Reading the Comics, February 3, 2017: Counting Edition

And now I can close out last week’s mathematically-themed comic strips. Two of them are even about counting, which is enough for me to make that the name of this set.

John Allen’s Nest Heads for the 2nd mentions a probability and statistics class and something it’s supposed to be good for. I would agree that probability and statistics are probably (I can’t find a better way to write this) the most practically useful mathematics one can learn. At least once you’re past arithmetic. They’re practical by birth; humans began studying them because they offer guidance in uncertain situations. And one can use many of their tools without needing more than arithmetic.

I’m not so staunchly anti-lottery as many mathematics people are. I’ll admit I play it myself, when the jackpot is large enough. When the expectation value of the prize gets to be positive, it’s harder to rationalize not playing. This happens only once or twice a year, but it’s fun to watch and see when it happens. I grant it’s a foolish way to use two dollars (two tickets are my limit), but you know? My budget is not so tight I can’t spend four dollars foolishly a year. Besides, I don’t insist on winning one of those half-billion-dollar prizes. I imagine I’d be satisfied if I brought in a mere $10,000.

Rick Detorie’s One Big Happy for the 3rd of February, 2017. A ‘gazillion’ is actually a surprisingly low number, hovering as it does somewhere around 212. Fun fact!

Rick Detorie’s One Big Happy for the 3rd continues my previous essay’s bit of incompetence at basic mathematics, here, counting. But working out that her age is between 22 an a gazillion may be worth doing. It’s a common mathematical challenge to find a correct number starting from little information about it. Usually we find it by locating bounds: the number must be larger than this and smaller than that. And then get the bounds closer together. Stop when they’re close enough for our needs, if we’re numerical mathematicians. Stop when the bounds are equal to each other, if we’re analytic mathematicians. That can take a lot of work. Many problems in number theory amount to “improve our estimate of the lowest (or highest) number for which this is true”. We have to start somewhere.

Samson’s Dark Side of the Horse for the 3rd is a counting-sheep joke and I was amused that the counting went so awry here. On looking over the strip again for this essay, though, I realize I read it wrong. It’s the fences that are getting counted, not the sheep. Well, it’s a cute little sheep having the same problems counting that Horace has. We don’t tend to do well counting more than around seven things at a glance. We can get a bit farther if we can group things together and spot that, say, we have four groups of four fences each. That works and it’s legitimate; we’re counting and we get the right count out of it. But it does feel like we’re doing something different from how we count, say, three things at a glance.

Mick Mastroianni and Mason MastroianniDogs of C Kennel for the 3rd is about the world’s favorite piece of statistical mechanics, entropy. There’s room for quibbling about what exactly we mean by thermodynamics saying all matter is slowly breaking down. But the gist is fair enough. It’s still mysterious, though. To say that the disorder of things is always increasing forces us to think about what we mean by disorder. It’s easy to think we have an idea what we mean by it. It’s hard to make that a completely satisfying definition. In this way it’s much like randomness, which is another idea often treated as the same as disorder.

Bill Amend’s FoxTrot Classics for the 3rd reprinted the comic from the 10th of February, 2006. Mathematics teachers always want to see how you get your answers. Why? … Well, there are different categories of mistakes someone can make. One can set out trying to solve the wrong problem. One can set out trying to solve the right problem in a wrong way. One can set out solving the right problem in the right way and get lost somewhere in the process. Or one can be doing just fine and somewhere along the line change an addition to a subtraction and get what looks like the wrong answer. Each of these is a different kind of mistake. Knowing what kinds of mistakes people make is key to helping them not make these mistakes. They can get on to making more exciting mistakes.

Reading the Comics, November 4, 2015: Gambling Edition

I don’t presume to guess why. But Comic Strip Master Command sent out orders one lead-time ago to have everybody do jokes that relate to gambling. We see the consequences here.

