## Theorem Thursday: The Intermediate Value Theorem

I am still taking requests for this Theorem Thursdays sequence. I intend to post each Thursday in June and July an essay talking about some theorem and what it means and why it’s important. I have gotten a couple of requests in, but I’m happy to take more; please just give me a little lead time. But I want to start with one that delights me.

# The Intermediate Value Theorem

I own a Scion tC. It’s a pleasant car, about 2400 percent more sporty than I am in real life. I got it because it met my most important criteria: it wasn’t expensive and it had a sun roof. That it looks stylish is an unsought bonus.

But being a car, and a black one at that, it has a common problem. Leave it parked a while, then get inside. In the winter, it gets so cold that snow can fall inside it. In the summer, it gets so hot that the interior, never mind the passengers, risk melting. While pondering this slight inconvenience I wondered, isn’t there any outside temperature that leaves my car comfortable?

My Scion tC, here, not too warm.

Of course there is. We know this before thinking about it. The sun heats the car, yes. When the outside temperature is low enough, there’s enough heat flowing out that the car gets cold. When the outside temperature’s high enough, not enough heat flows out. The car stays warm. There must be some middle temperature where just enough heat flows out that the interior doesn’t get particularly warm or cold. Not just one middle temperature, come to that. There is a range of temperatures that are comfortable to sit in. But that just means there’s a range of outside temperatures for which the car’s interior stays comfortable. We know this range as late April, early May, here. Most years, anyway.

The reasoning that lets us know there is a comfort-producing outside temperature we can see as a use of the Intermediate Value Theorem. It addresses a function f with domain [a, b], and range of the real numbers. The domain is closed; that is, the numbers we call ‘a’ and ‘b’ are both in the set. And f has to be a continuous function. If you want to draw it, you can do so without having to lift pen from paper. (WARNING: Do not attempt to pass your Real Analysis course with that definition. But that’s what the proper definition means.)

So look at the numbers f(a) and f(b). Pick some number between them, and I’ll call that number ‘g’. There must be at least one number ‘c’, that’s between ‘a’ and ‘b’, and for which f(c) equals g.

Bernard Bolzano, an early-19th century mathematician/logician/theologist/priest, gets the credit for first proving this theorem. Bolzano’s version was a little different. It supposes that f(a) and f(b) are of opposite sign. That is, f(a) is a positive and f(b) a negative number. Or f(a) is negative and f(b) is positive. And Bolzano’s theorem says there must be some number ‘c’ for which f(c) is zero.

You can prove this by drawing any wiggly curve at all and then a horizontal line in the middle of it. Well, that doesn’t prove it to mathematician’s satisfaction. But it will prove the matter in the sense that you’ll be convinced. It’ll also convince anyone you try explaining this to.

Any old real-valued function, drawn in blue. The number ‘g’ is something between the number f(a) and f(b). And somewhere there’s at least one number, between a and b, for where the function’s equal to g.

You might wonder why anyone needed this proved at all. It’s a bit like proving that as you pour water into the sink there’ll come a time the last dish gets covered with water. So it is. The need for a proof came about from the ongoing attempt to make mathematics rigorous. We have an intuitive idea of what it means for functions to be continuous; see my above comment about lifting pens from paper. Can that be put in terms that don’t depend on physical intuition? … Yes, it can. And we can divorce the Intermediate Value Theorem from our physical intuitions. We can know something that’s true even if we never see a car or a sink.

This theorem might leave you feeling a little hollow inside. Proving that there is some ‘c’ for which f(c) equals g, or even equals zero, doesn’t seem to tell us much about how to find it. It doesn’t even tell us that there’s only one ‘c’, rather than two or three or a hundred million candidates that meet our criteria. Fair enough. The Intermediate Value Theorem is more about proving the existence of solutions, rather than how to find them.

But knowing there is a solution can help us find them. The Intermediate Value Theorem as we know it grew out of finding roots for polynomials. One numerical method, easy to set up for any problem, is the bisection method. If you know that somewhere between ‘a’ and ‘b’ the function goes from positive to negative, then find the midpoint, ‘c’. The function is equal to zero either between ‘a’ and ‘c’, or between ‘c’ and ‘b’. Pick the side that it’s on, and bisect that. Pick the half of that which the zero must be in. Bisect that half. And repeat until you get close enough to the answer for your needs. (The same reasoning applies to a lot of problems in which you divide the search range in two each time until the answer appears.)

We can get some pretty heady results from the Intermediate Value Theorem, too, even if we don’t know where any of them are. An example you’ll see everywhere is that there must be spots on the opposite sides of the globe with the exact same temperature. Or humidity, or daily rainfall, or any other quantity like that. I had thought everyone was ripping that example off from Richard Courant and Herbert Robbins’s masterpiece What Is Mathematics?. But I can’t find this particular example in there. I wonder what we are all ripping it off from.

