a deep, thought-out theory of something. In this case, Randy’s theory of how to interest women. (He has rather a large number of romantic events around him.) It’s easy to suppose that people can be modeled mathematically. Even a crude model, one supposing that people have things they like and dislike, can give us good interesting results. This gets into psychology and sociology though. And probably requires computer modeling to get slightly useful results.

Rudy: 'You're wearing your lab coat. What's up?' Randy: 'Something big. Amending my unified theory of picking up chicks. Check it out.' (It's a blackboard filled with physics equations, as well as a sketch of a woman in a bikini.) Rudy: 'Explain, Doctor.' Randy: 'To start, you'll need a notepad and a gym membership.'
Darrin Bell and Theron Heir’s Rudy Park for the 10th of December, 2018. This strip is a rerun. It originally ran the 11th of January, 2010. Essays mentioning topics raised by Rudy Park are at this link.

Randy’s blackboard has a good number of legitimate equations on it. They’re maybe not so useful to his problem of modeling people, though. The lower left corner, for example, are three of Maxwell’s Equations, describing electromagnetism. I’m not sure about all of these, in part because I think some might be transcribed incorrectly. The second equation in the upper left, for example, looks like it’s getting at the curl of a conserved force field being zero, but it’s idiosyncratic to write that with a ‘d’ to start with. The symbols all over the right with both subscripts and superscripts look to me like tensor work. This turns up in electromagnetism, certainly. Tensors turn up anytime something, such as electrical conductivity, is different in different directions. But I’ve never worked deeply in those fields so all I can confidently say is that they look like they parse.

Nate's story: 'Barky the sheepdog stared in horror at the bloody foot on the barn floor. It was the fifth piece of Farmer Wobblewheel he'd found today. 'And don't forget about the three pieces we found yesterday!' said Winky the wonder monkey.' Franklin: 'What's a monkey doing on a farm?' Nate: 'Helping Barky discover who dismembered Farmer Wobblewheel *and* teaching us about numbers!' Story: ''Five pieces plus three pieces,' barked Barkey. 'That makes ... ' 'Eight,' chuckled Winky.' Francis: 'Ew.'
Lincoln Pierce’s Big Nate for the 14th of December, 2018. Other essays mentioning topics raised by Big Nate, both the current run — like this — and vintage 1990 are at this link.

Lincoln Pierce’s Big Nate for the 14th is part of a bit where Nate’s trying to write a gruesome detective mystery for kids. I’m not sure that’s a ridiculous idea, at least if the gore could be done at a level that wouldn’t be too visceral. Anyway, Nate has here got the idea of merging some educational value into the whole affair. It’s not presented as a story problem, just as characters explaining stuff to one another. There probably would be some room for an actual problem where Barky and Winky wanted to know something and had to work out how to find it from what they knew, though.

Playing in a cardboard box labelled SS Nora Dish. Jingles: 'Take the controls while I make the calculations for hyperspace.' Cecil: 'Wookie noise.' Jingles: 'Let's see. Bob has two bananas. He gives one to Joe who eats half and returns the remainder along with half a cantaloupe ... this ship needs a modern supercomputer.' Cecil: 'Wookie noise.'
Mel Henze’s Gentle Creatures for the 14th of December, 2018. All the essays where I’ve discussed Gentle Creatures are at this link although I suspect it’s mostly the same three comics discussed over and over.

Mel Henze’s Gentle Creatures for the 14th uses a story problem to stand in for science fictional calculations. The strip’s in reruns and I’ve included it here at least four times, I discover, so that’s probably enough for the comic until it gets out of reruns.


And since it was a low-volume week, let me mention strips I didn’t decide fit. Ray Kassinger asked about Tim Rickard’s Brewster Rockit for the 12th. Might it be a play on Schrödinger’s Cat, the famous thought-experiment about how to understand the mathematics of quantum mechanics? It’s possible, but I think it’s more likely just that cats like sitting in boxes. Thaves’s Frank and Ernest for the 13th looks like it should be an anthropomorphic numerals joke. But it’s playing on the idiom about three being a crowd, and the whole of the mathematical content is that three is a number. John Zakour and Scott Roberts’s Maria’s Day for the 15th mentions mathematics. Particularly, Maria wishing they weren’t studying it. It’s a cameo appearance; it could be any subject whose value a student doesn’t see. That’s all I can make of it.