John Rose’s Barney Google and Snuffy Smith for the 2nd of November builds its joke on the idea that the mathematics of gambling is all anyone really needs. It’s a better-than-average crack about the usefulness of mathematics. It’s also truer than average. Much of how we make decisions is built on the expectation value, a core concept of probability. If we do this, what can we expect to gain or lose? If we do that instead, what would we expect? If we can place a value — even a loose, approximate value — on our time, our money, our experiences, we gain a new tool for making decisions.

John Rose’s Barney Google and Snuffy Smith for the 2nd of November, 2015. I am actually more comfortable with Snuffy’s sentiments than you might think. However, I think he might want Jughaid to also get comfortable with card-counting techniques, and the conditional probabilities that they imply. (Given that a 10, an 8, and a 9 are already showing, what is the probability that the dealer will go over?)
Unanswered question: where did Snuffy Smith get a twenty-dollar bill from?

Probability runs through the history of mathematics. That’s euphemistic. Gambling runs through the history of mathematics. Quite a bit of what we call probability derives from people who wanted to better understand games of chance, and to get an edge in the bets they might place. A question like “how many ways can three dice come up?” is a good homework problem today. It was once a subject of serious study and argument. We realize it’s still a good question when we wonder if the first die coming up 6, the second 3, and and third 1 is a different outcome from the first die coming up 3, the second 1, and the third 6.

Fully understanding the mathematics of gambling requires not just counting and not just fractions. It will bring us to algebra, to calculus, and to all the tools that let us understand thermodynamics and quantum mechanics. If that isn’t everything, that is a good rough approximation.

Scott Adams’s Dilbert Classic for the 2nd of November originally ran the 8th of September, 1992. It’s about a sadly common kind of nerd behavior, the desire to one-up one’s stories of programming hardship. In this one the generic guy — a different figure from Adams’s current model of generic guy — asserts he goes back to before binary numbers, even. I admit skepticism. Certainly you could list different numbers by making the same symbol often enough. We do that when we resort to tally marks. But we need some second symbol to note the end of a number. With tally marks we can do that with physical space. A computer’s memory, though? That needs something else.

Kevin Fagan’s Drabble began a story about the logic of buying a lottery ticket this week. (The story goes on several days past this.) This is another probability, that is gambling, problem. Large jackpots present a pretty good philosophical challenge. It’s possible the jackpot will be so large that the expected value of buying a ticket is positive. This would seem to imply you should buy a ticket. But your chance of winning will be, as ever, vanishingly small. One chance in 200 million or more. You will not win. This would seem to imply you should not buy a ticket. Both are hard arguments to refute. I admit that when the jackpot gets sufficiently large, I’ll buy one or two tickets. I don’t expect to win the $200 million jackpot or anything like that, though. I’ll be content if I can secure a cozy little $25,000 minor prize. But I might just get a long john doughnut instead.

Larry Wright’s Motley for the 2nd of November originally ran that day in 1987. It name-drops E = mc² as shorthand for genius, the equation’s general role.

Doug Bratton’s Pop Culture Shock Therapy for the 3rd of November doesn’t mention E = mc², but it is an Albert Einstein joke. It doesn’t build on the comforting but dubious legend of Einstein being a poor student. That’s an unusual direction.

Eric the Circle for the 3rd of November is by “Shane”. It’s a cute joke: if Eric were in a horserace, how would his lead be measured? Obviously, by comparison to his diameter. I doubt the race caller would need so many digits past the decimal, though. If cartoons and old-time radio sitcoms about horseracing haven’t led me wrong, distances are measured in a couple common fractions of a horse length — a half, a quarter, three-quarters and so on. So surely Eric would be called “about seven radii” or “three and a half diameters” ahead. It would make sense if his lead were measured by circumferences, if he’s rolling along. But it can be surprisingly hard to estimate by eye what the circumference of a circle is. Diameters are easier.

Jonathan Lemon’s Rabbits Against Magic for the 4th of November has a M&oum;bius strip joke. Obviously, though, what’s taking so long is that Eightball’s spare tire isn’t even on the rim. This is bad.

