## Some fun with Latin Squares

I found a cute, playful bit paper on arXiv.org. Fun with Latin Squares, by Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam (Mira)Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui, and Cindy Wei, appears to be the result of a school research project. A Latin Square is an arrangement of numbers. Like, if you have the whole numbers from 1 to 5, you can make a five-row, five-column magic square, with each number appearing once in each row and column. If you have the numbers from 1 to 10 you can make a ten-row, ten-column magic square, again with each number appearing once per row and column. And so on.

So what the arXiv.org paper does is look at different types of Latin Squares, and whip up some new ones by imposing new rules. Latin Squares are one of those corners of mathematics I haven’t thought about much. But they do connect to other problems, such as sudoku, or knights-tour and similar problems of chess piece movement. So we get enlightenment in those from considering these. And from thinking how we might vary the rules about how to arrange numbers. It’s pleasant, fun exercise.

## Reading the Comics, November 7, 2019: The Casual Mentions Edition

As will sometimes happen it’s inconvenient for met to write up a paragraph or two on the particularly mathematically significant comic strips of the past week. Let me here share the comics that just mentioned mathematics, then, and save the heavy stuff for a bit later on.

Bill Watterson’s Calvin and Hobbes repeat for the 5th has Calvin seeking help with an arithmetic problem.

Greg Evans’s Luann Againn for the 6th — a repeat of Luann from the 6th of November, 1991 — sees Gunther thinking out walking to Luann’s house in the language of a word problem. It doesn’t help clear his thoughts.

Jason Chatfield’s Ginger Meggs for the 7th is the sudoku joke for the week.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 7th uses a knowledge of mathematics as shorthand for general knowledge. The strip does misspell “Pythagorean”. This could be a slip on the cartoonists’ part that got past their editors too. Or it could be an extra joke on how often the know-it-all, really, does not. He’ll just talk a confident game long after everyone else has stopped really listening. (I’m a recovering know-it-all myself. I know how our kind thinks.) Or it might be trolling know-it-alls into correcting them.

And this covers things through to Friday’s comics. I write this not having had the chance to read Saturday’s yet. When I do, and when I have the whole week’s strips to discuss, I’ll have it at this link. Furthermore, this week sees the last quarter of the Fall 2019 A to Z under way. I’m excited to learn what I’m doing for the letter ‘U’ also.

## Reading the Comics, April 24, 2019: Mic Drop Edition Edition

I can’t tell you how hard it is not to just end this review of last week’s mathematically-themed comic strips after the first panel here. It really feels like the rest is anticlimax. But here goes.

John Deering’s Strange Brew for the 20th is one of those strips that’s not on your mathematics professor’s office door only because she hasn’t seen it yet. The intended joke is obvious, mixing the tropes of the Old West with modern research-laboratory talk. “Theoretical reckoning” is a nice bit of word juxtaposition. “Institoot” is a bit classist in its rendering, but I suppose it’s meant as eye-dialect.

What gets it a place on office doors is the whiteboard, though. They’re writing out mathematics which looks legitimate enough to me. It doesn’t look like mathematics though. What’s being written is something any mathematician would recognize. It’s typesetting instructions. Mathematics requires all sorts of strange symbols and exotic formatting. In the old days, we’d handle this by giving the typesetters hazard pay. Or, if you were a poor grad student and couldn’t afford that, deal with workarounds. Maybe just leave space in your paper and draw symbols in later. If your university library has old enough papers you can see them. Maybe do your best to approximate mathematical symbols using ASCII art. So you get expressions that look something like this:

  / 2 pi
|   2
|  x cos(theta) dx - 2 F(theta) == R(theta)
|
/ 0


This gets old real fast. Mercifully, Donald Knuth, decades ago, worked out a great solution. It uses formatting instructions that can all be rendered in standard, ASCII-available text. And then by dark incantations and summoning of Linotype demons, re-renders that as formatted text. It handles all your basic book formatting needs — much the way HTML, used for web pages, will — and does mathematics much more easily. For example, I would enter a line like:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)


And this would be rendered in print as:

$\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)$

There are many, many expansions available to this, to handle specialized needs, hardcore mathematics among them.

Anyway, the point that makes me realize John Deering was aiming at everybody with an advanced degree in mathematics ever with this joke, using a string of typesetting instead of the usual equations here?

The typesetting language is named TeX.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. It’s about one of those questions that nags at you as a kid, and again periodically as an adult. The perimeter is the boundary around a shape. The circumference is the boundary around a circle. Why do we have two words for this? And why do we sound all right talking about either the circumference or the perimeter of a circle, while we sound weird talking about the circumference of a rhombus? We sound weird talking about the perimeter of a rhombus too, but that’s the rhombus’s fault.

The easy part is why there’s two words. Perimeter is a word of Greek origin; circumference, of Latin. Perimeter entered the English language in the early 15th century; circumference in the 14th. Why we have both I don’t know; my suspicion is either two groups of people translating different geometry textbooks, or some eager young Scholastic with a nickname like ‘Doctor Magnifico Triangulorum’ thought Latin sounded better. Perimeter stuck with circules early; every etymology I see about why we use the symbol π describes it as shorthand for the perimeter of the circle. Why `circumference’ ended up the word for circles or, maybe, ellipses and ovals and such is probably the arbitrariness of language. I suspect that opening “circ” sound cues people to think of it for circles and circle-like shapes, in a way that perimeter doesn’t. But that is my speculation and should not be mistaken for information.

Steve McGarry’s KidTown for the 21st is a kids’s information panel with a bit of talk about representing numbers. And, in discussing things like how long it takes to count to a million or a billion, or how long it would take to type them out, it gets into how big these numbers can be. Les Stewart typed out the English names of numbers, in words, by the way. He’d also broken the Australian record for treading water, and for continuous swimming.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th is a sudoku comic. Betty makes the common, and understandable, conflation of arithmetic with mathematics. But she’s right in identifying sudoku as a logical rather than an arithmetic problem. You can — and sometimes will see — sudoku type puzzles rendered with icons like stars and circles rather than numerals. That you can make that substitution should clear up whether there’s arithmetic involved. Commenters at GoComics meanwhile show a conflation of mathematics with logic. Certainly every mathematician uses logic, and some of them study logic. But is logic mathematics? I’m not sure it is, and our friends in the philosophy department are certain it isn’t. But then, if something that a recognizable set of mathematicians study as part of their mathematics work isn’t mathematics, then we have a bit of a logic problem, it seems.