This and my other Reading the Comics posts should all be available at this link. And please check back in Tuesday to see whether I make deadline for the letter ‘Y’ in my Fall 2018 Mathematics A To Z glossary.

Reading the Comics, December 8, 2018: Sam and Son Edition


That there were twelve comic strips making my cut as mention-worthy this week should have let me do three essays of four comics each. But the desire to include all the comics from the same day in one essay leaves me one short here. So be it. Three of the four cartoonists featured here have a name of Sansom or Samson, so, that’s an edition title for you. No, Sam and Silo do not appear here.

Art Sansom and Chip Sansom’s Born Loser for the 6th uses arithmetic as a test of deference. Will someone deny a true thing in order to demonstrate loyalty? Arithmetic is full of things that are inarguably true. If we take the ordinary meanings of one, plus, equals, and three, it can’t be that one plus one equals three. Most fields of human endeavor are vulnerable to personal taste, or can get lost in definitions and technicalities. Or the advance of knowledge: my love and I were talking last night how we remembered hearing, as kids, the trivia that panda bears were not really bears, but a kind of raccoon. (Genetic evidence has us now put giant pandas with the bears, and red pandas as part of the same superfamily as raccoons, but barely.) Or even be subject to sarcasm. Arithmetic has a harder time of that. Mathematical ideas do evolve in time, certainly. But basic arithmetic is pretty stable. Logic is also a reliable source of things we can be confident are true. But arithmetic is more familiar than most logical propositions.

Thornapple: 'You wanted to see me, chief?' Boss: 'Yes, Thornapple. One plus one equals three, am I correct?' Thornapple: 'Yes, sir!' Boss: 'Very good! You may leave.' Thornapple, to audience: 'Every so often, I need to check that the employees are still appropriately subservient.'
Art Sansom and Chip Sansom’s Born Loser for the 6th of December, 2018. Essays about the topics raised by The Born Loser should be at this link. I’m startled to discover this is apparently a new tag, though.

Samson’s Dark Side of the Horse for the 8th is the Roman Numerals joke for the week. It’s also a bit of a wordplay joke, although the music wordplay rather tha mathematics. Me, I still haven’t heard a clear reason why ‘MIC’ wouldn’t be a legitimate Roman numeral representation of 1099. I’m not sure whether ‘MIC’ would step on or augment the final joke, though.

Horace, trying to get to sleep, imagining sheep jumping a fence: MXCVII (1098). MXCIX (1099). MC (1100); it's a rapper sheep with a huge medallion and microphone.
Samson’s Dark Side of the Horse for the 8th of December, 2018. This and other essays mentioning Dark Side Of The Horse are at this link. This is certainly not a new tag.

Pab Sungenis’s New Adventures of Queen Victoria for the 8th has a comedia dell’arte-based structure for its joke. (The strip does that, now and then.) The comic uses a story problem, with the calculated answer rejected for the nonsense it would be. I suppose it must be possible for someone to eat eighty apples over a long enough time that it’s not distressing, and yet another twenty apples wouldn’t spoil. I wouldn’t try it, though.

Funnies dell'Arte. Arlecchino: 'A man has 100 apples. He eats 80. What does he have?' Newton: '20.' Arlecchino: 'No! A stomach ache! Ha ha ha ha ha!' Newton, leaving: 'I'm not surprised.' Arlecchino, calling after: 'Comedy is just something that happens to other people are far as you're concerned, huh?!'
Pab Sungenis’s New Adventures of Queen Victoria for the 8th of December, 2018. Essays based on stuff mentioned in New Adventures of Queen Victoria should be at this link. This also seems to be a new tag, somehow, and that doesn’t make sense to me.

This and my other Reading the Comics posts should all be available at this link.

My 2018 Mathematics A To Z: Extreme Value Theorem


The letter ‘X’ is a problem. For all that the letter ‘x’ is important to mathematics there aren’t many mathematical terms starting with it. Mr Wu, mathematics tutor and author of the MathTuition88 blog, had a suggestion. Why not 90s it up a little and write about an Extreme theorem? I’m game.