John Zakour and Scott Roberts’s Working Daze for the 4th of November is a variation on the joke about mathematicians being lousy at arithmetic. Here it’s an accountant who’s bad. I am reminded of the science fiction great Arthur C Clarke mentioning his time as an accounts auditor. He supposed that as long as figures added up approximately, to something like one percent, then there probably wasn’t anything requiring further scrutiny going on. He was able to finish his day’s work quickly, and went on to other jobs in time. Bob Newhart also claimed to not demand too much precision in the accounts he was overseeing. He then went on to sell comedy records to radio stations for a fair bit less than they cost to produce, so perhaps he was better off not working on the money side of things.

Reading The Comics, December 22, 2015: National Mathematics Day Edition

It was a busy week — well, it’s a season for busy weeks, after all — which is why the mathematics comics pile grew so large before I could do anything about it this time around. I’m not sure which I’d pick as my favorite; the Truth Facts tickles me by playing symbols up for confusion and ambiguity, but Quincy is surely the best-drawn of this collection, and good comic strip art deserves attention. Happily that’s a vintage strip from King Features so I feel comfortable including the comic strip for you to see easily.

Tony Murphy’s It’s All About You (December 15), a comic strip about people not being quite couples, tells a “what happens in Vegas” joke themed to mathematics. The particular topic — a “seminar on gap unification theory” — is something that might actually be a mathematics department specialty. The topic appears in number theory, and particularly in the field of partitions, the study of ways to subdivide collections of discrete things. At this point the subject starts getting specialized enough I can’t say very much intelligible about it; apparently there’s a way of studying these divisions by looking at the distances (the gaps) between where divisions are made (the partitions), but my attempts to find a clear explanation for this all turn up papers in number theory journals that I haven’t got access to and that, I confess, would take me a long while to understand. If anyone from the number theory group wanted to explain things I’d be glad to offer the space.

Reading The Comics, November 14, 2014: Rectangular States Edition

I have no idea why Comic Strip Master Command decided this week should see everybody do some mathematics-themed comic strips, but, so they did, and here’s my collection of the, I estimate, six hundred comic strips that touched on something recently. Good luck reading it all.

Samsons Dark Side of the Horse (November 10) is another entry on the theme of not answering the word problem.

Scott Adams’s Dilbert Classics (November 10) started a sequence in which Dilbert gets told the big boss was a geometry major, so, what can he say about rectangles? Further rumors indicate he’s more a geography fan, shifting Dilbert’s topic to the “many” rectangular states of the United States. Of course, there’s only two literally rectangular states, but — and Mark Stein’s How The States Got Their Shapes contains a lot of good explanations of this — many of the states are approximately rectangular. After all, when many of the state boundaries were laid out, the federal government had only vague if any idea what the landscapes looked like in detail, and there weren’t many existing indigenous boundaries the white governments cared about. So setting a proposed territory’s bounds to be within particular lines of latitude and longitude, with some modification for rivers or shorelines or mountain ranges known to exist, is easy, and can be done with rather little of the ambiguity or contradictory nonsense that plagued the eastern states (where, say, a colony’s boundary might be defined as where a river intersects a line of latitude that in fact it never touches). And while perfect rectangularity may be achieved only by Colorado and Wyoming, quite a few states — the Dakotas, Washington, Oregon, Missisippi, Alabama, Iowa — are rectangular enough.

Mikael Wulff and Anders Morgenthaler’s WuMo (November 10) shows that their interest in pi isn’t just a casual thing. They think about what those neglected and non-famous numbers get up to.

Jim Toomey’s Sherman’s Lagoon for the 11th of November, 2014. He’s got a point about pictures helping with this kind of problem.

Jim Toomey’s Sherman’s Lagoon starts a “struggling with mathematics homework” story on the 11th, with Sherman himself stumped by a problem that “looks more like a short story” than a math problem. By the 14th Megan points out that it’s a problem that really doesn’t make sense when applied to sharks. Such is the natural hazard in writing a perfectly good word problem without considering the audience.

Jim Toomey’s Sherman’s Lagoon for the 14th of November, 2014.