Does this magenta line bisect both the red and the greyish blobs simultaneously? … Probably not, unless I’ve been way lucky. But there is some line that does.

So here’s a neat example that is ripped off from them. Draw two blobs on the plane. Is there a straight line that bisects both of them at once? Bisecting here means there’s exactly as much of one blob on one side of the line as on the other. There certainly is. The trick is there are any number of lines that will bisect one blob, and then look at what that does to the other.

A similar ripped-off result you can do with a single blob of any shape you like. Draw any line that bisects it. There are a lot of candidates. Can you draw a line perpendicular to that so that the blob gets quartered, divided into four spots of equal area? Yes. Try it.

Does this pair of magenta lines split this blue blob into four pieces of exactly the same area? … Probably not, unless I’ve been lucky. But there is some pair of perpendicular lines that will do it. Also, is it me or does that blob look kind of like a butterfly?

But surely the best use of the Intermediate Value Theorem is in the problem of wobbly tables. If the table has four legs, all the same length, and the problem is the floor isn’t level it’s all right. There is some way to adjust the table so it won’t wobble. (Well, the ground can’t be angled more than a bit over 35 degrees, but that’s all right. If the ground has a 35 degree angle you aren’t setting a table on it. You’re rolling down it.) Finally a mathematical proof can save us from despair!

Except that the proof doesn’t work if the table legs are uneven which, alas, they often are. But we can’t get everything.

Courant and Robbins put forth one more example that’s fantastic, although it doesn’t quite work. But it’s a train problem unlike those you’ve seen before. Let me give it to you as they set it out:

Suppose a train travels from station A to station B along a straight section of track. The journey need not be of uniform speed or acceleration. The train may act in any manner, speeding up, slowing down, coming to a halt, or even backing up for a while, before reaching B. But the exact motion of the train is supposed to be known in advance; that is, the function s = f(t) is given, where s is the distance of the train from station A, and t is the time, measured from the instant of departure.

On the floor of one of the cars a rod is pivoted so that it may move without friction either forward or backward until it touches the floor. If it does touch the floor, we assume that it remains on the floor henceforth; this wil be the case if the rod does not bounce.

Is it possible to place the rod in such a position that, if it is released at the instant when the train starts and allowed to move solely under the influence of gravity and the motion of the train, it will not fall to the floor during the entire journey from A to B?

They argue it is possible, and use the Intermediate Value Theorem to show it. They admit the range of angles it’s safe to start the rod from may be too small to be useful.

But they’re not quite right. Ian Stewart, in the revision of What Is Mathematics?, includes an appendix about this. Stewart credits Tim Poston with pointing out, in 1976, the flaw. It’s possible to imagine a path which causes the rod, from one angle, to just graze tipping over, let’s say forward, and then get yanked back and fall over flat backwards. This would leave no room for any starting angles that avoid falling over entirely.

It’s a subtle flaw. You might expect so. Nobody mentioned it between the book’s original publication in 1941, after which everyone liking mathematics read it, and 1976. And it is one that touches on the complications of spaces. This little Intermediate Value Theorem problem draws us close to chaos theory. It’s one of those ideas that weaves through all mathematics.

## Reading the Comics, October 17, 2015: Rerun Edition

I hate to make it sound like I’m running out of things to say about mathematical comics. But the most recent bunch of strips have been reruns, as with Bill Amend’s FoxTrot or Tom Toles’s Randolph Itch, 2 am. And there’s some figurative reruns too, as a couple of things I’ve talked about before come around again. Also I’m not sure but I think I might have used this Edition Title before. It feels like one I might have. I hope you’ll enjoy anyway, please.

Bill Amend’s FoxTrot Classics for the 15th of October, originally run in 2004, is about binary numerals. It’s built on the fact the numeral ‘100’ represents a rather smaller number in base-two arithmetic than it does in base-ten. This is the sort of thing that’s funny to a mathematically-inclined nerd, such as Jason here. It’s the numerical equivalent of a pun, playing on how if you pretend something is in a different context, it would have a different meaning.

Dave Blazek’s Loose Parts for the 15th of October puts a shape other than a triangle into the orchestra pit. I’m amused, and it puts me in mind of the classic question, “Can One Hear The Shape Of A Drum?” The answer is tricky.

Bob Scott’s Molly and the Bear for the 15th of October is a Pi Day joke. I don’t believe it’s a rerun, but the engagingly-drawn strip is in reruns terribly often.

Tom Toles’s Randolph Itch, 2 am for the 15th of October is a rerun, not just from 1999 but from earlier this year. I don’t know if the strip is being run out of order or if the strip ran a shorter time than I thought. Anyway, it’s still a funny drawing and “r” doesn’t figure into it at all.

