Reading the Comics, April 18, 2017: Give Me Some Word Problems Edition


I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.

Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.

Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.

(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)

J C Duffy’s Lug Nuts for the 16th is a Pi Day joke that ran late.

Jef Mallett’s Frazz for the 17th starts a little thread about obsolete references in story problems. It’s continued on the 18th. I’m sympathetic in principle to both sides of the story problem debate.

Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, $2.25 per pound had to be divided by three.

But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.

And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.

So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?

Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.

Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.

Lee: 'I'm developing a new theory about avocado intelligence.' Joules: 'You can't be serious.' Lee: 'Avocado, what is the square root of 8,649?' Avocado: 'That's easy. It's 92?' Lee: 'Wrong. It's 93.' Joules: 'See? It's just a dumb piece of fruit.' Lee: 'I honestly thought I was on to something.'
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th of April, 2017. Before you ask what exactly the old theory of avocado intelligence was remember that Edison Lee’s lab partner there is a talking rat. Just saying.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.

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Reading the Comics, July 12, 2015: Chuckling At Hart Edition


I haven’t had the chance to read the Gocomics.com comics yet today, but I’d had enough strips to bring up anyway. And I might need something to talk about on Tuesday. Two of today’s strips are from the legacy of Johnny Hart. Hart’s last decades at especially B.C., when he most often wrote about his fundamentalist religious views, hurt his reputation and obscured the fact that his comics were really, really funny when they start. His heirs and successors have been doing fairly well at reviving the deliberately anachronistic and lightly satirical edge that made the strips funny to begin with, and one of them’s a perennial around here. The other, Wizard of Id Classics, is literally reprints from the earliest days of the comic strip’s run. That shows the strip when it was earning its place on every comics page everywhere, and made a good case for it.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (July 8) shows how a compass, without straightedge, can be used to ensure one’s survival. I suppose it’s really only loosely mathematical but I giggled quite a bit.

Ken Cursoe’s Tiny Sepuku (July 9) talks about luck as being just the result of probability. That’s fair enough. Random chance will produce strings of particularly good, or bad, results. Those strings of results can look so long or impressive that we suppose they have to represent something real. Look to any sport and the argument about whether there are “hot hands” or “clutch performers”. And Maneki-Neko is right that a probability manipulator would help. You can get a string of ten tails in a row on a fair coin, but you’ll get many more if the coin has an eighty percent chance of coming up tails.

Brant Parker and Johnny Hart’s Wizard of Id Classics (July 9, rerun from July 12, 1965) is a fun bit of volume-guessing and logic. So, yes, I giggled pretty solidly at both B.C. and The Wizard of Id this week.

Mell Lazarus’s Momma (July 11) identifies “long division” as the first thing a person has to master to be an engineer. I don’t know that this is literally true. It’s certainly true that liking doing arithmetic helps one in a career that depends on calculation, though. But you can be a skilled songwriter without being any good at writing sheet music. I wouldn’t be surprised if there are skilled engineers who are helpless at dividing fourteen into 588.

In the panel of interest, Loretta says the numbers (presumably the bills) don't add up, but they subtract down fine.
Bunny Hoest and John Reiner’s Lockhorns for the 12th of July, 2015.

Bunny Hoest and John Reiner’s Lockhorns (July 12) includes an example of using “adding up” to mean “make sense”. It’s a slight thing. But the same idiom was used last week, in Eric Teitelbaum and Bill Teitelbaum’s Bottomliners. I don’t think Comic Strip Master Command is ordering this punch line yet, but you never know.

And finally, I do want to try something a tiny bit new, and explicitly invite you-the-readers to say what strip most amused you. Please feel free to comment about your choices, r warn me that I set up the poll wrong. I haven’t tried this before.

Reading the Comics, April 27, 2015: Anthropomorphic Mathematics Edition


They’re not running at the frantic pace of April 21st, but there’s still been a fair clip of comic strips that mention some kind of mathematical topic. I imagine Comic Strip Master Command wants to be sure to use as many of these jokes up as possible before the (United States) summer vacation sets in.