Mike Peters’s Mother Goose and Grimm (November 12) takes one of its (frequent) breaks from the title characters for a panel-strip-style gag about Roman numerals.

Mike Peters’s Mother Goose and Grimm for the 12th of November, 2014.

Darrin Bell’s Candorville (November 12) starts talking about Zeno’s paradox — not the first time this month that a comic strip’s gotten to the apparent problem of covering any distance when distance is infinitely divisible. On November 13th it’s extended to covering stretches of time, which has exactly the same problem. Now it’s worth reminding people, because a stunning number of them don’t seem to understand this, that Zeno was not suggesting that there’s no such thing as motion (or that he couldn’t imagine an infinite convergent sequence; it’s easy to think of a geometric construction that would satisfy any ancient geometer); he was pointing out that there’s things that don’t make perfect sense about it. Either distance (and time) are infinitely divisible into indistinguishable units, or they are not; and either way has implications that seem contrary to the way motion works. Perhaps they can be rationalized; perhaps they can’t; but when you can find a question that’s easy to pose and hard to answer, you’re probably looking at something really worth thinking hard about.

Bill Amend’s FoxTrot Classics (November 12, a rerun) puns on the various meanings of “irrational”. A fun little fact you might want to try proving sometime, though I wouldn’t fault you if you only tried it out for a couple specific numbers and decided the general case too much to do: any whole number — like 2, 3, 4, or so on — has a square root that’s either another whole number, or else has a square root that’s irrational. There’s not a case where, say, the square root is exactly 45.144 or something like that, though it might be close.

Sandra Bell-Lundy’s Between Friends for the 13th of November, 2014.

Sandra Bell-Lundy’sBetween Friends (November 13) shows one of those cases where mental arithmetic really is useful, as Susan tries to work out — actually, staring at it, I’m not precisely sure what she is trying to work out. Her and her coffee partner’s ages in Grade Ten, probably, or else just when Grade Ten was. That’s most likely her real problem: if you don’t know what you’re looking for it’s very difficult to find it. Don’t start calculating before you know what you’re trying to work out.

If I wanted to work out what year was 35 years ago I’d probably just use a hack: 35 years before 2014 is one year before “35 years before 2015”, which is a much easier problem to do. 35 years before 2015 is also 20 years before 2000, which is 1980, so subtract one and you get 1979. (Alternatively, I might remember it was 35 years ago that the Buggles’ “Video Killed The Radio Star” first appeared, which I admit is not a method that would work for everyone, or for all years.) If I wanted to work out my (and my partner’s) age in Grade Ten … well, I’d use a slightly different hack: I remember very well that I was ten years old in Grade Five (seriously, the fact that twice my grade was my age overwhelmed my thinking on my tenth birthday, which is probably why I had to stay in mathematics), so, add five to that and I’d be 15 in Grade Ten.

Bill Whitehead’s Free Range (November 13) brings up one of the most-quoted equations in the world in order to show off how kids will insult each other, which is fair enough.

Rick Detorie’s One Big Happy (November 13), this one a rerun from a couple years ago because that’s how his strip works on Gocomics, goes to one of its regular bits of the kid Ruthie teaching anyone she can get in range, and while there’s a bit more to arithmetic than just adding two numbers to get a bigger number, she is showing off an understanding of a useful sanity check: if you add together two (positive) numbers, you have to get a result that’s bigger than either of the ones you started with. As for the 14th, and counting higher, well, there’s not much she could do about that.

Steve McGarry’s Badlands (November 14) talks about the kind of problem people wish to have: how to win a lottery where nobody else picks the same numbers, so that the prize goes undivided? The answer, of course, is to have a set of numbers that nobody else picked, but is there any way to guarantee that? And this gets into the curious psychology of random numbers: there is absolutely no reason that 1-2-3-4-5-6, or for that matter 7-8-9-10-11-12, would not come up just as often as, say, 11-37-39-51-52-55, but the latter set looks more random. But we see some strings of numbers as obviously a pattern, while others we don’t see, and we tend to confuse “we don’t know the pattern” with “there is no pattern”. I have heard the lore that actually a disproportionate number of people pick such obvious patterns like 1-2-3-4-5-6, or numbers that form neat pictures on a lottery card, no doubt cackling at how much more clever they are than the average person, and guaranteeing that if such a string ever does come out there’ll a large number of very surprised lottery winners. All silliness, really; the thing to do, obviously, is buy two tickets with the exact same set of numbers, so that if you do win, you get twice the share of anyone else, unless they’ve figured out the same trick.