Rick Detorie’s One Big Happy for the 16th of October, 2015.

Rick Detorie’s One Big Happy for the 16th of October shows Ruthie teaching her stuffed dolls about the number 1. Ruthie is a bit confused about the difference between the number one and the numeral, the way we represent the number. That’s common enough.

She does kind of have a point, though. The number one gets represented as a vertical stroke in the Arabic numerals we commonly use; also in Roman numerals used in making dates harder to read; also in Ancient Egyptian numerals; also in Chinese numerals. One almost suspects everyone is copying each other, or just started off with a tally mark and kept with it. Things get more complicated around ‘three’ or ‘four’. But it isn’t really universal, of course. The Mayans used a single dot, which is admittedly pretty close as a scheme. The Babylonians used a vertical wedge, a little triangle atop a stem that was presumably easy to carve with the tools available.

Ruben Bolling’s Super-Fun-Pak Comix for the 16th of October reprings a Chaos Butterfly installment. And the reminder that a system can be deterministic yet unpredictable sets me up for …

The rerun of Tom Toles’s Randolph Itch, 2 am that appeared on the 17th. The page of horoscopes saying “what happens to you today will be random, based on laws of probability” is funny, although, “random”? There is, it appears, randomness deeply encoded in the universe. There seems to be no way that atoms and molecules could work if they could not be random. But randomness follows laws. Those laws are so fundamental, and imply averages so relentlessly, that they create a human-scale world which might as well be deterministic. (I am deliberately bundling up the question of whether beings have free will and putting it off to the corner, in a little box, where I will not bother it.) In principle, we should be able to predict the day; we just need enough information, and time to compute.

Of course in practice we can’t, and can’t even come close. We may be able to predict the broad strokes of the day, but it is filled with the unpredictable. We call that random, but that is really a confession of ignorance. It’s much the way we might say there is a “probability” of one in seven that you were born on a Tuesday. There’s no such thing. The probability is either 1, because you were born on a Tuesday, or 0, because you were not. What day any given date in the Julian or Gregorian calendar occurred is a determined thing. What we mean by “a probability of one in seven” is that we are ignorant of your birthday, or have not done the work of finding out what day of the week that was. Thus the day of the week appears random.

John Graziano’s Ripley’s Believe It or Not for the 17th of October claims that Les Stewart wrote out “every number from one to one million in words’, using seven typewriters, in a project that took sixteen years and seven months. Sixteen years and seven months is something close to half a billion seconds. So if we take this, he was averaging about fifty seconds to write out each number. This sounds unimpressive, but after all, he had to take some time to sleep and probably had other projects to work on as well. Perhaps he was also working on putting the numbers in alphabetical order.

## Reading the Comics, September 16, 2015: Celebrity Appearance Edition

I couldn’t go on calling this Back To School Editions. A couple of the comic strips the past week have given me reason to mention people famous in mathematics or physics circles, and one who’s even famous in the real world too. That’ll do for a title.

Jeff Corriveau’s Deflocked for the 15th of September tells what I want to call an old joke about geese formations. The thing is that I’m not sure it is an old joke. At least I can’t think of it being done much. It seems like it should have been.

The formations that geese, or other birds, form has been a neat corner of mathematics. The question they inspire is “how do birds know what to do?” How can they form complicated groupings and, more, change their flight patterns at a moment’s notice? (Geese flying in V shapes don’t need to do that, but other flocking birds will.) One surprising answer is that if each bird is just trying to follow a couple of simple rules, then if you have enough birds, the group will do amazingly complex things. This is good for people who want to say how complex things come about. It suggests you don’t need very much to have robust and flexible systems. It’s also bad for people who want to say how complex things come about. It suggests that many things that would be interesting can’t be studied in simpler models. Use a smaller number of birds or fewer rules or such and the interesting behavior doesn’t appear.

Jeff Corriveau’s Deflocked for the 15th of September, 2015.

Scott Adams’s Dilbert Classics from the 15th and 16th of September (originally run the 22nd and 23rd of July, 1992) are about mathematical forecasts of the future. This is a hard field. It’s one people have been dreaming of doing for a long while. J Willard Gibbs, the renowned 19th century physicist who put the mathematics of thermodynamics in essentially its modern form, pondered whether a thermodynamics of history could be made. But attempts at making such predictions top out at demographic or rough economic forecasts, and for obvious reason.