Dan Thompson’s Brevity (April 23) is a straightforward pun strip. It also shows a correct understanding of how to draw a proper Venn Diagram. And after all why shouldn’t an anthropomorphized Venn Diagram star in movies too?

John Atkinson’sWrong Hands (April 23) gets into more comfortable territory with plain old numbers being anthropomorphized. The 1 is fair to call this a problem. What kind of problem depends on whether you read the x as a multiplication sign or as a variable x. If it’s a multiplication sign then I can’t think of any true statement that can be made from that bundle of symbols. If it’s the variable x then there are surprisingly many problems which could be made, particularly if you’re willing to count something like “x = 718” as a problem. I think that it works out to 24 problems but would accept contrary views. This one ended up being the most interesting to me once I started working out how many problems you could make with just those symbols. There’s a fun question for your combinatorics exam in that.

Continue reading “Reading the Comics, April 27, 2015: Anthropomorphic Mathematics Edition”

Reading the Comics, April 22, 2015: April 21, 2015 Edition


I try to avoid doing Reading The Comics entries back-to-back since I know they can get a bit repetitive. How many ways can I say something is a student-resisting-the-word-problem joke? But if Comic Strip Master Command is going to send a half-dozen strips at least mentioning mathematical topics in a single day, how can I resist the challenge? Worse, what might they have waiting for me tomorrow? So here’s a bunch of comic strips from the 21st of April, 2015:

Mark Anderson’s Andertoons plays on the idea of a number being used up. I’m most tickled by this one. I have heard that the New York Yankees may be running short on uniform numbers after having so many retired. It appears they’ve only retired 17 numbers, but they do need numbers for a 40-player roster as well as managers and coaches and other participants. Also, and this delights me, two numbers are retired for two people each. (Number 8, for Yogi Berra and Bill Dickey, and Number 42, for Jackie Robinson and Mariano Rivera.)

Continue reading “Reading the Comics, April 22, 2015: April 21, 2015 Edition”

Reading the Comics, April 10, 2015: Getting Into The Story Problem Edition


I know it’s been like forever, or four days, since the last time I had a half-dozen or so mathematically themed comic strips to write about, but if Comic Strip Master Command is going to order cartoonists to give me stuff to write about I’m not going to turn them away. Several seemed to me about the struggle to get someone to buy into a story — the thing being asked after in a word problem, perhaps, or about the ways mathematics is worth knowing, or just how the mathematics in a joke’s setup are presented — and how skepticism about these things can turn up. So I’ll declare that the theme of this collection.

Steve Sicula’s Home And Away started a sequence on April 7th about “is math really important?”, with the father trying to argue that it’s so very useful. I’m not sure anyone’s ever really been convinced by the argument that “this is useful, therefore it’s important, therefore it’s interesting”. Lots of things are useful or important while staying fantastically dull to all but a select few souls. I would like to think a better argument for learning mathematics is that it’s beautiful, and astounding, and it allows you to discover new ways of studying the world; it can offer all the joy of any art, even as it has a practical side. Anyway, the sequence goes on for several days, and while I can’t say the arguments get very convincing on any side, they do allow for a little play with the fourth wall that I usually find amusing in comics which don’t do that much.

Continue reading “Reading the Comics, April 10, 2015: Getting Into The Story Problem Edition”

Reading the Comics, March 31, 2015: Closing Out March Edition


It’s been another week of Comic Strip Master Command supporting my most popular regular feature around here. As sometimes happens there were so many comics in a row that I can’t catch them all up in a single post. Actually, there were enough just on the 29th of March to justify another Reading The Comics post, but I didn’t want to overload what was already a pretty busy month with more postings. This is a Gocomics.com-heavy entry, so I’m afraid folks have to click the links to see images. I hope you’ll be all right.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (March 29) is a bit of a geography joke built around the idea that a circle hasn’t got a side. Whether it does or not — besides “inside” and “outside”, source for another joke — requires thinking carefully what you mean by a shape’s side: does it have to be straight? If it can be curved, can it curve so sharply that it looks like it’s a corner? For that matter, can you tell a circle apart from, for example, the chiliagon, a regular polygon with a thousand equal sides? (If you can, then, how about a regular polygon with a million, or a billion, or more equal sides, to the point that you can’t tell the difference?) If you can’t, then how do you know a circle was in the story at all?