Come Sunday I should have a fresh Reading the Comics essay available at this link.

## Reading the Comics, November 20, 2018: What Mathematics Is For Edition

The first half of last week’s comics offered another bunch of chances to think about what mathematics is for. Before I do get into all that, though, may I mention the most recent update of Gregory Taylor’s serial:

It does conclude with a vote about the next direction to take. So it’s a good chance for people who like to see authors twisting to their audience’s demands.

Mort Walker and Dik Browne’s Hi and Lois for the 23rd of May, 1961 builds off a major use of arithmetic. Budgeting doesn’t get much attention from mathematicians. I suppose it seems to us like all the basic problems are solved: adding? Subtracting? Multiplication? All familiar things. Especially now with decimal currency. There are great unsolved problems in mathematics, but they get into specialized areas of financial mathematics and just don’t matter for ordinary household budgeting.

Hi comes across a bit harsh here. I’m going to suppose he was taken so by surprise by Lois’s problem that he spoke without thinking.

Scott Hilburn’s The Argyle Sweater for the 19th is the anthropomorphic numerals strip for the week. With the title of “improper fractions” it’s wordplay on the common meaning for a mathematical term. Two times over, come to it. That negative refers to a class of numbers as well as disapproval of something is ordinary enough. I’ve mentioned it, I estimate, 840 times this month alone.

Jokes about the technical and common meanings of “improper” are rarer. In a proper fraction, the numerator is a smaller number than the denominator. In an improper fraction, we don’t count on that. I remember a modest bit of time in elementary and middle school working on converting improper fractions into mixed fractions — a whole number plus a proper fraction. And also don’t remember anyone caring about that after calculus. In most arithmetic work, there’s not much that’s easier about “1 + 1/2” than about “3/2”. The one major convenience “1 + 1/2” has is that it’s easy to tell at a glance how big the number is. It’s not mysterious how big a number 3/2 is, but that’s because of long familiarity. If I asked you whether 54/17 or 46/13 was the larger number, you’d be fairly stumped and maybe cranky. So there’s not much reason to worry about improper fractions while you’re doing work. For the final presentation of an answer, proper or mixed fractions may well be better.

Whoever colored that minus symbol before the 5 screwed up and confused the joke. Syndicated cartoonists give precise coloring instructions for Sunday strips. But many of them don’t, or aren’t able to, give coloring instructions for weekday strips like this. And mistakes like that are the unfortunate result.

Pascal Wyse and Joe Berger’s Berger and Wyse for the 19th features a sudoku appearance. It’s labelled a diversion, and so it is, as many mathematics and logic puzzles will be. The lone commenter at GoComics claims to have solved the puzzle, so I will suppose they’re being honest about this.

Brian Fies’s Mom’s Cancer for the 19th I have mentioned before, although not since I started including images for all mentioned comics. It’s set a moment when treatment for Mom’s cancer has been declared a great success.

The trouble is, as Feis lays out, volume is three-dimensional. We are pretty good at measuring the length, or at least the greatest width of something. You might call that the “characteristic length”. A linear dimension. But volume scales as the cube of this characteristic length. And the sad thing is that 0.8 times 0.8 times 0.8 is, roughly, 0.5. This means that the characteristic length dropping by 20% drops the volume by 50%. Or, as Feis is disappointed to see in this strip and its successor, the great news of a 50% reduction in the turmor’s mass is that it’s just 20% less big in every direction. It doesn’t look like enough.

Bill Holbrook’s On The Fastrack for the 20th presents one of Fi’s seminars about why mathematics is a good thing. The offscreen student’s question about why one should learn mathematics goes unanswered. As often happens the question is presented as though it’s too absurd to deserve answering. The questioner is conflating “mathematics” with “calculating arithmetic”, yes. And a computer will be better at these calculations. A related question, sometimes asked (and rarely on-topic for my essays here), is why one needs to learn any specific facts when a computer is so much better at finding them.

Knowing facts is not understanding them, no. But it is hard to understand a thing without knowing facts. More, without loving the knowing of facts. If we don’t need to be good at calculating, we do still need to know what to have calculated. And why to calculate that instead of something else. In calculating we can learn things of great beauty. And some of us do go on to mathematics which cannot be calculated. There is software that will do very well at computing, say, the indefinite integral of functions. I don’t know of any that will even start on a problem like “find the kernel of this ring”. But these are problems we see, and think interesting, because our experience in arithmetic trains us to notice them. Perhaps there is new interesting mathematics that we would notice if we didn’t have preconceptions set by times tables and long division. But it is hard to believe that we can’t find it because we’re not ignorant enough. I wouldn’t risk it.

This and more Reading the Comics posts should all be at this link.

And for the rest of the calendar year my Fall 2018 Mathematics A To Z should continue posting new essays. I’m still looking for topics for the last half-dozen letters of the alphabet. Give your mathematics term a try.

## Reading the Comics, March 25, 2017: Slow Week Edition

Slow week around here for mathematically-themed comic strips. These happen. I suspect Comic Strip Master Command is warning me to stop doing two-a-week essays on reacting to comic strips and get back to more original content. Message received. If I can get ahead of some projects Monday and Tuesday we’ll get more going.

Patrick Roberts’s Todd the Dinosaur for the 20th is a typical example of mathematics being something one gets in over one’s head about. Of course it’s fractions. Is there anything in elementary school that’s a clearer example of something with strange-looking rules and processes for some purpose students don’t even know what they are? In middle school and high school we get algebra. In high school there’s trigonometry. In high school and college there’s calculus. In grad school there’s grad school. There’s always something.

Jeff Stahler’s Moderately Confused for the 21st is the usual bad-mathematics-of-politicians joke. It may be a little more on point considering the Future Disgraced Former President it names, but the joke is surely as old as politicians and hits all politicians with the same flimsiness.