The Extreme Value Theorem, which I chose to write about, is a fundamental bit of analysis. There is also a similarly-named but completely unrelated Extreme Value Theory. This exists in the world of statistics. That’s about outliers, and about how likely it is you’ll find an even more extreme outlier if you continue sampling. This is valuable in risk assessment: put another way, it’s the question of what neighborhoods you expect to flood based on how the river’s overflowed the last hundred years. Or be in a wildfire, or be hit by a major earthquake, or whatever. The more I think about it the more I realize that’s worth discussing too. Maybe in the new year, if I decide to do some A To Z extras.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Extreme Value Theorem.

There are some mathematical theorems which defy intuition. You can encounter one and conclude that can’t be so. This can inspire one to study mathematics, to understand how it could be. Famously, the philosopher Thomas Hobbes encountered the Pythagorean Theorem and disbelieved it. He then fell into a controversial love with the subject. Some you can encounter, and study, and understand, and never come to believe. This would be the Banach-Tarski Paradox. It’s the realization that one can split a ball into as few as five pieces, and reassemble the pieces, and have two complete balls. They can even be wildly larger or smaller than the one you started with. It’s dazzling.

And then there are theorems that seem the opposite. Ones that seem so obvious, and so obviously true, that they hardly seem like mathematics. If they’re not axioms, they might as well be. The extreme value theorem is one of these.

It’s a theorem about functions. Here, functions that have a domain and a range that are both real numbers. Even more specifically, about continuous functions. “Continuous” is a tricky idea to make precise, but we don’t have to do it. A century of mathematicians worked out meanings that correspond pretty well to what you’d imagine it should mean. It means you can draw a graph representing the function without lifting the pen. (Do not attempt to use this definition at your thesis defense. I’m skipping what a century’s worth of hard thinking about the subject.)

And it’s a theorem about “extreme” values. “Extreme” is a convenient word. It means “maximum or minimum”. We’re often interested in the greatest or least value of a function. Having a scheme to find the maximum is as good as having one to find a minimum. So there’s little point talking about them as separate things. But that forces us to use a bunch of syllables. Or to adopt a convention that “by maximum we always mean maximum or minimum”. We could say we mean that, but I’ll bet a good number of mathematicians, and 95% of mathematics students, would forget the “or minimum” within ten minutes. “Extreme”, then. It’s short and punchy and doesn’t commit us to a maximum or a minimum. It’s simply the most outstanding value we can find.

The Extreme Value Theorem doesn’t help us find them. It only proves to us there is an extreme to find. Particularly, it says that if a continuous function has a domain that’s a closed interval, then it has to have a maximum and a minimum. And it has to attain the maximum and the minimum at least once each. That is, something in the domain matches to the maximum. And something in the domain matches to the minimum. Could be multiple times, yes.

This might not seem like much of a theorem. Existence proofs rarely do. It’s a bias, I suppose. We like to think we’re out looking for solutions. So we suppose there’s a solution to find. Checking that there is an answer before we start looking? That seems excessive. Before heading to the airport we might check the flight wasn’t delayed. But we almost never check that there is still a Newark to fly to. I’m not sure, in working out problems, that we check it explicitly. We decide early on that we’re working with continuous functions and so we can try out the usual approaches. That we use the theorem becomes invisible.

And that’s sort of the history of this theorem. The Extreme Value Theorem, for example, is part of how we now prove Rolle’s Theorem. Rolle’s theorem is about functions continuous and differentiable on the interval from a to b. And functions that have the same value for a and for b. The conclusion is the function hass got a local maximum or minimum in-between these. It’s the theorem depicted in that xkcd comic you maybe didn’t check out a few paragraphs ago. Rolle’s Theorem is named for Michael Rolle, who prove the theorem (for polynomials) in 1691. The Indian mathematician Bhaskara II, in the 12th century, is credited with stating the theorem too. The Extreme Value Theorem was proven around 1860. (There was an earlier proof, by Bernard Bolzano, whose name you’ll find all over talk about limits and functions and continuity and all. But that was unpublished until 1930. The proofs known about at the time were done by Karl Weierstrass. His is the other name you’ll find all over talk about limits and functions and continuity and all. Go on, now, guess who it was proved the Extreme Value Theorem. And guess what theorem, bearing the name of two important 19th-century mathematicians, is at the core of proving that. You need at most two chances!) That is, mathematicians were comfortable using the theorem before it had a clear identity.