Continue reading “Reading the Comics, March 31, 2015: Closing Out March Edition”

Reading the Comics, March 26, 2015: Kind Of Hanging Around Edition


I’m sorry to have fallen silent the last few days; it’s been a bit busy and I’ve been working on follow-ups to a couple of threads. Fortunately Comic Strip Master Command is still around and working to make sure I don’t disappear altogether, and I have a selection of comic strips which at least include a Jumble world puzzle, which should be a fun little diversion.

Tony Rubino and Gary Markstein’s Daddy’s Home (March 23) asks what seems like a confused question to me, “if you believe in infinity, does that mean anything is possible?” As I say, I’m not sure I understand how belief in infinity comes into play, but that might just reflect my background: I’ve been thoroughly convinced that one can describe collections of things that have infinitely many elements — the counting numbers, rectangles, continuous functions — as well as that one can subdivide things — like segments of a number line — infinitely many times — as well as of quantities that are larger than any finite number and so must be infinitely large; so, what’s to not believe in? (I’m aware that there are philosophical and theological questions that get into things termed “potential” and “actual” infinities, but I don’t understand the questions those terms are meant to address.) The phrasing of “anything is possible” seems obviously flawed to me. But if we take it to mean instead “anything not logically inconsistent or physically prohibited is possible” then we seem to have a reasonable question, if that hasn’t just reduced to “anything not impossible is possible”. I guess ultimately I just wonder if the kid is actually trying to understand anything or if he’s just procrastinating.

Continue reading “Reading the Comics, March 26, 2015: Kind Of Hanging Around Edition”

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.

Continue reading “Reading the Comics, March 22, 2015: Word Problems Edition”

Reading the Comics, March 10, 2015: Shapes Of Things Edition


If there’s a theme running through today’s collection of mathematics-themed comic strips it’s shapes: I have good reason to talk about a way of viewing circles and spheres and even squares and boxes; and then both Euclid and men’s ties get some attention.

Eric the Circle (March 5), this one by “regina342”, does a bit of shape-name-calling. I trust that it’s not controversial that a rectangle is also a parallelogram, but people might be a bit put off by describing a circle as a sphere, what with circles being two-dimensional figures and spheres three-dimensional ones. For ordinary purposes of geometry that’s a fair enough distinction. Let me now make this complicated.

Continue reading “Reading the Comics, March 10, 2015: Shapes Of Things Edition”

Reading the Comics, March 4, 2015: Driving Me Crazy Edition


I like it when there are themes to these collections of mathematical comics, but since I don’t decide what subjects cartoonists write about — Comic Strip Master Command does — it depends on luck and my ability to dig out loose connections to find any. Sometimes, a theme just drops into my lap, though, as with today’s collection: several cartoonists tossed off bits that had me double-checking their work and trying to figure out what it was I wasn’t understanding. Ultimately I came to the conclusion that they just made mistakes, and that’s unnerving since how could a mathematical error slip through the rigorous editing and checking of modern comic strips?

Mac and Bill King’s Magic in a Minute (March 1) tries to show off how to do a magic trick based on parity, using the spots on a die to tell whether it was turned in one direction or another. It’s a good gimmick, and parity — whether something is odd or even — can be a great way to encode information or to do simple checks against slight errors. That said, I believe the Kings made a mistake in describing the system: I can’t figure out how the parity of the three sides of a die facing you could not change, from odd to even or from even to odd, as the die is rotated one turn. I believe they mean that you should just count the dots on the vertical sides, so that for example in the “Howdy Do It?” panel in the lower right corner, add two and one to make three. But with that corrected it should be a good trick.