John Graziano’s Ripley’s Believe It Or Not for the 22nd names Greek mathematician Pythagoras. That’s close enough to on-point to include here, especially considering what a slow week it’s been. It may not be fair to call Pythagoras a mathematician. My understanding is we don’t know that actually did anything in mathematics, significant or otherwise. His cult attributed any of its individuals’ discoveries to him, and may have busied themselves finding other, unrelated work to credit to their founder. But there’s so much rumor and gossip about Pythagoras that it’s probably not fair to automatically dismiss any claim about him. The beans thing I don’t know about. I would be skeptical of anyone who said they were completely sure.

Vic Lee’s Pardon My Planet for the 23rd is the usual sort of not-understanding-mathematics joke. In this case it’s about percentages, which are good for baffling people who otherwise have a fair grasp on fractions. I wonder if people would be better at percentages if they learned to say “percent” as “out of a hundred” instead. I’m sure everyone who teaches percentages teaches that meaning, but that doesn’t mean the warning communicates.

Stephan Pastis’s Pearls Before Swine for the 24th jams a bunch of angle puns into its six panels. I think it gets most of the basic set in there.

Samson’s Dark Side Of The Horse for the 25th mentions sudokus, and that’s enough for a slow week like this. I thought Horace was reaching for a calculator in the last panel myself, and was going to say that wouldn’t help any. But then I checked the numbers in the boxes and that made it all better.

## Reading the Comics, November 5, 2016: Surprisingly Few Halloween Costumes Edition

Comic Strip Master Command gave me a light load this week, which suit me fine. I’ve been trying to get the End 2016 Mathematics A To Z comfortably under way instead. It does strike me that there were fewer Halloween-themed jokes than I’d have expected. For all the jokes there are to make about Halloween I’d imagine some with some mathematical relevance would come up. But they didn’t and, huh. So it goes. The one big exception is the one I’d have guessed would be the exception.

Bill Amend’s FoxTrot for the 30th — a new strip — plays with the scariness of mathematics. Trigonometry specifically. Trig is probably second only to algebra for the scariest mathematics normal people encounter. And that’s probably more because people get to algebra before they might get to trigonometry. Which is madness, in its way. Trigonometry is about how we can relate angles, arcs, and linear distances. It’s about stuff anyone would like to know, like how to go from an easy-to-make observation of the angle spanned by a thing to how big the thing must be. But the field does require a bunch of exotic new functions like sine and tangent and novelty acts like “arc-cosecant”. And the numbers involved can be terrible things. The sine of an angle, for example, is almost always going to be some irrational number. For common angles we use a lot it’ll be an irrational number with an easy-to-understand form. For example the sine of 45 degrees, mentioned here, is “one-half the square root of two”. Anyone not trying to be intimidating will use that instead. But the sine of, say, 50 degrees? I don’t know what that is either except that it’s some never-ending sequence of digits. People love to have digits, but when they’re asked to do something with them, they get afraid and I don’t blame them.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 30th uses sudoku as shorthand for “genius thinking”. I am aware some complain sudoku isn’t mathematics. It’s certainly logic, though, and if we’re going to rule out logic puzzles from mathematics we’re going to lose a lot of fun fields. One of the commenters provided what I suppose the solution to be. (I haven’t checked.) If wish to do the puzzle be careful about scrolling.

In Jef Mallet’s Frazz for the 2nd Caulfield notices something cute about 100. A perfect square is a familiar enough idea; it’s a whole number that’s the square of another whole number. The “roundest of round numbers” is a value judgement I’m not sure I can get behind. It’s a good round number, anyway, at least for stuff that’s sensibly between about 50 and 150. Or maybe between 50 and 500 if you’re just interested in about how big something might be. An irrational number, well, you know where that joke’s going.

Mrs Olsen doesn’t seem impressed by Caulfield’s discovery, although in fairness we don’t see the actual aftermath. Sometimes you notice stuff like that and it is only good for a “huh”. But sometimes you get into some good recreational mathematics. It’s the sort of thinking that leads to discovering magic squares and amicable numbers and palindromic prime numbers and the like. Do they lead to important mathematics? Some of them do. Or at least into interesting mathematics. Sometimes they’re just passingly amusing.

Greg Curfman’s Meg rerun for the 12th quotes Einstein’s famous equation as the sort of thing you could just expect would be asked in school. I’m not sure I ever had a class where knowing E = mc2 was the right answer to a question, though. Maybe as I got into physics since we did spend a bit of time on special relativity and E = mc2 turns up naturally there. Maybe I’ve been out of elementary school too long to remember.

Mark Tatulli’s Heart of the City for the 4th has Heart and Dean talking about postapocalyptic society. Heart doubts that postapocalyptic society would need people like him, “with long-division experience”. Ah, but, grant the loss of computing devices. People will still need to compute. Before the days of electrical, and practical mechanical, computing people who could compute accurately were in demand. The example mathematicians learn to remember is Zacharias Dase, a German mental calculator. He was able to do astounding work and in his head. But he didn’t earn so much money as pro-mental-arithmetic propaganda would like us to believe. And why work entirely in your head if you don’t need to?

Larry Wright’s Motley Classics rerun for the 5th is a word problem joke. And it’s mixed with labor relations humor for the sake of … I’m not quite sure, actually. Anyway I would have sworn I’d featured this strip in a long-ago Reading The Comics post, but I don’t see it on a casual search. So, go figure.

## Reading the Comics, August 19, 2016: Mathematics Signifier Edition

I know it seems like when I write these essays I spend the most time on the first comic in the bunch and give the last ones a sentence, maybe two at most. I admit when there’s a lot of comics I have to write up at once my energy will droop. But Comic Strip Master Command apparently wants the juiciest topics sent out earlier in the week. I have to follow their lead.

Stephen Beals’s Adult Children for the 14th uses mathematics to signify deep thinking. In this case Claremont, the dog, is thinking of the Riemann Zeta function. It’s something important in number theory, so longtime readers should know this means it leads right to an unsolved problem. In this case it’s the Riemann Hypothesis. That’s the most popular candidate for “what is the most important unsolved problem in mathematics right now?” So you know Claremont is a deep-thinking dog.