The next day Dilbert’s garbageman, the smartest person in the world, asserts the problem is chaos theory, that “any complex iterative model is no better than a wild guess”. I wouldn’t put it that way, although I’m not sure what would convey the idea within the space available. One problem with predicting complicated systems, even if they are deterministic, is that there is a difference between what we can measure a system to be and what the system actually is. And for some systems that slight error will be magnified quickly to the point that a prediction based on our measurement is useless. (Fortunately this seems to affect only interesting systems, so we can still do things like study physics in high school usefully.)

Maria Scrivan’s Half Full for the 16th of September makes the Common Core joke. A generation ago this was a New Math joke. It’s got me curious about the history of attempts to reform mathematics teaching, and how poorly they get received. Surely someone’s written a popular or at least semipopular book about the process? I need some friends in the anthropology or sociology departments to tell, I suppose.

In Mark Tatulli’s Heart of the City for the 16th of September, Heart is already feeling lost in mathematics. She’s in enough trouble she doesn’t recognize mathematics terms. That is an old joke, too, although I think the best version of it was done in a Bloom County with no mathematical content. (Milo Bloom met his idol Betty Crocker and learned that she was a marketing icon who knew nothing of cooking. She didn’t even recognize “shish kebob” as a cooking term.)

Mell Lazarus’s Momma for the 16th of September sneers at the idea of predicting where specks of dust will land. But the motion of dust particles is interesting. What can be said about the way dust moves when the dust is being battered by air molecules that are moving as good as randomly? This becomes a problem in statistical mechanics, and one that depends on many things, including just how fast air particles move and how big molecules are. Now for the celebrity part of this story.

Albert Einstein published four papers in his “Annus mirabilis” year of 1905. One of them was the Special Theory of Relativity, and another the mass-energy equivalence. Those, and the General Theory of Relativity, are surely why he became and still is a familiar name to people. One of his others was on the photoelectric effect. It’s a cornerstone of quantum mechanics. If Einstein had done nothing in relativity he’d still be renowned among physicists for that. The last paper, though, that was on Brownian motion, the movement of particles buffeted by random forces like this. And if he’d done nothing in relativity or quantum mechanics, he’d still probably be known in statistical mechanics circles for this work. Among other things this work gave the first good estimates for the size of atoms and molecules, and gave easily observable, macroscopic-scale evidence that molecules must exist. That took some work, though.

Dave Whamond’s Reality Check for the 16th of September shows off the Metropolitan Museum of Symmetry. This is probably meant to be an art museum. Symmetries are studied in mathematics too, though. Many symmetries, the ways you can swap shapes around, form interesting groups or rings. And in mathematical physics, symmetries give us useful information about the behavior of systems. That’s enough for me to claim this comic is mathematically linked.

## Reading the Comics, September 10, 2015: Back To School Edition

I assume that Comic Strip Master Command ordered many mathematically-themed comic strips to coincide with the United States school system getting back up to full. That or they knew I’d have a busy week. This is only the first part of comic strips that have appeared since Tuesday.

Mel Henze’s Gentle Creatures for the 7th and the 8th of September use mathematical talk to fill out the technobabble. It’s a cute enough notion. These particular strips ran last year, and I talked about them then. The talk of a “Lagrangian model” interests me. It name-checks a real and important and interesting scientist who’s not Einstein or Stephen Hawking. But I’m still not aware of any “Lagrangian model” that would be relevant to starship operations.

Jon Rosenberg’s Scenes from a Multiverse for the 7th of September speaks of a society of “powerful thaumaturgic diagrammers” who used Venn diagrams not wisely but too well. The diagrammers got into trouble when one made “a Venn diagram that showed the intersection of all the Venns and all the diagrams”. I imagine this not to be a rigorous description of what happened. But Venn diagrams match up well with many logic problems. And self-referential logic, logic statements that describe their own truth or falsity, is often problematic. So I would accept a story in which Venn diagrams about Venn diagrams leads to trouble. The motif of tying logic and mathematics into magic is an old one. I understand it. A clever mathematical argument often feels like magic, especially the surprising ones. To me, the magical theorems are those that prove a set of seemingly irrelevant lemmas. Then, with that stock in hand, the theorem goes on to the main point in a few wondrous lines. If you can do that, why not transmute lead, or accidentally retcon a society out of existence?

Mark Anderson’s Andertoons for the 8th of September just delights me. Occasionally I feel a bit like Mark Anderson’s volunteer publicity department. A panel like this, though, makes me feel that he deserves it.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 8th of September is the first anthropomorphic-geometric-figures joke we’ve had here in a while.

Mike Baldwin’s Cornered for the 9th of September is a drug testing joke, and a gambling joke. Both are subjects driven by probabilities. Any truly interesting system is always changing. If we want to know whether something affects the system we have to know whether we can make a change that’s bigger than the system does on its own. And this gives us drug-testing and other statistical inference tests. If we apply a drug, or some treatment, or whatever, how does the system change? Does it change enough, consistently, that it’s not plausible that the change just happened by chance? Or by some other influence?