Continue reading “Reading the Comics, March 4, 2015: Driving Me Crazy Edition”

Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?


Well, I thought it’d be unlikely to get too many more mathematics comics before the end of the year, but Comic Strip Master Command apparently sent out orders to clear out the backlog before the new calendar year starts. I think Dark Side of the Horse is my favorite of the strips, blending a good joke with appealing artwork, although The Buckets gives me the most to talk about.

Greg Cravens’s The Buckets (December 28) is about what might seem only loosely a mathematical topic: that the calendar is really a pretty screwy creation. And it is, as anyone who’s tried to program a computer to show dates has realized. The core problem, I suppose, is that the calendar tries to meet several goals simultaneously: it’s supposed to use our 24-hour days to keep track of the astronomical year, which is an approximation to the cycle of seasons of the year, and there’s not a whole number of days in a year. It’s also supposed to be used to track short-term events (weeks) and medium-term events (months and seasons). The number of days that best approximate the year, 365 and 366, aren’t numbers that lend themselves to many useful arrangements. The months try to divide that 365 or 366 reasonably uniformly, with historial artifacts that can be traced back to the Roman calendar was just an unspeakable mess; and, something rarely appreciated, the calendar also has to make sure that the date of Easter is something reasonable. And, of course, any reforming of the calendar has to be done with the agreement of a wide swath of the world simultaneously. Given all these constraints it’s probably remarkable that it’s only as messed up as it is.

To the best of my knowledge, January starts the New Year because Tarquin Priscus, King of Rome from 616 – 579 BC, found that convenient after he did some calendar-rejiggering (particularly, swapping the order of February and January), though I don’t know why he thought that particularly convenient. New Years have appeared all over the calendar year, though, with the start of January, the start of September, Christmas Day, and the 25th of March being popular options, and if you think it’s messed up to have a new year start midweek, think about having a new year start in the middle of late March. It all could be worse.

Continue reading “Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?”

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.)

Veronica was out of town for a week; Archie's test scores improved. This demonstrates that test scores aren't everything.
Henry Scarpelli and Crag Boldman’s Archie for the 28th of November, 2014. Clearly we should subject this phenomenon to scientific inquiry!

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, September 15, 2014: Are You Trying To Overload Me Edition


One of the little challenges in writing about mathematics-themed comics is one of pacing: how often should I do a roundup? Posting weekly, say, helps figure out a reasonable posting schedule for those rare moments when I’m working ahead of deadline, but that leaves the problem of weeks that just don’t have anything. Waiting for a certain number of comics before writing about them seems more reasonable, but then I have to figure how many comics are enough. I’ve settled into five-to-six as my threshold for a new post, but that can mean I have weeks where it seems like I’m doing nothing but comic strips posts. And then there’s conditions like this one where Comic Strip Master Command had its cartoonists put up just enough that I’d started composing a fresh post, and then tossed in a whole bunch more the next day. It’s like they’re trying to shake me by having too many strips to write about. I’d have though they’d be flattered to have me writing about them so.

Tiger studied times tables today. Studied it, not learned it.
Bud Blake’s _Tiger_ for the 11th of September, 2014.

Bud Blake’s Tiger (September 11, rerun) mentions Tiger as studying the times tables and points out the difference between studying a thing and learning it.

Marc Anderson’s Andertoons (September 12) belongs to that vein of humor about using technology words to explain stuff to kids. I admit I’m vague enough on the concept of mashups that I can accept that it might be a way of explaining addition, but it feels like it might also be a way of describing multiplication or for that matter the composition of functions. I suppose the kids would be drawn as older in those cases, though.