Once you know that it’s there, though, the Extreme Value Theorem’s a great one. It’s useful. Rolle’s Theorem I just went through. There’s also the quite similar Mean Value Theorem. This one is about functions continuous and differentiable on an interval. It tells us there’s at least one point where the derivative is equal to the mean slope of the function on that interval. This is another theorem that’s a quick proof once you have the Extreme Value Theorem. Or we can get more esoteric. There’s a technique known as Lagrange Multipliers. It’s a way to find where on a constrained surface a function is at its maximum or minimum. It’s a clever technique, one that I needed time to accept as a thing that could possibly work. And why should it work? Go ahead, guess what the centerpiece of at least one method of proving it is.

Step back from calculus and into real analysis. That’s the study of why calculus works, and how real numbers work. The Extreme Value Theorem turns up again and again. Like, one technique for defining the integral itself is to approximate a function with a “stepwise” function. This is one that looks like a pixellated, rectangular approximation of the function. The definition depends on having a stepwise rectangular approximation that’s as close as you can get to a function while always staying less than it. And another stepwise rectangular approximation that’s as close as you can get while always staying greater than it.

And then other results. Often in real analysis we want to know about whether sets are closed and bounded. The Extreme Value Theorem has a neat corollary. Start with a continuous function with domain that’s a closed and bounded interval. Then, this theorem demonstrates, the range is also a closed and bounded interval. I know this sounds like a technical point. But it is the sort of technical point that makes life easier.

The Extreme Value Theorem even takes on meaning when we don’t look at real numbers. We can rewrite it in topological spaces. These are sets of points for which we have an idea of a “neighborhood” of points. We don’t demand that we know what distance is exactly, though. What had been a closed and bounded interval becomes a mathematical construct called a “compact set”. The idea of a continuous function changes into one about the image of an open set being another open set. And there is still something recognizably the Extreme Value Theorem. It tells us about things called the supremum and infimum, which are slightly different from the maximum and minimum. Just enough to confuse the student taking real analysis the first time through.

Topological spaces are an abstracted concept. Real numbers are topological spaces, yes. But many other things also are. Neighborhoods and compact sets and open sets are also abstracted concepts. And so this theorem has its same quiet utility in these many spaces. It’s just there quietly supporting more challenging work.


And now I get to really relax: I already have a Reading the Comics post ready for tomorrow, and Sunday’s is partly written. Now I just have to find a mathematical term starting with ‘Y’ that’s interesting enough to write about.

Reading the Comics, December 5, 2018: December 5, 2018 Edition


And then I noticed there were a bunch of comic strips with some kind of mathematical theme on the same day. Always fun when that happens.

Bill Holbrook’s On The Fastrack uses one of Holbrook’s common motifs. That’s the depicting as literal some common metaphor. in this case it’s “massaging the numbers”, which might seem not strictly mathematics. But while numbers are interesting, they’re also useful. To be useful they must connect to something we want to know. They need context. That context is always something of human judgement. If the context seems inappropriate to the listener, she thinks the presenter is massaging the numbers. If the context seems fine, we trust the numbers as showing something truth.

A man making a report touches a figure 8, reducing it to a wobbly mess. He finally has several wrinkled, flattened numbers dangling over the 'screen' edge. Fi: 'You massage these numbers, didn't you?' Man: 'No! They're naturally relaxed!'
Bill Holbrook’s On The Fastrack for the 5th of December, 2018. Essays inspired by On The Fastrack appear at this link.