You might have noticed a controversy going around psychology journals. A fair number of experiments were re-run, by new experimenters following the original protocols as closely as possible. Quite a few of the reported results didn’t happen again, or happened in a weaker way. That’s produced some handwringing. No one thinks deliberate experimental fraud is that widespread in the field. There may be accidental fraud, people choosing data or analyses that heighten the effect they want to prove, or that pick out any effect. However, it may also simply be chance again. Psychology experiments tend to have a lower threshold of “this is sufficiently improbable that it indicates something is happening” than, say, physics has. Psychology has a harder time getting the raw data. A supercollider has enormous startup costs, but you can run the thing for as long as you like. And every electron is the same thing. A test of how sleep deprivation affects driving skills? That’s hard. No two sleepers or drivers are quite alike, even at different times of the day. There’s not an obvious cure. Independent replication of previously done experiments helps. That’s work that isn’t exciting — necessary as it is, it’s also repeating what others did — and it’s harder to get people to do it, or pay for it. But in the meantime it’s harder to be sure what interesting results to trust.

Ruben Bolling’s Super-Fun-Pak Comix for the 9th of September is another Chaos Butterfly installment. I don’t want to get folks too excited for posts I technically haven’t written yet, but there is more Chaos Butterfly soon.

Rick Stromoski’s Soup To Nutz for the 10th of September has Royboy guess the odds of winning a lottery are 50-50. Silly, yes, but only because we know that anyone is much more likely to lose a lottery than to win it. But then how do we know that?

Since the rules of a lottery are laid out clearly we can reason about the probability of winning. We can calculate the number of possible outcomes of the game, and how many of them count as winning. Suppose each of those possible outcomes are equally likely. Then the probability of winning is the number of winning outcomes divided by the number of probable outcomes. Quite easy.

— Of course, that’s exactly what Royboy did. There’s two possible outcomes, winning or losing. Lacking reason to think they aren’t equally likely he concluded a win and a loss were just as probable.

We have to be careful what we mean by “an outcome”. What we probably mean for a drawn-numbers lottery is the number of ways the lottery numbers can be drawn. For a scratch-off card we mean the number of tickets that can be printed. But we’re still stuck with this idea of “equally likely” outcomes. I suspect we know what we mean by this, but trying to say what that is clearly, and without question-begging, is hard. And even this works only because we know the rules by which the lottery operates. Or we can look them up. If we didn’t know the details of the lottery’s workings, past the assumption that it has consistently followed rules, what could we do?

Well, that’s what we have probability classes for, and particularly the field of Bayesian probability. This field tries to estimate the probabilities of things based on what actually happens. Suppose Royboy played the lottery fifty times and lost every time. That would smash the idea that his chances were 50-50, although that would not yet tell him what the chances really are.

• #### ivasallay 5:33 pm on Tuesday, 15 September, 2015 Permalink | Reply

Soup to Nutz could make a worthwhile classroom discussion.

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• #### Joseph Nebus 12:18 am on Friday, 18 September, 2015 Permalink | Reply

Not just a discussion — you could almost hang a whole course in probability on this one! I had to restrain myself from writing forever about it and just publish already.

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## Reading the Comics, August 14, 2015: Name-Dropping Edition

There have been fewer mathematically-themed comic strips than usual the past week, but they have been coming in yet. This week seems to have included a fair number of name-drops of interesting mathematical concepts.

David L Hoyt and Jeff Knurek’s Jumble (August 10) name-drops the abacus. It has got me wondering about how abacuses were made in the pre-industrial age. On the one hand they could in principle be made by anybody who has beads and rods. On the other hand, a skillfully made abacus will make the tool so much more effective. Who made and who sold them? I honestly don’t know.

David L Hoyt and Jeff Knurek’s Jumble for the 10th of August, 2015. The link will likely expire around the 10th of September.

Mick Mastroianni and Mason Mastroianni’s Dogs of C Kennel (August 11) has Tucker reveal that most of the mathematics he scrawls is just to make his work look harder. I suspect Tucker overdid his performance. My experience is you can get the audience’s eyes to glaze over with much less mathematics on the board.

Leigh Rubin’s Rubes (August 11) mentions chaos theory. It’s not properly speaking a Chaos Butterfly comic strip. But certainly it’s in the vicinity.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 11) name-drops Banach-Tarski. This is a reference to a famous-in-some-circles theorem, or paradox. The theorem, published in 1924 by Stefan Banach and Alfred Tarski, shows something astounding. It’s possible to take a ball, and disassemble it into a number of pieces. Then, doing nothing more than sliding and rotating the pieces, one can reassemble the pieces to get two balls each with the same volume of the original. If that doesn’t sound ridiculous enough, consider that it’s possible to do this trick by cutting the ball into as few as five pieces. (Four, if you’re willing to exclude the exact center of the original ball.) So you can see why this is called a paradox, and why this joke works for people who know the background.

Scott Hilburn’s The Argyle Sweater (August 12) illustrates that joke about rounding up the cattle you might have seen going around.

• #### sarcasticgoat 3:50 pm on Saturday, 15 August, 2015 Permalink | Reply

Are the scrambled words mathematical? Because I do this kind of thing a lot, and I could only get the first one ‘Query’??

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• #### Joseph Nebus 4:47 am on Tuesday, 18 August, 2015 Permalink | Reply

They’re not mathematical. Well, the third one I could stretch to be mathematical, if I tried, and I might use it if my Reading the Comics post was a little short that week. Only the punch line has a fairly direct mathematical link. The Jumble words don’t tend to be thematically linked. ‘QUERY’ comes up a lot in the puzzles, too.

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• #### sarcasticgoat 1:36 pm on Tuesday, 18 August, 2015 Permalink | Reply

I still haven’t got any of them….

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• #### Joseph Nebus 9:01 pm on Saturday, 22 August, 2015 Permalink | Reply

I find it often helps if you write out the letters of the jumbled word in alphabetical order and then try combinations from there. This destroys accidental patterns (like the appearance of ‘SUN’ in the third word) that might keep you from getting to the real word.

And if that doesn’t help they did print the answers the next day. And if that link’s expired since then there’s the Internet Anagram Server to the rescue.

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## Reading the Comics, June 30, 2015: Fumigating The Theater Edition

One of my favorite ever episodes of The Muppet Show when I was a kid had the premise the Muppet Theater was being fumigated and so they had to put on a show from the train station instead. (It was the Loretta Lynn episode, third season, number eight.) I loved seeing them try to carry on as normal when not a single thing was as it should be. Since then — probably before, too, but I don’t remember that — I’ve loved seeing stuff trying to carry on in adverse circumstances.

Why this is mentioned here is that Sunday night my computer had a nasty freeze and some video card mishaps. I discovered that my early-2011 MacBook Pro might be among those recalled earlier this year for a service glitch. My computer is in for what I hope is a simple, free, and quick repair. But obviously I’m not at my best right now. I might be even longer than usual answering people and goodness knows how the statistics survey of June will go.

Anyway. Rick Kirkman and Jerry Scott’s Baby Blues (June 26) is a joke about motivating kids to do mathematics. And about how you can’t do mathematics over summer vacation.

Ruben Bolling’s Tom The Dancing Bug (June 26) features a return appearance of Chaos Butterfly. Chaos Butterfly does what Chaos Butterfly does best.

Charles Schulz’s Peanuts Begins (June 26; actually just the Peanuts of March 23, 1951) uses arithmetic as a test of smartness. And as an example of something impractical.

Alex Hallatt’s Arctic Circle (June 28) is a riff on the Good Will Hunting premise. That movie’s particular premise — the janitor solves an impossible problem left on the board — is, so far as I know, something that hasn’t happened. But it’s not impossible. Training will help one develop reasoning ability. Training will provide context and definitions and models to work from. But that’s not essential. All that’s essential is the ability to reason. Everyone has that ability; everyone can do mathematics. Someone coming from outside the academy could do first-rate work. However, I’d bet on the person with the advanced degree in mathematics. There is value in training.

Alex Hallatt’s Arctic Circle for the 28th of June, 2015.

But as many note, the Good Will Hunting premise has got a kernel of truth in it. In 1939, George Dantzig, a grad student in mathematics at University of California/Berkeley, came in late to class. He didn’t know that two problems on the board were examples of unproven theorems, and assumed them to be homework. So he did them, though he apologized for taking so long to do them. Before you draw too much inspiration from this, though, remember that Dantzig was a graduate student almost ready to start work on a PhD thesis. And the problems were not thought unsolvable, just conjectures not yet proven. Snopes, as ever, provides some explanation of the legend and some of the variant ways the story is told.

Mac King and Bill King’s Magic In A Minute (June 28) shows off a magic trick that you could recast as a permutations problem. If you’ve been studying group theory, and many of my Mathematics A To Z terms have readied you for group theory, you can prove why this trick works.

Guy Gilchrist’s Nancy (June 28) carries on Baby Blues‘s theme of mathematics during summer vacation being simply undoable.

Piers Baker’s Ollie and Quentin for December 28, 2014, and repeated on June 28, 2015.

Piers Baker’s Ollie and Quentin (June 28) is a gambler’s fallacy-themed joke. It was run — on ComicsKingdom, back then — back in December, and I talked some more about it then.

Mike Twohy’s That’s Life (June 28) is about the perils of putting too much attention into mental arithmetic. It’s also about how perilously hypnotic decimals are: if the pitcher had realized “fourteen million over three years” must be “four and two-thirds million per year” he’d surely have been less distracted.

• #### Thumbup 2:42 pm on Tuesday, 30 June, 2015 Permalink | Reply

Smart kid. It definitely doesn’t mix!

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• #### Joseph Nebus 6:44 pm on Saturday, 4 July, 2015 Permalink | Reply

I’m actually a little surprised Hammy didn’t think mathematics and candy couldn’t mix. The use seems quite dear to a kid’s heart. At least I would’ve thought it dear when I was a kid.

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• #### Thumbup 7:33 pm on Saturday, 4 July, 2015 Permalink | Reply

Joseph Nebus, Really?! Cool. Mathematics rather an amazing thing. Yeah. You have a good fourth.

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• #### Joseph Nebus 6:32 pm on Sunday, 5 July, 2015 Permalink | Reply

Really so. Of course, I do enjoy calculating stuff more than the norm.

I did indeed have a good fourth; hope you enjoyed yours.

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• #### Thumbup 12:21 am on Monday, 6 July, 2015 Permalink | Reply

Joseph Nebus,
Yes, I did. Thank you.

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• #### Joseph Nebus 3:19 am on Monday, 6 July, 2015 Permalink | Reply

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• #### Thumbup 4:55 am on Monday, 6 July, 2015 Permalink | Reply

:)

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• #### scifihammy 3:41 pm on Tuesday, 30 June, 2015 Permalink | Reply

Love the cartoons – especially the “smiley face” answer and the 50%! So many people I know don’t get that either. :)

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• #### Joseph Nebus 6:47 pm on Saturday, 4 July, 2015 Permalink | Reply

Well, the 50 percent thing is a tricky point. I mean, all probabilities are tricky; that’s why you should never, ever, ever trust your instinctive response to a probability question. But applying a probability is even more tricky. I imagine that’s because we have a feeling for how things “ought” to be, but they never actually are that way. It’s very hard to feel confident in the application of something when every example seems wrong in one way or another.

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• #### scifihammy 7:08 am on Sunday, 5 July, 2015 Permalink | Reply

That’s it exactly! Took me a while to grasp that you can toss a coin 99 times and get 99 Heads in a row, but that still doesn’t mean that the next toss will be Tails; probably more likely the coin is weighted! ;) Still, as you say, it is hard to ignore your instinct on an outcome.

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• #### Joseph Nebus 6:38 pm on Sunday, 5 July, 2015 Permalink | Reply

At 99 heads in a row I’d certainly bet on the coin being weighted.

Still, yeah; I forget where (possibly one of John D Cook’s Twitter feeds) I saw the warning from. But it’s good to remember that there are entire fields of psychology dedicated to studying how bad people’s intuitive feeling for probability problems are.

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• #### sheldonk2014 10:35 pm on Tuesday, 30 June, 2015 Permalink | Reply

I love thus as a theory things being dine under adverse circumstances
Didn’t even think that was a actual theory
That’s great

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• #### Joseph Nebus 6:50 pm on Saturday, 4 July, 2015 Permalink | Reply

I’m not so sure it’s a theory so much as it is making sure the show goes on. But it did produce a great Muppet Show episode. Of course, aren’t they all great? Even the bad ones still have Fozzie Bear.

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• #### ivasallay 8:58 am on Saturday, 4 July, 2015 Permalink | Reply

I thought Baby Blues was cute, but Nancy offended me.

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• #### Joseph Nebus 6:53 pm on Saturday, 4 July, 2015 Permalink | Reply

That’s an interesting split of feelings, given they’re basically the same joke. What do you suppose makes the difference?

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• #### ivasallay 6:24 am on Wednesday, 8 July, 2015 Permalink | Reply

The kid in Baby Blues was only speaking for himself. Nancy implied that the whole world should feel that way.

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• #### Joseph Nebus 5:49 am on Friday, 10 July, 2015 Permalink | Reply

That’s a fair enough distinction.

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## 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.

• #### ivasallay 1:39 am on Thursday, 5 March, 2015 Permalink | Reply

Mutt and Jeff made me laugh the loudest, but Eric the Circle appeals to both my intellect and my humorous side.

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• #### Joseph Nebus 11:53 pm on Thursday, 5 March, 2015 Permalink | Reply

Oh, I didn’t think about which strips I particularly liked. Well, Mutt and Jeff has that classic old-fashioned joke structure that I enjoy, and Eric the Circle probably the best laugh even if it’s a bit over how awful a pickup line that is.

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• #### abyssbrain 6:01 am on Thursday, 5 March, 2015 Permalink | Reply

The parity trick with the dice is interesting. I remembered a trick in one of Martin Gardner’s book that involves 3 cups which also relies on the concept of parity.

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• #### Joseph Nebus 11:55 pm on Thursday, 5 March, 2015 Permalink | Reply

I don’t know the trick you mean — I’m surprisingly poorly-read in Martin Gardner — but I think I can imagine the sorts of parity-based magic tricks to be done with several cups instead.

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• #### abyssbrain 12:24 am on Friday, 6 March, 2015 Permalink | Reply

I have read two variations that involve cups. Gardner also discussed parities in general on several occasions

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## Reading the Comics, January 29, 2015: Returned Motifs Edition

I do occasionally worry that my little blog is going to become nothing but a review of mathematics-themed comic strips, especially when Comic Strip Master Command sends out abundant crops like it has the past few weeks. This week’s offerings bring out the return of a lot of familiar motifs, like fighting with word problems and anthropomorphized numbers; and there’s one strip that suggests a pair of articles I wrote a while back might be useful yet.

Bill Amend’s FoxTrot (January 25, and not a rerun) puts out a little word problem, about what grade one needs to get a B in this class, in the sort of passive-aggressive sniping teachers long to get away with. As Paige notes, it really isn’t a geometry problem, although I wonder if there’s a sensible way to represent it as a geometry problem.

Ruben Bolling’s Super-Fun-Pax Comix superstar Chaos Butterfly appears not just in the January 25th installment but also gets a passing mention in Mark Heath’sSpot the Frog (January 29, rerun). Chaos Butterfly in all its forms seems to be popping up a lot lately; I wonder if it’s something in the air.

• #### Carrie Rubin 5:22 pm on Friday, 30 January, 2015 Permalink | Reply

Love the Break of Day one, but you’re right, March 14th would have been the perfect day to publish it.

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• #### ivasallay 8:23 am on Saturday, 31 January, 2015 Permalink | Reply

Long ago I learned to read with the Dick and Jane books.
My favorite comics this time were Baldo and Break of the Day.

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• #### ioanaiuliana 9:28 pm on Friday, 6 February, 2015 Permalink | Reply

I love ‘Break of Day’, but I find it really nice now, especially because it also works for Valentine’s Day (14th Feb) :))) Sooo gooooood! :)

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• #### Joseph Nebus 8:00 pm on Saturday, 17 January, 2015 Permalink | Reply Tags: accounting ( 8 ), answers, anthropomorphism ( 6 ), chaos, KenKen ( 2 ), navigation ( 4 ), pi ( 40 ), robert benchley ( 4 ), textbooks ( 2 )

This week’s collection of mathematics-themed comic strips includes one of the best examples of using mathematics in real life, because it describes how to find your position if you’re lost in, in this case, an uncharted island. I’m only saddened that I couldn’t find a natural way to work in how to use an analog watch as a makeshift compass, so I’m shoehorning it in up here, as well as pointing out that if you don’t have an analog clock to use, you can still approximate it by drawing the hands of the clock on a sheet of paper and using that as a pretend watch, and there is something awesome about using a sheet of paper with the time drawn on it as a way to finding north.

Dave Whamond’s Reality Check (January 12) is a guru-on-the-mountain joke, explaining that the answers to life are in the back of the math book. It’s certainly convention for a mathematics book, at least up through about Intro Differential Equations, to include answers to the problems, or at least a selection of problems, in the back, and on reflection it’s a bit of an odd convention. You don’t see that in, say, a history book even where the questions can be reduced to picking out trivia from the main text. I suppose the math-answers convention reflects an idea that there’s a correct way to go about solving a problem, and therefore, you can check whether you picked the correct way and followed it correctly with no more answer than a printed “15/2” as guide. In this way, I suppose, a mathematics textbook can be self-teaching — at least, the eager student can do some of her own pass/fail grading — which was probably invaluable back in the days when finding a skilled mathematics teacher was so much harder than it is today.

• #### ivasallay 7:32 pm on Monday, 19 January, 2015 Permalink | Reply

I love that the answers to all of the questions of the universe are in the back of a math book, but I suspect it’s only the odd questions.
Kids learning about division would probably enjoy the Maria’s Day strip if their teacher showed it to them.
Thanks for reading so many comics and sharing these with us!

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• #### Joseph Nebus 10:01 pm on Tuesday, 20 January, 2015 Permalink | Reply

Happy to serve.

I wonder how it is that it’s most often the odd problems that have answers. I’ve seen some books in which the even-numbered problems have the given answers, and the occasional freak case in which there’s no obvious pattern, but it seems to me that odd is more popular. Although I probably should actually check some books and report back before declaring it’s so.

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