The big Σ ordinary people might recognize as representing “sum”. The notation means to evaluate, for each legitimate value of the thing underneath — here it’s ‘n’ — the value of the expression to the right of the Sigma. Here that’s $\frac{1}{n^s}$. Then add up all those terms. It’s not explicit here, but context would make clear, n is positive whole numbers: 1, 2, 3, and so on. s would be a positive number, possibly a whole number.

The big capital Pi is more mysterious. It’s Sigma’s less popular brother. It means “product”. For each legitimate value of the thing underneath it — here it’s “p” — evaluate the expression on the right. Here that’s $\frac{1}{1 - \frac{1}{p^s}}$. Then multiply all that together. In the context of the Riemann Zeta function, “p” here isn’t just any old number, or even any old whole number. It’s only the prime numbers. Hence the “p”. Good notation, right? Yeah.

This particular equation, once shored up with the context the symbols live in, was proved by Leonhard Euler, who proved so much you sometimes wonder if later mathematicians were needed at all. It ties in to how often whole numbers are going to be prime, and what the chances are that some set of numbers are going to have no factors in common. (Other than 1, which is too boring a number to call a factor.) But even if Claremont did know that Euler got there first, it’s almost impossible to do good new work without understanding the old.

Charlos Gary’s Working It Out for the 14th is this essay’s riff on pie charts. Or bar charts. Somewhere around here the past week I read that a French idiom for the pie chart is the “cheese chart”. That’s a good enough bit I don’t want to look more closely and find out whether it’s true. If it turned out to be false I’d be heartbroken.

Ryan North’s Dinosaur Comics for the 15th talks about everyone’s favorite physics term, entropy. Everyone knows that it tends to increase. Few advanced physics concepts feel so important to everyday life. I almost made one expression of this — Boltzmann’s H-Theorem — a Theorem Thursday post. I might do a proper essay on it yet. Utahraptor describes this as one of “the few statistical laws of physics”, which I think is a bit unfair. There’s a lot about physics that is statistical; it’s often easier to deal with averages and distributions than the mass of real messy data.

Utahraptor’s right to point out that it isn’t impossible for entropy to decrease. It can be expected not to, in time. Indeed decent scientists thinking as philosophers have proposed that “increasing entropy” might be the only way to meaningfully define the flow of time. (I do not know how decent the philosophy of this is. This is far outside my expertise.) However: we would expect at least one tails to come up if we simultaneously flipped infinitely many coins fairly. But there is no reason that it couldn’t happen, that infinitely many fairly-tossed coins might all come up heads. The probability of this ever happening is zero. If we try it enough times, it will happen. Such is the intuition-destroying nature of probability and of infinitely large things.

Tony Cochran’s Agnes on the 16th proposes to decode the Voynich Manuscript. Mathematics comes in as something with answers that one can check for comparison. It’s a familiar role. As I seem to write three times a month, this is fair enough to say to an extent. Coming up with an answer to a mathematical question is hard. Checking the answer is typically easier. Well, there are many things we can try to find an answer. To see whether a proposed answer works usually we just need to go through it and see if the logic holds. This might be tedious to do, especially in those enormous brute-force problems where the proof amounts to showing there are a hundred zillion special cases and here’s an answer for each one of them. But it’s usually a much less hard thing to do.

Johnny Hart and Brant Parker’s Wizard of Id Classics for the 17th uses what seems like should be an old joke about bad accountants and nepotism. Well, you all know how important bookkeeping is to the history of mathematics, even if I’m never that specific about it because it never gets mentioned in the histories of mathematics I read. And apparently sometime between the strip’s original appearance (the 20th of August, 1966) and my childhood the Royal Accountant character got forgotten. That seems odd given the comic potential I’d imagine him to have. Sometimes a character’s only good for a short while is all.

Mark Anderson’s Andertoons for the 18th is the Andertoons representative for this essay. Fair enough. The kid speaks of exponents as a kind of repeating oneself. This is how exponents are inevitably introduced: as multiplying a number by itself many times over. That’s a solid way to introduce raising a number to a whole number. It gets a little strained to describe raising a number to a rational number. It’s a confusing mess to describe raising a number to an irrational number. But you can make that logical enough, with effort. And that’s how we do make the idea rigorous. A number raised to (say) the square root of two is something greater than the number raised to 1.4, but less than the number raised to 1.5. More than the number raised to 1.41, less than the number raised to 1.42. More than the number raised to 1.414, less than the number raised to 1.415. This takes work, but it all hangs together. And then we ask about raising numbers to an imaginary or complex-valued number and we wave that off to a higher-level mathematics class.

Nate Fakes’s Break of Day for the 18th is the anthropomorphic-numerals joke for this essay.

Lachowski’s Get A Life for the 18th is the sudoku joke for this essay. It’s also a representative of the idea that any mathematical thing is some deep, complicated puzzle at least as challenging as calculating one’s taxes. I feel like this is a rerun, but I don’t see any copyright dates. Sudoku jokes like this feel old, but comic strips have been known to make dated references before.

Samson’s Dark Side Of The Horse for the 19th is this essay’s Dark Side Of The Horse gag. I thought initially this was a counting-sheep in a lab coat. I’m going to stick to that mistaken interpretation because it’s more adorable that way.

## Reading the Comics, March 9, 2016: Mathematics Recreation Edition

I haven’t been skipping the comics, even with the effort of keeping up on the Leap Day 2016 A To Z Glossary. I just try to keep to the pace which Comic Strip Master Command sets.

The kids-information feature Short Cuts, by Jeff Harris, got ahead of “Pi Day” last Sunday. I imagine the feature gets run mid-week in some features, so that it’s better to run a full week before March 14th. But here’s a bundle of trivia, some jokes, some activities, that sort of thing. I am curious about one of Harris’s trivias, that Pi “plays an important role in some of the equations used in Einstein’s famous general theory of relativity”. That’s true, but it’s not as if general relativity is a rare appearance for pi in physics. Maybe Harris chose it on aesthetic grounds. General relativity has a familiar name and exotic concepts. And it allowed him to put in an equation that’s mysterious yet attractive-looking.