Scott Hilburn’s The Argyle Sweater is a seasonal pun that couldn’t wait for a day closer to Christmas. I’m a little curious why not. It would be the same joke with any subject, certainly. The strip did make me wonder if Ebeneezer Scrooge, in-universe, might have taken calculus. This lead me to see that it’s a bit vague what, precisely, Scrooge, or Scrooge-and-Marley, did. The movies are glad to position him as having a warehouse, and importing and exporting things, and making and collecting on loans and whatnot. These are all trades that mathematicians would like to think benefit from knowing advanced mathematics. The logic of making loans implies attention be paid to compounding interest, risks, and expectation values, as well as projecting cash-flow a fair bit into the future. But in the original text he doesn’t make any stated loans, and the only warehouse anyone enters is Fezziwig’s. Well, the Scrooge and Marley sign stands “above the warehouse door”, but we only ever go in to the counting-house. And yes, what Scrooge does besides gather money and misery is irrelevant to the setting of the story.

Caption: 'Young Ebeneezer Scrooge gets a visit from the Ghost of Calculus Passed.' The Ghost holds up a D+ paper, terrifying Scrooge in his bed. The Ghost: 'If you'd only studied you could've gotten a C.'
Scott Hilburn’s The Argyle Sweater for the 5th of December, 2018. Some of the many times I’ve talked about The Argyle Sweater appear at this link.

Teresa Burritt’s Dadaist strip Frog Applause uses knowledge of mathematics as an emblem of intelligence. “Multivariate analysis” is a term of art from statistics. It’s about measuring how one variable changes depending on two or more other variables. The goal is obvious: we know there are many things that influence anything of interest. Can we find what things have the strongest effects? The weakest effects? There are several ways we might mean “strongest” effect, too. It might mean that a small change in the independent variable produces a big change in the dependent one. Or it might mean that there’s very little noise, that a change in the independent variable produces a reliable change in the dependent one. Or we might have several variables that are difficult to measure precisely on their own, but with a combination that’s noticeable. The basic calculations for this look a lot like those for single-variable analysis. But there’s much more calculation. It’s more tedious, at least. My reading suggests that multivariate analysis didn’t develop much until there were computers cheap enough to do the calculations. Might be coincidence, though. Many machine-learning techniques can be described as multivariate analysis problems.

Caption: 'She knew she was smarter than he was, but she married him anyway.' Clip-art woman comforting a seated man; 'Look at it this way. If you don't know what a multivariate analysis is, you probably can't do one.'
Teresa Burritt’s Frog Applause for the 5th of December, 2018. Essays with reason to mention Frog Applause should be at this link.

Greg Evans’s Luann Againn is a Pi Day joke from before the time when Pi Day was a thing. Brad’s magazine flipping like that is an unusual bit of throwaway background humor for the comic strip.

Luann: 'Brad, how much is 'pi'?' Brad: 'A whole one or just a slice?'
Greg Evans’s Luann Againn for the 5th of December, 2018. This originally ran the 5th of December, 1990. Essays which mention Luann, current-day or 1990-vintage rerun, appear at this link.

Doug Savage’s Savage Chickens is a bunch of shape jokes. Since I was talking about tiling the plane so recently the rhombus seemed on-point enough. I’m think the irregular heptagon shown here won’t tile the plane. But given how much it turns out I didn’t know, I wouldn’t want to commit to that.

Title: Ninja Weapons in descending Order of Effectiveness. Ninja Star; Ninja Rhombus; Ninja Irregular Heptagon; Ninja Sturgeon.
Doug Savage’s Savage Chickens for the 5th of December, 2018. Essays that mention Savage Chickens are at this link.

I’m working hard on a latter ‘X’ essay for my Fall 2018 Mathematics A To Z glossary. That should appear on Friday. And there should be another Reading the Comics post later this week, at this link.

My 2018 Mathematics A To Z: Witch of Agnesi


Nobody had a suggested topic starting with ‘W’ for me! So I’ll take that as a free choice, and get lightly autobiogrpahical.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Witch of Agnesi.

I know I encountered the Witch of Agnesi while in middle school. Eighth grade, if I’m not mistaken. It was a footnote in a textbook. I don’t remember much of the textbook. What I mostly remember of the course was how much I did not fit with the teacher. The only relief from boredom that year was the month we had a substitute and the occasional interesting footnote.