Bill Amend’s FoxTrot (September 13, rerun) does a word problem joke, but it does have the nice beat in the penultimate panel of Paige running a sanity check and telling at a glance that “two dollars” can’t possibly be the right answer. Sanity checks are nice things to have; they don’t guarantee against making mistakes, but they at least provide some protection against the easiest mistakes, and having some idea of what an answer could plausibly be might help in working out the answer. For example, if Paige had absolutely no idea how to set up equations for this problem, she could reason that the apple and the orange have to cost something from 1 to 29 cents, and could try out prices until finding something that satisfies both requirements. This is an exhausting method, but it would eventually work, too, and sometimes “working eventually” is better than “working cleverly”.

Bill Schorr’s The Grizzwells (September 13) starts out by playing on the fact that “yard” has multiple meanings; it also circles around one of those things that distinguishes word problems from normal mathematics. A word problem, by convention, normally contains exactly the information needed to solve what’s being asked — there’s neither useless information included nor necessary information omitted, except if the question-writer has made a mistake. In a real world application, figuring out what you need, and what you don’t need, is part of the work, possibly the most important part of the work. So to answer how many feet are in a yard, Gunther (the bear) is right to ask more questions about how big the yard is, as a start.

Ed would rather snack bars came in 100-calorie forms, rather than 70-calorie ones.
Steve Kelly and Jeff Parker’s _Dustin_ for the 14th of September, 2014.

Steve Kelley and Jeff Parker’s Dustin (September 14) is about one of the applications for mental arithmetic that people find awfully practical: counting the number of food calories that you eat. Ed’s point about it being convenient to have food servings be nice round numbers, as they’re easier to work with, is a pretty good one, and it’s already kind of accounted for in food labelling: it’s permitted (in the United States) to round off calorie counts to the nearest ten or so, on the rather sure grounds that if you are counting calories you’d rather add 70 to the daily total than 68 or 73. Don’t read the comments thread, which includes the usual whining about the Common Core and the wild idea that mental arithmetic might be well done by working out a calculation that’s close to the one you want but easier to do and then refining it to get the accuracy you need.

Mac and Bill King’s Magic In A Minute kids activity panel (September 14) presents a magic trick that depends on a bit of mental arithmetic. It’s a nice stunt, although it is certainly going to require kids to practice things because, besides dividing numbers by 4, it also requires adding 6, and that’s an annoying number to deal with. There’s also a nice little high school algebra problem to be done in explaining why the trick works.

Bill Watterson’s Calvin and Hobbes (September 15, rerun) includes one of Hobbes’s brilliant explanations of how arithmetic works, and if I haven’t wasted the time spent memorizing the strips where Calvin tries to do arithmetic homework then Hobbes follows up tomorrow with imaginary numbers. Can’t wait.

Jef Mallet’s Frazz (September 15) expresses skepticism about a projection being made for the year 2040. Extrapolations and interpolations are a big part of numerical mathematics and there’s fair grounds to be skeptical: even having a model of whatever your phenomenon is that accurately matches past data isn’t a guarantee that there isn’t some important factor that’s been trivial so far but will become important and will make the reality very different from the calculations. But that hardly makes extrapolations useless: for one, the fact that there might be something unknown which becomes important is hardly a guarantee that there is. If the modelling is good and the reasoning sound, what else are you supposed to use for a plan? And of course you should watch for evidence that the model and the reality aren’t too very different as time goes on.

Gary Wise and Lance Aldrich’s Real Life Adventures (September 15) describes mathematics as “insufferable and enigmatic”, which is a shame, as mathematics hasn’t said anything nasty about them, now has it?

Reading the Comics, August 25, 2014: Summer Must Be Ending Edition


I’m sorry to admit that I can’t think of a unifying theme for the most recent round of comic strips which mention mathematical topics, other than that this is one of those rare instances of nobody mentioning infinite numbers of typing monkeys. I have to guess Comic Strip Master Command sent around a notice that summer vacation (in the United States) will be ending soon, so cartoonists should start practicing their mathematics jokes.

Tom Toles’s Randolph Itch, 2 a.m. (August 22, rerun) presents what’s surely the lowest-probability outcome of a toss of a fair coin: its landing on the edge. (I remember this as also the gimmick starting a genial episode of The Twilight Zone.) It’s a nice reminder that you do have to consider all the things that might affect an experiment’s outcome before concluding what are likely and unlikely results.

It also inspires, in me, a side question: a single coin, obviously, has a tiny chance of landing on its side. A roll of coins has a tiny chance of not landing on its side. How thick a roll has to be assembled before the chance of landing on the side and the chance of landing on either edge become equal? (Without working it out, my guess is it’s about when the roll of coins is as tall as it is across, but I wouldn’t be surprised if it were some slightly oddball thing like the roll has to be the square root of two times the diameter of the coins.)

Doug Savage’s Savage Chickens (August 22) presents an “advanced Sudoku”, in a puzzle that’s either trivially easy or utterly impossible: there’s so few constraints on the numbers in the presented puzzle that it’s not hard to write in digits that will satisfy the results, but, if there’s one right answer, there’s not nearly enough information to tell which one it is. I do find interesting the problem of satisfiability — giving just enough information to solve the puzzle, without allowing more than one solution to be valid — an interesting one. I imagine there’s a very similar problem at work in composing Ivasallay’s Find The Factors puzzles.

Phil Frank and Joe Troise’s The Elderberries (August 24, rerun) presents a “mind aerobics” puzzle in the classic mathematical form of drawing socks out of a drawer. Talking about pulling socks out of drawers suggests a probability puzzle, but the question actually takes it a different direction, into a different sort of logic, and asks about how many socks need to be taken out in order to be sure you have one of each color. The easiest way to apply this is, I believe, to use what’s termed the “pigeon hole principle”, which is one of those mathematical concepts so clear it’s hard to actually notice it. The principle is just that if you have fewer pigeon holes than you have pigeons, and put every pigeon in a pigeon hole, then there’s got to be at least one pigeon hole with more than one pigeons. (Wolfram’s MathWorld credits the statement to Peter Gustav Lejeune Dirichlet, a 19th century German mathematician with a long record of things named for him in number theory, probability, analysis, and differential equations.)

Dave Whamond’s Reality Check (August 24) pulls out the old little pun about algebra and former romantic partners. You’ve probably seen this joke passed around your friends’ Twitter or Facebook feeds too.

Julie Larson’s The Dinette Set (August 25) presents some terrible people’s definition of calculus, as “useless math with letters instead of numbers”, which I have to gripe about because that seems like a more on-point definition of algebra. I’m actually sympathetic to the complaint that calculus is useless, at least if you don’t go into a field that requires it (although that’s rather a circular definition, isn’t it?), but I don’t hold to the idea that whether something is “useful” should determine whether it’s worth learning. My suspicion is that things you find interesting are worth learning, either because you’ll find uses for them, or just because you’ll be surrounding yourself with things you find interesting.

Shifting from numbers to letters, as are used in algebra and calculus, is a great advantage. It allows you to prove things that are true for many problems at once, rather than just the one you’re interested in at the moment. This generality may be too much work to bother with, at least for some problems, but it’s easy to see what’s attractive in solving a problem once and for all.

Mikael Wulff and Anders Morgenthaler’s WuMo (August 25) uses a couple of motifs none of which I’m sure are precisely mathematical, but that seem close enough for my needs. First there’s the motif of Albert Einstein as just being so spectacularly brilliant that he can form an argument in favor of anything, regardless of whether it’s right or wrong. Surely that derives from Einstein’s general reputation of utter brilliance, perhaps flavored by the point that he was able to show how common-sense intuitive ideas about things like “it’s possible to say whether this event happened before or after that event” go wrong. And then there’s the motif of a sophistic argument being so massive and impressive in its bulk that it’s easier to just give in to it rather than try to understand or refute it.

It’s fair of the strip to present Einstein as beginning with questions about how one perceives the universe, though: his relativity work in many ways depends on questions like “how can you tell whether time has passed?” and “how can you tell whether two things happened at the same time?” These are questions which straddle physics, mathematics, and philosophy, and trying to find answers which are logically coherent and testable produced much of the work that’s given him such lasting fame.