Samson’s Dark Side Of The Horse for the 7th of March made me wonder how many sudoku puzzles there are. The answer is — well, you have to start thinking carefully about what you mean by “how many”. For example: start with one puzzle. Swap out every appearance of a 1 with a 2, and a 2 with a 1. Is this new one actually a different puzzle? You can make a case for yes or for no. And that’s before we get into the question of how many clues to give to solve the puzzle. If I’m not misreading Wikipedia’s “Mathematics of Sudoku” page, the number of different nine-by-nine combinations of digits that can be legitimate sudoku puzzle solutions is 6,670,903,752,021,072,936,960. This was worked out in 2005 by Bertram Felgenhauer and Frazer Jarvis. They worked it out partly by logic, partly by brute force. Brute force is trying all the possibilities to see what works. It’s a method that rewards endurance. We like that we can turn it over to computers now. Or cartoon horses, whichever. They’re good at endurance.

Jef Mallett’s Frazz started a sequence about problem-writing on the 7th of March. Caulfield’s setup, complaining about trains and apple bushels, suggests he was annoyed by mathematics problems. I understand. Much of real mathematics starts with curiosity about something (how many sudoku puzzles are there?). Then it’s working out what computation might answer that question. Then it’s doing that calculation. And then it’s verifying that the calculation is right. Mathematics educators have to teach ways to do a calculation, and test that. And to teach how to know what calculation to do, and test that. That’s challenging enough. Add to that working out something to be curious about and you understand the appeal of stock setups. Maybe mathematics should include some courses in creative writing and short-short fiction. (Verification is, in my experience, the part nobody cares about. This is a shame. The hardest part of doing numerical mathematics is making sure your computation makes any sense.)

Richard Thompson’s Richard’s Poor Almanac rerun the 7th of March features the Non-Euclidean Creeper. It’s a plant perhaps related to the Cubist Fir Christmas tree and to the Otterloops’ troublesome non-Euclidean tree. Non-Euclidean geometry will probably always sound more intimidating and exotic. Euclidean geometry describes the way objects on the human scale behave. Shapes that fit on the table, or in your garden, follow Euclidean rules. But non-Euclidean isn’t magic; it’s the way that shapes on the surface of a globe work, for example. And the idea of drawing a thing like a square on the surface of the Earth isn’t so bizarre.

Paul Trap’s Thatababy for the 7th makes sport of geometry.

My love and I were talking the other day about Jim Toomey’s Sherman’s Lagoon. It’s a bit odd as comic strips go. It’s been around forever, for one, but nobody talks about it. It’s stayed reliably funny. Comic strips that’ve been around forever tend to … you know … not be. The strip’s done as a work-and-home strip except the cast is all sea life. And the thing is, Toomey keeps paying attention to new discoveries in sea life, and other animal research. And this is a fantastic era for discoveries in sea life, aside from how humans have now eaten all of it and we don’t have any left. I am not joking when I say the comic strip is an effortless way to keep up with new discoveries about the oceans.

I missed it when in December the discovery was announced to the world. But the setup, about the common name being given by a group of kids, is apparently quite correct. So we should expect from Toomey. (The scientific name is Etmopterus benchleyi. The last name refers to Peter Benchley, repentant Jaws novelist.) LiveScience.com’s article says lead author Dr Vicky Vásquez had to “scale them back” from their starting point, the “super ninja”. This differs from Hawthorne’s claim that the kids started from the “math stinks” shark, but it’s still a delight anyway.

## Reading the Comics, June 21, 2015: Blatantly Padded Edition, Part 2

I said yesterday I was padding one mathematics-comics post into two for silly reasons. And I was. But there were enough Sunday comics on point that splitting one entry into two has turned out to be legitimate. Nice how that works out sometimes.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (June 19) uses mathematics as something to heap upon a person until they yield to your argument. It’s a fallacious way to argue, but it does work. Even at a mathematical conference the terror produced by a screen full of symbols can chase follow-up questions away. On the 21st, they present mathematics as a more obviously useful thing. Well, mathematics with a bit of physics.

Nate Frakes’s Break Of Day (June 19) is this week’s anthropomorphic algebra joke.

Niklas Eriksson’s Carpe Diem (June 20) is captioned “Life at the Quantum Level”. And it’s built on the idea that quantum particles could be in multiple places at once. Whether something can be in two places at once depends on coming up with a clear idea about what you mean by “thing” and “places” and for that matter “at once”; when you try to pin the ideas down they prove to be slippery. But the mathematics of quantum mechanics is fascinating. It cries out for treating things we would like to know about, such as positions and momentums and energies of particles, as distributions instead of fixed values. That is, we know how likely it is a particle is in some region of space compared to how likely it is somewhere else. In statistical mechanics we resort to this because we want to study so many particles, or so many interactions, that it’s impractical to keep track of them all. In quantum mechanics we need to resort to this because it appears this is just how the world works.

(It’s even less on point, but Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 21st of June has a bit of riffing on Schrödinger’s Cat.)

Brian and Ron Boychuk’s Chuckle Brothers (June 20) name-drops algebra as the kind of mathematics kids still living with their parents have trouble with. That’s probably required by the desire to make a joking definition of “aftermath”, so that some specific subject has to be named. And it needs parents to still be watching closely over their kids, something that doesn’t quite fit for college-level classes like Intro to Differential Equations. So algebra, geometry, or trigonometry it must be. I am curious whether algebra reads as the funniest of that set of words, or if it just fits better in the space available. ‘Geometry’ is as long a word as ‘algebra’, but it may not have the same connotation of being an impossibly hard class.

And from the world of vintage comic strips, Jimmy Hatlo’s Little Iodine (June 21, originally run the 18th of April, 1954) reminds us that anybody can do any amount of arithmetic if it’s something they really want to calculate.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney (June 21) is another strip using the idea of mathematics — and particularly word problems — to signify great intelligence. I suppose it’s easier to recognize the form of a word problem than it is to recognize a good paper on the humanities if you only have two dozen words to show it in.