It was in a chapter about graphing equations. That is, finding curves whose points have coordinates that satisfy some equation. In a bit of relief from lines and parabolas the footnote offered this:

y = \frac{8a^3}{x^2 + 4a^2}

In a weird tantalizing moment the footnote didn’t offer a picture. Or say what an ‘a’ was doing in there. In retrospect I recognize ‘a’ as a parameter, and that different values of it give different but related shapes. No hint what the ‘8’ or the ‘4’ were doing there. Nor why ‘a’ gets raised to the third power in the numerator or the second in the denominator. I did my best with the tools I had at the time. Picked a nice easy boring ‘a’. Picked out values of ‘x’ and found the corresponding ‘y’ which made the equation true, and tried connecting the dots. The result didn’t look anything like a witch. Nor a witch’s hat.

It was one of a handful of biographical notes in the book. These were a little attempt to add some historical context to mathematics. It wasn’t much. But it was an attempt to show that mathematics came from people. Including, here, from Maria Gaëtana Agnesi. She was, I’m certain, the only woman mentioned in the textbook I’ve otherwise completely forgotten.

We have few names of ancient mathematicians. Those we have are often compilers like Euclid whose fame obliterated the people whose work they explained. Or they’re like Pythagoras, credited with discoveries by people who obliterated their own identities. In later times we have the mathematics done by, mostly, people whose social positions gave them time to write mathematics results. So we see centuries where every mathematician is doing it as their side hustle to being a priest or lawyer or physician or combination of these. Women don’t get the chance to stand out here.

Today of course we can name many women who did, and do, mathematics. We can name Emmy Noether, Ada Lovelace, and Marie-Sophie Germain. Challenged to do a bit more, we can offer Florence Nightingale and Sofia Kovalevskaya. Well, and also Grace Hopper and Margaret Hamilton if we decide computer scientists count. Katherine Johnson looks likely to make that cut. But in any case none of these people are known for work understandable in a pre-algebra textbook. This must be why Agnesi earned a place in this book. She’s among the earliest women we can specifically credit with doing noteworthy mathematics. (Also physics, but that’s off point for me.) Her curve might be a little advanced for that textbook’s intended audience. But it’s not far off, and pondering questions like “why 8a^3 ? Why not a^3 ?” is more pleasant, to a certain personality, than pondering what a directrix might be and why we might use one.

The equation might be a lousy way to visualize the curve described. The curve is one of that group of interesting shapes you get by constructions. That is, following some novel process. Constructions are fun. They’re almost a craft project.

For this we start with a circle. And two parallel tangent lines. Without loss of generality, suppose they’re horizontal, so, there’s lines at the top and the bottom of the curve.

Take one of the two tangent points. Again without loss of generality, let’s say the bottom one. Draw a line from that point over to the other line. Anywhere on the other line. There’s a point where the line you drew intersects the circle. There’s another point where it intersects the other parallel line. We’ll find a new point by combining pieces of these two points. The point is on the same horizontal as wherever your line intersects the circle. It’s on the same vertical as wherever your line intersects the other parallel line. This point is on the Witch of Agnesi curve.

Now draw another line. Again, starting from the lower tangent point and going up to the other parallel line. Again it intersects the circle somewhere. This gives another point on the Witch of Agnesi curve. Draw another line. Another intersection with the circle, another intersection with the opposite parallel line. Another point on the Witch of Agnesi curve. And so on. Keep doing this. When you’ve drawn all the lines that reach from the tangent point to the other line, you’ll have generated the full Witch of Agnesi curve. This takes more work than writing out y = \frac{8a^3}{x^2 + 4a^2} , yes. But it’s more fun. It makes for neat animations. And I think it prepares us to expect the shape of the curve.

It’s a neat curve. Between it and the lower parallel line is an area four times that of the circle that generated it. The shape is one we would get from looking at the derivative of the arctangent. So there’s some reasons someone working in calculus might find it interesting. And people did. Pierre de Fermat studied it, and found this area. Isaac Newton and Luigi Guido Grandi studied the shape, using this circle-and-parallel-lines construction. Maria Agnesi’s name attached to it after she published a calculus textbook which examined this curve. She showed, according to people who present themselves as having read her book, the curve and how to find it. And she showed its equation and found the vertex and asymptote line and the inflection points. The inflection points, here, are where the curve chances from being cupped upward to cupping downward, or vice-versa.

It’s a neat function. It’s got some uses. It’s a natural smooth-hill shape, for example. So this makes a good generic landscape feature if you’re modeling the flow over a surface. I read that solitary waves can have this curve’s shape, too.

And the curve turns up as a probability distribution. Take a fixed point. Pick lines at random that pass through this point. See where those lines reach a separate, straight line. Some regions are more likely to be intersected than are others. Chart how often any particular line is the new intersection point. That chart will (given some assumptions I ask you to pretend you agree with) be a Witch of Agnesi curve. This might not surprise you. It seems inevitable from the circle-and-intersecting-line construction process. And that’s nice enough. As a distribution it looks like the usual Gaussian bell curve.

It’s different, though. And it’s different in strange ways. Like, for a probability distribution we can find an expected value. That’s … well, what it sounds like. But this is the strange probability distribution for which the law of large numbers does not work. Imagine an experiment that produces real numbers, with the frequency of each number given by this distribution. Run the experiment zillions of times. What’s the mean value of all the zillions of generated numbers? And it … doesn’t … have one. I mean, we know it ought to, it should be the center of that hill. But the calculations for that don’t work right. Taking a bigger sample makes the sample mean jump around more, not less, the way every other distribution should work. It’s a weird idea.

Imagine carving a block of wood in the shape of this curve, with a horizontal lower bound and the Witch of Agnesi curve as the upper bound. Where would it balance? … The normal mathematical tools don’t say, even though the shape has an obvious line of symmetry. And a finite area. You don’t get this kind of weirdness with parabolas.

(Yes, you’ll get a balancing point if you actually carve a real one. This is because you work with finitely-long blocks of wood. Imagine you had a block of wood infinite in length. Then you would see some strange behavior.)

It teaches us more strange things, though. Consider interpolations, that is, taking a couple data points and fitting a curve to them. We usually start out looking for polynomials when we interpolate data points. This is because everything is polynomials. Toss in more data points. We need a higher-order polynomial, but we can usually fit all the given points. But sometimes polynomials won’t work. A problem called Runge’s Phenomenon can happen, where the more data points you have the worse your polynomial interpolation is. The Witch of Agnesi curve is one of those. Carl Runge used points on this curve, and trying to fit polynomials to those points, to discover the problem. More data and higher-order polynomials make for worse interpolations. You get curves that look less and less like the original Witch. Runge is himself famous to mathematicians, known for “Runge-Kutta”. That’s a family of techniques to solve differential equations numerically. I don’t know whether Runge came to the weirdness of the Witch of Agnesi curve from considering how errors build in numerical integration. I can imagine it, though. The topics feel related to me.

I understand how none of this could fit that textbook’s slender footnote. I’m not sure any of the really good parts of the Witch of Agnesi could even fit thematically in that textbook. At least beyond the fact of its interesting name, which any good blog about the curve will explain. That there was no picture, and that the equation was beyond what the textbook had been describing, made it a challenge. Maybe not seeing what the shape was teased the mathematician out of this bored student.


And next is ‘X’. Will I take Mr Wu’s suggestion and use that to describe something “extreme”? Or will I take another topic or suggestion? We’ll see on Friday, barring unpleasant surprises. Thanks for reading.

Reading the Comics, December 4, 2018: Christmas Specials Edition


This installment took longer to write than you’d figure, because it’s the time of year we’re watching a lot of mostly Rankin/Bass Christmas specials around here. So I have to squeeze words out in-between baffling moments of animation and, like, arguing whether there’s any possibility that Jack Frost was not meant to be a Groundhog Day special that got rewritten to Christmas because the networks weren’t having it otherwise.

Graham Nolan’s Sunshine State for the 3rd is a misplaced Pi Day strip. I did check the copyright to see if it might be a rerun from when it was more seasonal.