Reading the Comics, July 28, 2014: Homework in an Amusement Park Edition

I don’t think my standards for mathematics content in comic strips are seriously lowering, but the strips do seem to be coming pretty often for the summer break. I admit I’m including one of these strips just because it lets me talk about something I saw at an amusement park, though. I have my weaknesses.

Harley Schwadron’s 9 to 5 (July 25) builds its joke around the ambiguity of saying a salary is six (or some other number) of figures, if you don’t specify what side of the decimal they’re on. That’s an ordinary enough gag, although the size of a number can itself be an interesting thing to know. The number of digits it takes to write a number down corresponds, roughly, with the logarithm of a number, and in the olden days a lot of computations depended on logarithms: multiplying two numbers is equivalent to adding their logarithms; dividing two numbers, subtracting their logarithms. And addition and subtraction are normally easier than multiplication and division. Similarly, raising one number to a power becomes multiplying one number by the logarithm of another, and multiplication is easier than exponentiation. So counting the number of digits in a number might be something anyway.

Steve Breen and Mike Thompson’s Grand Avenue (July 25) has the kids mention something as being “like going to an amusement park to do math homework”, which gives me a chance to share this incident. Last year my love and I were in the Cedar Point amusement park (in Sandusky, Ohio), and went to the coffee shop. We saw one guy sitting at a counter, with his laptop and a bunch of papers sprawled out, looking pretty much like we do when we’re grading papers, and we thought initially that it was so very sad that someone would be so busy at work that (we presumed) he couldn’t even really participate in the family expedition to the amusement park.

And then we remembered: not everybody lives a couple hours away from an amusement park. If we lived, say, fifteen minutes from a park we had season passes to, we’d certainly at least sometimes take our grading work to the park, so we could get it done in an environment we liked and reward ourselves for getting done with a couple roller coasters and maybe the Cedar Downs carousel (which is worth an entry around these parts anyway). To grade, anyway; I’d never have the courage to bring my laptop to the coffee shop. So I guess all I’m saying is, I have a context in which yes, I could imagine going to an amusement park to grade math homework at least.

Wulff and Morgenthaler Truth Facts (July 25) makes a Venn diagram joke in service of asserting that only people who don’t understand statistics would play the lottery. This is an understandable attitude of Wulff and Morgenthaler, and of many, many people who make the same claim. The expectation value — the amount you expect to win some amount, times the probability you will win that amount, minus the cost of the ticket — is negative for all but the most extremely oversized lottery payouts, and the most extremely oversized lottery payouts still give you odds of winning so tiny that you really aren’t hurting your chances by not buying a ticket. However, the smugness behind the attitude bothers me — I’m generally bothered by smugness — and jokes like this one contain the assumption that the only sensible way to live is a ruthless profit-and-loss calculation to life that even Jeremy Bentham might say is a bit much. For the typical person, buying a lottery ticket is a bit of a lark, a couple dollars of disposable income spent because, what the heck, it’s about what you’d spend on one and a third sodas and you aren’t that thirsty. Lottery pools with coworkers or friends make it a small but fun social activity, too. That something is a net loss of money does not mean it is necessarily foolish. (This isn’t to say it’s wise, either, but I’d generally like a little more sympathy for people’s minor bits of recreational foolishness.)

Marc Anderson’s Andertoons (July 27) does a spot of wordplay about the meaning of “aftermath”. I can’t think of much to say about this, so let me just mention that Florian Cajori’s A History of Mathematical Notations reports (section 201) that the + symbol for addition appears to trace from writing “et”, meaning and, a good deal and the letters merging together and simplifying from that. This seems plausible enough on its face, but it does cause me to reflect that the & symbol also is credited as a symbol born from writing “et” a lot. (Here, picture writing Et and letting the middle and lower horizontal strokes of the E merge with the cross bar and the lowest point of the t.)