Juba’s Viivi and Wagner (June 21) is a timely reminder that while sudokus may be fun logic puzzles, they are ultimately the puzzle you decide to make of them.

## Reading the Comics, March 22, 2015: Word Problems Edition

After the flurry of comic strips that did Pi Day jokes last time around, and that one had worked in a March Madness joke, I’d expected there to be at least a couple of mathematically-mind college basketball tournament strips coming up this week. If they did, they didn’t appear on the comics sites I normally read, though. This time around turned out to be much more about word problems and the problem-answerer resisting the actual answering of the word problems. It’s possible that Comic Strip Master Command didn’t notice that this would be the weekend that United States readers would spend the most of their time complaining about how their bracket picks weren’t working right.

Phil Frank and Joe Troise’s The Elderberries (March 17, rerun) mentions sudoku, and how to play it, and also shows off how explaining things really is a pleasure, at least as long as you have someone who wants to know listening to the explanation. The strip’s also made me realize I don’t remember what the Professor’s background was. Certainly anyone of any background might enjoy sudoku puzzles, or at least know them well enough to explain how to do them, though I wonder if there’s not a use of the motif here that “professors are smart people, mathematics-or-logic puzzles require smartness, so professors are skilled at mathematics-or-logic puzzles”. (For what it’s worth, I’m not much on this sort of puzzle, though I believe that just reflects that I don’t care to do them very much, so I don’t have the experience needed to do them impressively well.)

Dan Thompson’s Rip Haywire (March 17) features a word problem as part of an aptitude test. Interesting to me is that the test is a multiple-choice, which means one should be able to pick the right answer without doing the whole multiplication of “3.29 times 6.5”: 3.29 is pretty near 3.30, so the answer will be about 3 times 6.5 plus a tenth of 3 times 6.5. And 3 times 6.5 is going to be 3 times 6 plus 3 times a half, or 18 plus 1.5. So, look for the answer that’s about 19.5 plus 1.95, which will be around 21.45. In particular, look for an answer a little bit less than that (to be exact, 0.01 times 6.5 less than that.) Of course, if the exam-writer was clever, 21.45 was included as a plausible yet incorrect answer, but at least the problem can be worked out in one’s head.

## Reading the Comics, February 28, 2015: Calendar Reform Edition

It’s the last day of the shortest month of the year, a day that always makes me think about whether the calendar could be different. I was bit by the calendar-reform bug as a child and I’ve mostly recovered from the infection, but some things can make it flare up again and I’ve never stopped being fascinated by the problem of keeping track of days, which you’d think would not be so difficult.

That’s why I’m leading this review of comics with Jef Mallet’s Frazz (February 27) even if it’s not transparently a mathematics topic. The biggest problem with calendar reform is there really aren’t fully satisfactory ways to do it. If you want every month to be as equal as possible, yeah, 13 months of 28 days each, plus one day (in leap years, two days) that doesn’t belong to any month or week is probably the least obnoxious, if you don’t mind 13 months to the year meaning there’s no good way to make a year-at-a-glance calendar tolerably symmetric. If you don’t want the unlucky, prime number of 13 months, you can go with four blocks of months with 31-30-30 days and toss in a leap day that’s again, not in any month or week. But people don’t seem perfectly comfortable with days that belong to no month — suggest it to folks, see how they get weirded out — and a month that doesn’t belong to any week is right out. Ask them. Changing the default map projection in schools is an easier task to complete.

There are several problems with the calendar, starting with the year being more nearly 365 days than a nice, round, supremely divisible 360. Also a factor is that the calendar tries to hack together the moon-based months with the sun-based year, and those don’t fit together on any cycle that’s convenient to human use. Add to that the need for Easter to be close to the vernal equinox without being right at Passover and you have a muddle of requirements, and the best we can hope for is that the system doesn’t get too bad.

## Reading the Comics, November 28, 2014: Greatest Hits Edition?

I don’t ever try speaking for Comic Strip Master Command, and it almost never speaks to me, but it does seem like this week’s strips mentioning mathematical themes was trying to stick to the classic subjects: anthropomorphized numbers, word problems, ways to measure time and space, under-defined probability questions, and sudoku. It feels almost like a reunion weekend to have all these topics come together.

Dan Thompson’s Brevity (November 23) is a return to the world-of-anthropomorphic-numbers kind of joke, and a pun on the arithmetic mean, which is after all the statistic which most lends itself to puns, just edging out the “range” and the “single-factor ANOVA F-Test”.

Phil Frank Joe Troise’s The Elderberries (November 23, rerun) brings out word problem humor, using train-leaves-the-station humor as a representative of the kinds of thinking academics do. Nagging slightly at me is that I think the strip had established the Professor as one of philosophy and while it’s certainly not unreasonable for a philosopher to be interested in mathematics I wouldn’t expect this kind of mathematics to strike him as very interesting. But then there is the need to get the idea across in two panels, too.

Jonathan Lemon’s Rabbits Against Magic (November 25) brings up a way of identifying the time — “half seven” — which recalls one of my earliest essays around here, “How Many Numbers Have We Named?”, because the construction is one that I find charming and that was glad to hear was still current. “Half seven” strikes me as similar in construction to saying a number as “five and twenty” instead of “twenty-five”, although I’m ignorant as to whether the actually is any similarity.

Scott Hilburn’s The Argyle Sweater (November 26) brings out a joke that I thought had faded out back around, oh, 1978, when the United States decided it wasn’t going to try converting to metric after all, now that we had two-liter bottles of soda. The curious thing about this sort of hyperconversion (it’s surely a satiric cousin to the hypercorrection that makes people mangle a sentence in the misguided hope of perfecting it) — besides that the “yard” in Scotland Yard is obviously not a unit of measure — is the notion that it’d be necessary to update idiomatic references that contain old-fashioned units of measurement. Part of what makes idioms anything interesting is that they can be old-fashioned while still making as much sense as possible; “in for a penny, in for a pound” is a sensible thing to say in the United States, where the pound hasn’t been legal tender since 1857; why would (say) “an ounce of prevention is worth a pound of cure” be any different? Other than that it’s about the only joke easily found on the ground once you’ve decided to look for jokes in the “systems of measurement” field.