Liz: 'I'm going to bake pies. What's your favorite?' 'Cherry!' 'Apple!' Liz 'Here comes Paul! Let's ask him, too.' Dink: 'He hates pie!' Paul: 'What are you talking about?' Dink: 'Nothing that would interest you.' Mel: 'We're talking about pie!' Paul: 'So you don't think I'm smart enough to discuss pi? Pi is the ratio of a circle's circumference to its diameter! It's a mathematical constant used in mathematics and physics! Its value is approximately 3.14159!' Mel: 'You forgot the most important thing about pie!' Paul: 'What's that?' Mel: 'It tastes delicious!' Dink: 'I hate pie!' Mel, Dink, and Liz: 'We know!'
Graham Nolan’s Sunshine State for the 3rd of December, 2018. This and other essays mentioning Sunshine State should be at this link. Or will be someday; it’s a new tag. Yeah, Paul’s so smart he almost knows the difference between it’s and its.

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 3rd is the anthropomorphic numerals joke for the week. … You know, I’ve always wondered in this sort of setting, what are two-digit numbers like? I mean, what’s the difference between a twelve and a one-and-two just standing near one another? How do people recognize a solitary number? This is a darned silly thing to wonder so there’s probably a good web comic about it.

An Old West town. an anthropomorphic 2 says to a 4, 'You know, Slim, I don't like the odds.' Standing opposite them, guns at the ready, are a hostile 5, 1, 3, and 7.
Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 3rd of December, 2018. Essays inspired by Yaffle should appear at this link. It’s also a new tag, so don’t go worrying that there’s only this one essay there yet.

John Hambrock’s The Brilliant Mind of Edison Lee for the 4th has Edison forecast the outcome of a basketball game. I can’t imagine anyone really believing in forecasting the outcome, though. The elements of forecasting a sporting event are plausible enough. We can suppose a game to be a string of events. Each of them has possible outcomes. Some of them score points. Some block the other team’s score. Some cause control of the ball (or whatever makes scoring possible) to change teams. Some take a player out, for a while or for the rest of the game. So it’s possible to run through a simulated game. If you know well enough how the people playing do various things? How they’re likely to respond to different states of things? You could certainly simulate that.

Harley: 'C'mon, Edison, let's play basketball.' Edison: 'If I take into account the size and weight of the ball, the diameter of the hoop and your height in relation to it, and the number of hours someone your age would've had time to practice ... I can conclude that I'd win by 22 points. Nice game. Better luck next time.' Harley: 'But ... '
John Hambrock’s The Brilliant Mind of Edison Lee for the 4th of December, 2018. More ideas raised by Edison Lee I discuss at this link. Also it turns out Edison’s friend here is named Harley, which I mention so I have an easier time finding his name next time I need to refer to this strip. This will not work.

But all sorts of crazy things will happen, one game or another. Run the same simulation again, with different random numbers. The final score will likely be different. The course of action certainly will. Run the same simulation many times over. Vary it a little; what happens if the best player is a little worse than average? A little better? What if the referees make a lot of mistakes? What if the weather affects the outcome? What if the weather is a little different? So each possible outcome of the sporting event has some chance. We have a distribution of the possible results. We can judge an expected value, and what the range of likely outcomes is. This demands a lot of data about the players, though. Edison Lee can have it, I suppose. The premise of the strip is that he’s a genius of unlimited competence. It would be more likely to expect for college and professional teams.

Rover, dog: 'Can I help with your homework?' Red, kid: 'How are you at long division?' Rover: 'OK, I guess. Lemme see the problem first.' (Red holds the notes out to Rover, who tears the page off and chews it up.) Red: 'That was actually short division, but it'll do nicely for now.'
Brian Basset’s Red and Rover for the 4th of December, 2018. And more Red and Rover discussions are at this link.

Brian Basset’s Red and Rover for the 4th uses arithmetic as the homework to get torn up. I’m not sure it’s just a cameo appearance. It makes a difference to the joke as told that there’s division and long division, after all. But it could really be any subject.


I’m figuring to get to the letter ‘W’ in my Fall 2018 Mathematics A To Z glossary for Tuesday. Reading the Comics posts this week. And I also figure there should be two more When posted, they’ll be at this link.