Berkeley Breathed’s Bloom County (July 27, rerun from, I believe, July of 1988) is one of the earliest appearances I can remember of the Grand Unification appearing in popular culture, certainly in comic strips. Unifications have a long and grand history in mathematics and physics in explaining things which look very different by the same principles, with the first to really draw attention probably being Descartes showing that algebra and geometry could be understood as a single thing, and problems difficult in one field could be easy in the other. In physics, the most thrilling unification was probably the explaining of electricity, magnetism, and light as the same thing in the 19th century; being able to explain many varied phenomena with some simple principles is just so compelling. General relativity shows that we can interpret accelerations and gravitation as the same thing; and in the late 20th century, physicists found that it’s possible to use a single framework to explain both electromagnetism and the forces that hold subatomic particles together and that break them apart.

It’s not yet known how to explain gravity and quantum mechanics in the same, coherent, frame. It’s generally assumed they can be reconciled, although I suppose there’s no logical reason they have to be. Finding a unification — or a proof they can’t be unified — would certainly be one of the great moments of mathematical physics.

The idea of the grand unification theory as an explanation for everything is … well, fair enough. A grand unification theory should be able to explain what particles in the universe exist, and what forces they use to interact, and from there it would seem like the rest of reality is details. Perhaps so, but it’s a long way to go from a simple starting point to explaining something as complicated as a penguin. I guess what I’m saying is I doubt Oliver would notice the non-existence of Opus in the first couple pages of his work.

Thom Bluemel’s Birdbrains (July 28) takes us back to the origin of numbers. It also makes me realize I don’t know what’s the first number that we know of people discovering. What I mean is, it seems likely that humans are just able to recognize a handful of numbers, like one and two and maybe up to six or so, based on how babies and animals can recognize something funny if the counts of small numbers of things don’t make sense. And larger numbers were certainly known to antiquity; probably the fact that numbers keep going on forever was known to antiquity. And some special numbers with interesting or difficult properties, like pi or the square root of two, were known so long ago we can’t say who discovered them. But then there are numbers like the Euler-Mascheroni constant, which are known and recognized as important things, and we can say reasonably well who discovered them. So what is the first number with a known discoverer?

Reading the Comics, September 26, 2012

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

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

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

Why Someone Should Not Take That Deal

My commenters, thank them, quite nicely outlined the major reasons that someone in the Deal or No Deal problem I posited would be wiser to take the Banker’s offer of a sure $11,750 rather than to keep a randomly selected one of $1, $10, $7,500, $25,000, or $35,000. Even though the expectation value, the average that the Contestant could expect from sticking with her suitcase if she played the game an enormous number of times is $13,502.20, fairly noticeably larger than the Banker’s offer, she is just playing the game the once. She’s more likely to do worse than the Banker’s offer, and is as likely to do much worse — $1 or $10 — rather than do any better.

If we suppose the contestant’s objective is to get as much money as possible from playing, her strategy is different if she plays just the once versus if she plays unlimitedly many times. I don’t know a name for this class of problems; maybe we can dub it the “lottery paradox”. It’s not rare for a lottery jackpot to rise high enough that the expected value of one’s winnings are more than the ticket price, which is typically when I’ll bother to buy one (well, two), but I know it’s effectively certain that all I’ll get from the purchase is one (well, two) dollars poorer.

It also strikes me that I have the article subjects for this and the previous entry reversed. Too bad.

Thinking about the lottery

I feel as though I ought to write something thoughtful about the lottery, since the Mega Millions draw has got to be an impressively high value, and there are fine things to be said about probability, expectation values, and what we really mean by the probability of an event when we can only take one attempt at doing it (since I can’t make myself think the lottery will go another week without a winner and it is likely to be years before such a big jackpot rises again).