Mark Heath’s Spot the Frog (November 26, rerun) I’m not sure actually counts as a mathematics joke, although it’s got me intrigued: Surly Toad claims to have a stick in his mouth to use to give the impression of a smile, or 37 (“Sorry, 38”) other facial expressions. The stick’s shown as a bundle of maple twigs, wound tightly together and designed to take shapes easily. This seems to me the kind of thing that’s grown as an application of knot theory, the study of, well, it’s almost right there in the name. Knots, the study of how strings of things can curl over and around and cross themselves (or other strings), seemed for a very long time to be a purely theoretical playground, not least because, to be addressable by theory, the knots had to be made of an imaginary material that could be stretched arbitrarily finely, and could be pushed frictionlessly through it, which allows for good theoretical work but doesn’t act a thing like a shoelace. Then I think everyone was caught by surprise when it turned out the mathematics of these very abstract knots also describe the way proteins and other long molecules fold, and unfold; and from there it’s not too far to discovering wonderful structures that can change almost by magic with slight bits of pressure. (For my money, the most astounding thing about knots is that you can describe thermodynamics — the way heat works — on them, but I’m inclined towards thermodynamic problems.)

Henry Scarpelli and Crag Boldman’s Archie (November 28, rerun) offers an interesting problem: when Veronica was out of town for a week, Archie’s test scores improved. Is there a link? This kind of thing is awfully interesting to study, and awfully difficult to: there’s no way to run a truly controlled experiment to see whether Veronica’s presence affects Archie’s test scores. After all, he never takes the same test twice, even if he re-takes a test on the same subject (and even if the re-test were the exact same questions, he would go into it the second time with relevant experience that he didn’t have the first time). And a couple good test scores might be relevant, or might just be luck, or it might be that something else happened to change that week that we haven’t noticed yet. How can you trace down plausible causal links in a complicated system?

One approach is an experimental design that, at least in the psychology textbooks I’ve read, gets called A-B-A, or A-B-A-B, experiment design: measure whatever it is you’re interested in during a normal time, “A”, before whatever it is whose influence you want to see has taken hold. Then measure it for a time “B” where something has changed, like, Veronica being out of town. Then go back as best as possible to the normal situation, “A” again; and, if your time and research budget allow, going back to another stretch of “B” (and, hey, maybe even “A” again) helps. If there is an influence, it ought to appear sometime after “B” starts, and fade out again after the return to “A”. The more you’re able to replicate this the sounder the evidence for a link is.

(We’re actually in the midst of something like this around home: our pet rabbit was diagnosed with a touch of arthritis in his last checkup, but mildly enough and in a strange place, so we couldn’t tell whether it’s worth putting him on medication. So we got a ten-day prescription and let that run its course and have tried to evaluate whether it’s affected his behavior. This has proved difficult to say because we don’t really have a clear way of measuring his behavior, although we can say that the arthritis medicine is apparently his favorite thing in the world, based on his racing up to take the liquid and his trying to grab it if we don’t feed it to him fast enough.)

Ralph Hagen’s The Barn (November 28) has Rory the sheep wonder about the chances he and Stan the bull should be together in the pasture, given how incredibly vast the universe is. That’s a subtly tricky question to ask, though. If you want to show that everything that ever existed is impossibly unlikely you can work out, say, how many pastures there are on Earth multiply it by an estimate of how many Earth-like planets there likely are in the universe, and take one divided by that number and marvel at Rory’s incredible luck. But that number’s fairly meaningless: among other obvious objections, wouldn’t Rory wonder the same thing if he were in a pasture with Dan the bull instead? And Rory wouldn’t be wondering anything at all if it weren’t for the accident by which he happened to be born; how impossibly unlikely was that? And that Stan was born too? (And, obviously, that all Rory and Stan’s ancestors were born and survived to the age of reproducing?)

Except that in this sort of question we seem to take it for granted, for instance, that all Stan’s ancestors would have done their part by existing and doing their part to bringing Stan around. And we’d take it for granted that the pasture should exist, rather than be a farmhouse or an outlet mall or a rocket base. To come up with odds that mean anything we have to work out what the probability space of all possible relevant outcomes is, and what the set of all conditions that satisfy the concept of “we’re stuck here together in this pasture” is.

Mark Pett’s Lucky Cow (November 28) brings up sudoku puzzles and the mystery of where they come from, exactly. This prompted me to wonder about the mechanics of making sudoku puzzles and while it certainly seems they could be automated pretty well, making your own amounts to just writing the digits one through nine nine times over, and then blanking out squares until the puzzle is hard. A casual search of the net suggests the most popular way of making sure you haven’t blanking out squares so that the puzzle becomes unsolvable (in this case, that there’s two or more puzzles that fit the revealed information) is to let an automated sudoku solver tell you. That’s true enough but I don’t see any mention of any algorithms by which one could check if you’re blanking out a solution-foiling set of squares. I don’t know whether that reflects there being no algorithm for this that’s more efficient than “try out possible solutions”, or just no algorithm being more practical. It’s relatively easy to make a computer try out possible solutions, after all.

A paper published by Mária Ercsey-Ravasz and Zoltán Toroczkai in Nature Scientific Reports in 2012 describes the recasting of the problem of solving sudoku into a deterministic, dynamical system, and matches the difficulty of a sudoku puzzle to chaotic behavior of that system. (If you’re looking at the article and despairing, don’t worry. Go to the ‘Puzzle hardness as transient chaotic dynamics’ section, and read the parts of the sentence that aren’t technical terms.) Ercsey-Ravasz and Toroczkai point out their chaos-theory-based definition of hardness matches pretty well, though not perfectly, the estimates of difficulty provided by sudoku editors and solvers. The most interesting (to me) result they report is that sudoku puzzles which give you the minimum information — 17 or 18 non-blank numbers to start — are generally not the hardest puzzles. 21 or 22 non-blank numbers seem to match the hardest of puzzles, though they point out that difficulty has got to depend on the positioning of the non-blank numbers and not just how many there are.

## Reading the Comics, October 25, 2014: No Images Again Edition

I had assumed it was a freak event last time that there weren’t any Comics Kingdom strips with mathematical topics to discuss, and which comics I include as pictures here because I don’t know that the links made to them will work for everyone arbitrarily far in the future. Apparently they’re just not in a very mathematical mood this month, though. Such happens; I’m sure they’ll reappear soon enough.

John Zakour and Scott Roberts’ Working Daze (October 22, a “best of” rerun) brings up one of my very many peeves-regarding-pedantry, the notion that you “can’t give more than 100 percent”. It depends on what 100 percent means. The metaphor of “giving 110 percent” is based on the one-would-think-obvious point that there is a standard quantity of effort, which is the 100 percent, and to give 110 percent is to give measurably more than the standard effort. The English language has enough illogical phrases in it; we don’t need to attack ones that are only senseless if you go out to pick a fight with them.

Mark Anderson’s Andertoons (October 23) shows a student attacking a problem with appreciable persistence. As the teacher says, though, there’s no way the student’s attempts at making 2 plus 2 equal 5 is ever not going to be wrong, at least unless we have different ideas about what is meant by 2, plus, equals, and 5. It’s easy to get from this point to some pretty heady territory: since it’s true that two plus two can’t equal five (using the ordinary definitions of these words), then this statement is true not just everywhere in this universe but in all possible universes. This — indeed, all — arithmetic would even be true if there were no universe. But if something can be true regardless of what the universe is like, or even if there is no universe, then how can it tell us anything about the specific universe that actually exists? And yet it seems to do so, quite well.

Tim Lachowski’s Get A Life (October 23) is really an accounting joke, or really more a “taxes they so mean” joke, but I thought it worth mentioning that, really, the majority of the mathematics the world has done have got to have been for the purposes of bookkeeping and accounting. I’m sorry that I’m not better-informed about this so as to better appreciate what is, in some ways, the dark matter of mathematical history.

Keith Tutt and Daniel Saunders’s chipper Lard’s World Peace Tips (October 23) recommends “be a genius” as one of the ways to bring about world peace, and uses mathematics as the comic shorthand for “genius activity”, not to mention sudoku as the comic shorthand for “mathematics”. People have tried to gripe that sudoku isn’t really mathematics; while it’s not arithmetic, though — you could replace the numerals with letters or with arbitrary symbols not to be repeated in one line, column, or subsquare and not change the problem at all — it’s certainly logic.

John Graziano’s Ripley’s Believe It or Not (October 23) besides giving me a spot of dizziness with that attribution line makes the claim that “elephants have been found to be better at some numerical tasks than chimps or even humans”. I can believe that, more or less, though I notice it doesn’t say exactly what tasks elephants are so good (or chimps and humans so bad) at. Counting and addition or subtraction seem most likely, though, because those are processes it seems possible to create tests for. At some stages in human and animal development the animals have a clear edge in speed or accuracy. I don’t remember reading evidence of elephant skills before but I can accept that they surely have some.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 24) applies the tools of infinite series — adding up infinitely many of a sequence of terms, often to a finite total — to parenting and the problem of one kid hitting another. This is held up as Real Analysis — – the field in which you learn why Calculus works — and it is, yeah, although this is the part of Real Analysis you can do in high school.

John Zakour and Scott Roberts’s Maria’s Day (October 25) picks up on the Math Wiz Monster in Maria’s closet mentioned last time I did one of these roundups. And it includes an attack on the “Common Core” standards, understandably: it’s unreasonable to today’s generation of parents that mathematics should be taught any differently from how it was taught to them, when they didn’t understand the mathematics they were being taught. Innovation in teaching never has a chance.

Dave Whamond’s Reality Check (October 25) reminds us that just because stock framing can be used to turn a subtraction problem into a word problem doesn’t mean that it can’t jump all the way out of mathematics into another field.

I haven’t included any comics from today — the 26th of October — in my reading yet but really, what are the odds there’s like a half-dozen comics of obvious relevance with nice, juicy topics to discuss?

## Reading the Comics, 16 May 2013

It’s a good time for another round of comic strip reading, particularly I haven’t had the time to think in detail about all the news in number theory that’s come out this past week, and that I’m not sure whether I should go into explaining arc lengths after I trapped at least one friend into trying to work out the circumference of an ellipse (you can’t do it either, but there are a lot of curves you could). I also notice I’m approaching that precious 10,000th blog hit here, so I can get back to work verifying that law about random data starting with the digit 1.

Berkeley Breathed’s Bloom County (May 2, rerun) throws up a bunch of mathematical symbols with the intention of producing a baffling result, so that Milo can make a clean getaway from Freida. The splendid thing to me, though, is that Milo’s answer — “log 10 times 10 to the derivative of 10,000” — actually does parse, if you read it a bit charitably. The “log 10” bit we can safely suppose to mean the logarithm base 10, because the strip originally ran in 1981 or so when there was still some use for the common logarithm. These days, we have calculators, and “log” is moving over to be the “natural logarithm”, base e, what was formerly denoted as “ln”.

## Reading the Comics, November 11, 2012

Since Scott Adams’s Dilbert hasn’t done anything to deserve my scrutiny let me carry on my quest to identify all the comic strips that mention some mathematical thing. I’m leaving a couple out; for example, today (the 11th) Rob Harrell’s Adam @ Home and Bill Amend’s FoxTrot mentioned the alignment of digits in the date’s representation in numerals, but that seems too marginal, and yet here I am talking about it. I can’t be bothered coming up with rules I can follow for my own amusement here, can I?

## Reading the Comics, September 26, 2012

I haven’t time to write a short piece today so let me go through a fresh batch of math-themed comic strips instead. There might be a change coming to these features soon, both in the strips I read and in how I present them, since Comics Kingdom, which provides the King Features Syndicate comic strips, has shown signs that they’re tightening up web access to their strips.

I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.

For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though: