Reading the Comics, April 4, 2020: Ruling Things Out Edition

This little essay should let me wrap up the rest of the comic strips from the past week. Most of them were casual mentions. At least I thought they were when I gathered them. But let’s see what happens when I actually write my paragraphs about them.

Darrin Bell and Theron Heir’s Rudy Park rerun for the 1st of April uses arithmetic as emblematic of things which we know with certainty to be true.

Thaves’s Frank and Ernest for the 2nd is a bit of wordplay, having Euclid and Galileo talking about parallel universes. I’m not sure that Galileo is the best fit for this, but I’m also not sure there’s another person connected who could be named. It’d have to be a name familiar to an average reader as having something to do with geometry. Pythagoras would seem obvious, but the joke is stronger if it’s two people who definitely did not live at the same time. Did Euclid and Pythagoras live at the same time? I am a mathematics Ph.D. and have been doing pop mathematics blogging for nearly a decade now, and I have not once considered the question until right now. Let me look it up.

It doesn’t make any difference. The comic strip has to read quickly. It might be better grounded to post Euclid meeting Gauss or Lobachevsky or Euler (although the similarity in names would be confusing) but being understood is better than being precise.

Stephan Pastis’s Pearls Before Swine for the 2nd is a strip about the foolhardiness of playing the lottery. And it is foolish to think that even a $100 purchase of lottery tickets will get one a win. But it is possible to buy enough lottery tickets as to assure a win, even if it is maybe shared with someone else. It’s neat that an action can be foolish if done in a small quantity, but sensible if done in enough bulk.

Mark Anderson’s Andertoons for the 3rd of April, 2020. This is actually the first time I’ve mentioned this strip in two months. But any time I discuss a topic raised by Andertoons should appear at this link.

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. Wavehead has made a bunch of failed attempts at subtracting seven from ten, but claims it’s at least progress that some thing have been ruled out. I’ll go along with him that there is some good in ruling out wrong answers. The tricky part is in how you rule them out. For example, obvious to my eye is that the correct answer can’t be more than ten; the problem is 10 minus a positive number. And it can’t be less than zero; it’s ten minus a number less than ten. It’s got to be a whole number. If I’m feeling confident about five and five making ten, then I’d rule out any answer that isn’t between 1 and 4 right away. I’ve got the answer down to four guesses and all I’ve really needed to know is that 7 is greater than five but less than ten. That it’s an even number minus an odd means the result has to be odd; so, it’s either one or three. Knowing that the next whole number higher than 7 is an 8 says that we can rule out 1 as the answer. So there’s the answer, done wholly by thinking of what we can rule out. Of course, knowing what to rule out takes some experience.

Mark Parisi’s Off The Mark for the 4th is roughly the anthropomorphic numerals joke for the week. It’s a dumb one, but, that’s what sketchbooks are for.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is the Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th for the week. It shows in joking but not wrong fashion a mathematical physicist’s encounters with orbital mechanics. Orbital mechanics are a great first physics problem. It’s obvious what they’re about, and why they might be interesting. And the mathematics of it is challenging in ways that masses on springs or balls shot from cannons aren’t.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th of April, 2020. This is actually the first time I’ve mentioned this strip ina week. But any time I discuss a topic raised in Saturday Morning Breakfast Cereal should appear at this link.

A few problems are very easy, like, one thing in circular orbit of another. A few problems are not bad, like, one thing in an elliptical or hyperbolic orbit of another. All our good luck runs out once we suppose the universe has three things in it. You’re left with problems that are doable if you suppose that one of the things moving is so tiny that it barely exists. This is near enough true for, for example, a satellite orbiting a planet. Or by supposing that we have a series of two-thing problems. Which is again near enough true for, for example, a satellite travelling from one planet to another. But these is all work that finds approximate solutions, often after considerable hard work. It feels like much more labor to smaller reward than we get for masses on springs or balls shot from cannons. Walking off to a presumably easier field is understandable. Unfortunately, none of the other fields is actually easier.

Pythagoras died somewhere around 495 BC. Euclid was born sometime around 325 BC. That’s 170 years apart. So Pythagoras was as far in Euclid’s past as, oh, Maria Gaetana Agnesi is to mine.

I did a little series looking into orbital mechanics, not necessarily ones that look like planetary orbits, a couple years ago. You might enjoy that. And I figure to have more mathematically-themed comic strips in the near future. Thanks for reading.

Reading the Comics, March 17, 2020: Random Edition

I thought last week’s comic strips mentioning mathematics in detail were still subjects easy to describe in one or two paragraphs each. I wasn’t quite right. So here’s a half of a week, even if it is a day later than I had wanted to post.

John Zakour and Scott Roberts’s Working Daze for the 15th is a straggler Pi Day joke, built on the nerd couple Roy and Kathy letting the date slip their minds. This is a very slight Pi Day reference but I feel the need to include it for completeness’s sake. It reminds me of the sequence where one year Schroeder forgot Beethoven’s birthday, and was devastated.

John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2020. Essays featuring Working Daze, which often turns up in Pi Day events, are at this link. And generally essays tied to Pi Day are at this link.

Lincoln Peirce’s Big Nate for the 15th is a wordy bit of Nate refusing the story problem. Nate complains about a lack of motivation for the characters in it. But then what we need for a story problem isn’t the characters to do something so much as it is the student to want to solve the problem. That’s hard work. Everyone’s fascinated by some mathematical problems, but it’s hard to think of something that will compel everyone to wonder what the answer could be.

At one point Nate wonders what happens if Todd stops for gas. Here he’s just ignoring the premise of the question: Todd is given as travelling an average 55 mph until he reaches Saint Louis, and that’s that. So this question at least is answered. But he might need advice to see how it’s implied.

Lincoln Peirce’s Big Nate for the 15th of March, 2020. Essays with something mentioned by either Big Nate or the 1990s-repeats Big Nate: First Class are gathered at this link.

So this problem is doable by long division: 1825 divided by 80, and 1192 divided by 55, and see what’s larger. Can we avoid dividing by 55 if we’re doing it by hand? I think so. Here’s what I see: 1825 divided by 80 is equal to 1600 divided by 80 plus 225 divided by 80. That first is 20; that second is … eh. It’s a little less than 240 divided by 80, which is 3. So Mandy will need a little under 23 hours.

Is 23 hours enough for Todd to get to Saint Louis? Well, 23 times 55 will be 23 times 50 plus 23 times 5. 23 times 50 is 22 times 50 plus 1 times 50. 22 times 50 is 11 times 100, or 1100. So 23 times 50 is 1150. And 23 times 5 has to be 150. That’s more than 1192. So Todd gets there first. I might want to figure just how much less than 23 hours Mandy needs, to be sure of my calculation, but this is how I do it without putting 55 into an ugly number like 1192.

Mark Leiknes’s Cow and Boy repeat for the 17th of March, 2020. The too-rare appearances of Cow and Boy Reruns in my essays are here.

Mark Leiknes’s Cow and Boy repeat for the 17th sees the Boy, Billy, trying to beat the lottery. He throws at it the terms chaos theory and nonlinear dynamical systems. They’re good and probably relevant systems. A “dynamical system” is what you’d guess from the name: a collection of things whose properties keep changing. They change because of other things in the collection. When “nonlinear” crops up in mathematics it means “oh but such a pain to deal with”. It has a more precise definition, but this is its meaning. More precisely: in a linear system, a change in the initial setup makes a proportional change in the outcome. If Todd drove to Saint Louis on a path two percent longer, he’d need two percent more time to get there. A nonlinear system doesn’t guarantee that; a two percent longer drive might take ten percent longer, or one-quarter the time, or some other weirdness. Nonlinear systems are really good for giving numbers that look random. There’ll be so many little factors that make non-negligible results that they can’t be predicted in any useful time. This is good for drawing number balls for a lottery.

Chaos theory turns up a lot in dynamical systems. Dynamical systems, even nonlinear ones, often have regions that behave in predictable patterns. We may not be able to say what tomorrow’s weather will be exactly, but we can say whether it’ll be hot or freezing. But dynamical systems can have regions where no prediction is possible. Not because they don’t follow predictable rules. But because any perturbation, however small, produces changes that overwhelm the forecast. This includes the difference between any possible real-world measurement and the real quantity.

Obvious question: how is there anything to study in chaos theory, then? Is it all just people looking at complicated systems and saying, yup, we’re done here? Usually the questions turn on problems such as how probable it is we’re in a chaotic region. Or what factors influence whether the system is chaotic, and how much of it is chaotic. Even if we can’t say what will happen, we can usually say something about when we can’t say what will happen, and why. Anyway if Billy does believe the lottery is chaotic, there’s not a lot he can be doing with predicting winning numbers from it. Cow’s skepticism is fair.

Ryan North’s Dinosaur Comics for the 17th of March, 2020. Essays that mention something brought up in Dinosaur Comics are gathered at this link.

Ryan North’s Dinosaur Comics for the 17th is one about people asked to summon random numbers. Utahraptor is absolutely right. People are terrible at calling out random numbers. We’re more likely to summon odd numbers than we should be. We shy away from generating strings of numbers. We’d feel weird offering, say, 1234, though that’s as good a four-digit number as 1753. And to offer 2222 would feel really weird. Part of this is that there’s not really such a thing as “a” random number; it’s sequences of numbers that are random. We just pick a number from a random sequence. And we’re terrible at producing random sequences. Here’s one study, challenging people to produce digits from 1 through 9. Are their sequences predictable? If the numbers were uniformly distributed from 1 through 9, then any prediction of the next digit in a sequence should have a one chance in nine of being right. It turns out human-generated sequences form patterns that could be forecast, on average, 27% of the time. Individual cases could get forecast 45% of the time.

There are some neat side results from that study too, particularly that they were able to pretty reliably tell the difference between two individuals by their “random” sequences. We may be bad at thinking up random numbers but the details of how we’re bad can be unique.

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And I’m not done yet. There’s some more comic strips from last week to discuss and I’ll have that post here soon. Thanks for reading.

Reading the Comics, January 27, 2020: Alley Oop Followup Edition

I apologize for missing Sunday. I wasn’t able to make the time to write about last week’s mathematically-themed comic strips. But I’m back in the swing of things. Here are some of the comic strips that got my attention.

Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 26th has something neat in the background. Oop and Garg walk past a vendor showing off New Numbers. This is, among other things, a cute callback to one of the first of Lemon and Sayers’s Little Oop strips.. (And has nothing to do with the daily storyline featuring the adult Alley Oop.) And it is a funny idea to think of “new numbers”. I imagine most of us trust that numbers are just … existing, somewhere, as concepts independent of our knowing them. We may not be too sure about the Platonic Forms. But, like, “eight” seems like something that could plausibly exist independently of our understanding of it.

Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 26th of January, 2020. The handful of times I’ve head to talk about Alley Oop or Little Oop are gathered at this link.

Still, we do keep discovering things we didn’t know were numbers before. The earliest number notations, in the western tradition, for example, used letters to represent numbers. This did well for counting numbers, up to a large enough total. But it required idiosyncratic treatment if you wanted to handle large numbers. Hindu-Arabic numerals make it easy to represent whole numbers as large as you like. But that’s at the cost of adding ten (well, I guess eight) symbols that have nothing to do with the concept represented. Not that, like, ‘J’ looks like the letter J either. (There is a folk etymology that the Arabic numerals correspond to the number of angles made if you write them out in a particular way. Or less implausibly, the number of strokes needed for the symbol. This is ingenious and maybe possibly has helped one person somewhere, ever, learn the symbols. But it requires writing, like, ‘7’ in a way nobody has ever done, and it’s ahistorical nonsense. See section 96, on page 64 of the book and 84 of the web presentation, in Florian Cajori’s History of Mathematical Notations.)

Still, in time we discovered, for example, that there were irrational numbers and those were useful to have. Negative numbers, and those are useful to have. That there are complex-valued numbers, and those are useful to have. That there are quaternions, and … I guess we can use them. And that we can set up systems that resemble arithmetic, and work a bit like numbers. Those are often quite useful. I expect Lemon and Sayers were having fun with the idea of new numbers. They are a thing that, effectively, happens.

Lincoln Peirce’s Big Nate: First Class for the 26th of January, 2020. It originally ran the 15th of January, 1995. Essays mentioning either Big Nate or the rerun Big Nate: First Class should be gathered at this link.

Lincoln Peirce’s Big Nate: First Class for the 26th has Nate badgering Francis for mathematics homework answers. Could be any subject, but arithmetic will let Peirce fit in a couple answers in one panel.

Art Sansom and Chip Sansom’s The Born Loser for the 26th of January, 2020. There are times that I discuss The Born Loser, and those essays are at this link.

Art Sansom and Chip Sansom’s The Born Loser for the 26th is another strip on the theme of people winning the lottery and being hit by lightning. And, as I’ve mentioned, there is at least one person known to have won a lottery and survived a lightning strike.

David Malki’s Wondermark for the 27th of January, 2020. I am surprised to learn that I already have a tag for this comic, but it turns out I’ve mentioned it as long ago as late December. So, essays mentioning Wondermark: they’re at this link.

David Malki’s Wondermark for the 27th describes engineering as “like math, but louder”, which is a pretty good line. And it uses backgrounds of long calculations to make the point of deep thought going on. I don’t recognize just what calculations are being done there, but they do look naggingly familiar. And, you know, that’s still a pretty lucky day.

Mark Anderson’s Andertoons for the 27th of January, 2020. And I have a lot of essays mentioning something from Andertoons gathered at this link.

Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. It depicts Wavehead having trouble figuring where to put the decimal point in the multiplication of two decimal numbers. Relatable issue. There are rules you can follow for where to put the decimal in this sort of operation. But the convention of dropping terminal zeroes after the decimal point can make that hazardous. It’s something that needs practice, or better: though. In this case, what catches my eye is that 2.95 times 3.2 has to be some number close to 3 times 3. So 9.440 is the plausible answer.

Mike Twohy’s That’s Life for the 27th of January, 2020. So I have some essays mentioning this comic strip, but from before I started tagging them. I’ll try to add tags to those old essays when I have the chance. In the meanwhile, this essay and maybe future ones mentioning That’s Life should be at this link.

Mike Twohy’s That’s Life for the 27th presents a couple of plausible enough word problems, framed as Sports Math. It’s funny because of the idea that the workers who create events worth billions of dollars a year should be paid correspondingly.

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This isn’t all for the week from me. I hope to have another Reading the Comics installment at this link, soon. Thanks for reading.

Reading the Comics, October 22, 2019: Bifurcated Week Edition

The past week started strong for mathematically-themed comics. Then it faded out into strips that just mentioned the existence of mathematics. I have no explanation for this phenomenon. It makes dividing up the week’s discussion material easy enough, though.

John Zakour and Scott Roberts’s Maria’s Day rerun for the 19th is a lottery joke. Maria’s come up with a scheme to certainly win the grand prize in a lottery. There’s no disputing that one could, on buying enough tickets, get an appreciable chance of winning. Even, in principle, get a certain win. There’s no guaranteeing a solo win, though. But sometimes lottery jackpots will grow large enough that even if you had to split the prize two or three ways it’d be worth it.

John Zakour and Scott Roberts’s Maria’s Day rerun for the 19th of October, 2019. It originally ran the 28th of July, 2012. The strip has gone into Sunday-only mode, I believe, but I’m still writing about Maria’s Day in essays gathered at this link.

Tom Horacek’s Foolish Mortals for the 21st plays on the common wisdom that mathematicians’ best work is done when they’re in their 20s. Or at least their most significant work. I don’t like to think that’s so, as someone who went through his 20s finding nothing significant. But my suspicion is that really significant work is done when someone with fresh eyes looks at a new problem. Young mathematicians are in a good place to learn, and are looking at most everything with fresh eyes, and every problem is new. Still, experienced mathematicians, bringing the habits of thought that served well one kind of problem, looking at something new will recreate this effect. We just need to find ideas to think about that we haven’t worn down.

Tom Horacek’s Foolish Mortals for the 21st of October, 2019. I wasn’t sure I ever wrote about this strip, but no, I have, and appearances by Foolish Mortals in these pages are here.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st has a petitioner asking god about whether P = NP. This is shorthand for a famous problem in the study of algorithms. It’s about finding solutions to problems, and how much time it takes to find the solution. This time usually depends on the size of whatever it is you’re studying. The question, interesting to mathematicians and computer scientists, is how fast this time grows. There are many classes of these problems. P stands for problems solvable in polynomial time. Here the number of steps it takes grows at, like, the square or the cube or the tenth power of the size of the thing. NP is non-polynomial problems, growing, like, with the exponential of the size of the thing. (Do not try to pass your computer science thesis defense with this description. I’m leaving out important points here.) We know a bunch of P problems, as well as NP problems.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st of October, 2019. This strip I sometimes think I write about every essay. But Saturday Morning Breakfast Cereal essays are at this link.

Like, in this comic, God talks about the problem of planning a long delivery route. Finding the shortest path that gets to a bunch of points is an NP problem. What we don’t know about NP problems is whether the problem is we haven’t found a good solution yet. Maybe next year some bright young 68-year-old mathematician will toss of a joke on a Reddit subthread and then realize, oh, this actually works. Which would be really worth knowing. One thing we know about NP problems is there’s a big class of them that are all, secretly, versions of each other. If we had a good solution for one we’d have a solution for all of them. So that’s why a mathematician or computer scientist would like to hear God’s judgement on how the world is made.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 22nd of October, 2019. This and other essays featuring Baldo are at this link.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 22nd has Baldo asking his sister to do some arithmetic. I fancy he’s teasing her. I like doing some mental arithmetic. If nothing else it’s worth having an expectation of the answer to judge whether you’ve asked the computer to do the calculation you actually wanted.

Mike Thompson’s Grand Avenue for the 22nd of October, 2019. And essays where I discuss Grand Avenue appear at this link.

Mike Thompson’s Grand Avenue for the 22nd has Gabby demanding to know the point of learning Roman numerals. As numerals, not much that I can see; they serve just historical and decorative purposes these days, mostly as a way to make an index look more fancy. As a way to learn that how we represent numbers is arbitrary, though? And that we can use different schemes if that’s more convenient? That’s worth learning, although it doesn’t have to be Roman numerals. They do have the advantage of using familiar symbols, though, which (say) the Babylonian sexagesimal system would not.

And that’s the comic strips with enough mathematics for me to discuss from the first half of last week. I plan tomorrow to at least mention the strips with just mentions of mathematics. And then Tuesday, The A-to-Z reaches the letter Q. I’m interested to see how that turns out too.

Reading the Comics, February 13, 2019: Light Geometry Edition

Comic Strip Master Command decided this would be a light week, with about six comic strips worth discussing. I’ll go into four of them here, and in a day or two wrap up the remainder. There were several strips that didn’t quite rate discussion, and I’ll share those too. I never can be sure what strips will be best taped to someone’s office door.

Alex Hallatt’s Arctic Circle for the 10th was inspired by a tabular iceberg that got some attention in October 2018. It looked surprisingly rectangular. Smoother than we expect natural things to be. My first thought about this strip was to write about crystals. The ways that molecules can fit together may be reflected in how the whole structure looks. And this gets us to studying symmetries.

Alex Hallatt’s Arctic Circle for the 10th of February, 2019. Essays in which I discuss Arctic Circle should be at this link.

But I got to another thought. We’re surprised to see lines in nature. We know what lines are, and understand properties of them pretty well. Even if we don’t specialize in geometry we can understand how we expect them to work. I don’t know how much of this is a cultural artifact: in the western mathematics tradition lines and polygons and circles are taught a lot, and from an early age. My impression is that enough different cultures have similar enough geometries, though. (Are there any societies that don’t seem aware of the Pythagorean Theorem?) So what is it that has got so many people making perfect lines and circles and triangles and squares out of crooked timbers?

Russell Myers’s Broom Hilda for the 13th of February, 2019. Essays inspired by Broom Hilda should be gathered at this link.

Russell Myers’s Broom Hilda for the 13th is a lottery joke. Also, really, an accounting joke. Most of the players of a lottery will not win, of course. Nearly none of them will win more than they’ve paid into the lottery. If they didn’t, there would be an official inquiry. So, yes, nearly all people, even those who win money at the lottery, would have had more money if they skipped playing altogether.

Where it becomes an accounting question is how much did Broom Hilda expect to have when the week was through? If she planned to spend $20 on lottery tickets, and got exactly that? It seems snobbish to me to say that’s a dumber way to spend twenty bucks than, say, buying twenty bucks worth of magazines that you’ll throw away in a month would be. Or having dinner at a fast-casual place. Or anything else that you like doing even though it won’t leave you, in the long run, any better off. Has she come out ahead? That depends where she figures she should be.

Eric the Circle for the 13th of February, 2019, this one by Alabama_Al. Appearances by Eric the Circle, whoever the writer, should be at this link.

Eric the Circle for the 13th, this one by Alabama_Al, is a plane- and solid-geometry joke. This gets it a bit more solidly on-topic than usual. But it’s still a strip focused on the connotations of mathematically-connected terms. There’s the metaphorical use of the ‘plane’ as in the thing people perceive as reality. There’s conflation between the idea of a ‘higher plane’ and ‘higher dimensions’. Also somewhere in here is the idea that ‘higher’ and ‘more’ dimensions of space are the same thing. ‘Transcendental’ here is used in the common English sense of surpassing something. ‘Transcendental’ has a mathematical definition too. That one relates to polynomials, because everything in mathematics is about polynomials. And, of course, one of the two numbers we know to be transcendental, and that people have any reason to care about, is π, which turns up all over circles.

Larry Wright’s Motley for the 13th of February, 2019. It originally ran in 1988, I believe on the same date. When I have written about Motley the results should appear at this link. In transcribing the strip for the alt-text here I was getting all ready to grumble that I didn’t know the kid’s name, and the strip is so old and minor that nobody has a cast list on it. Then I noticed, oh, yes, Mom says what the kid’s name is.

Larry Wright’s Motley for the 13th riffs on the form of a story problem. Joey’s mother does ask something that seems like a plausible addition problem. I’m a bit surprised he hadn’t counted all the day’s cookies already, but perhaps he doesn’t dwell on past snacks.

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This and all my Reading the Comics posts should appear at this link. Thanks for looking at my comments.

Reading the Comics, April 6, 2017: Abbreviated Week Edition

I’m writing this a little bit early because I’m not able to include the Saturday strips in the roundup. There won’t be enough to make a split week edition; I’ll just add the Saturday strips to next week’s report. In the meanwhile:

Mac King and Bill King’s Magic in a Minute for the 2nd is a magic trick, as the name suggests. It figures out a card by way of shuffling a (partial) deck and getting three (honest) answers from the other participant. If I’m not counting wrongly, you could do this trick with up to 27 cards and still get the right card after three answers. I feel like there should be a way to explain this that’s grounded in information theory, but I’m not able to put that together. I leave the suggestion here for people who see the obvious before I get to it.

Bil Keane and Jeff Keane’s Family Circus (probable) rerun for the 6th reassured me that this was not going to be a single-strip week. And a dubiously included single strip at that. I’m not sure that lotteries are the best use of the knowledge of numbers, but they’re a practical use anyway.

Bil Keane and Jeff Keane’s Family Circus for the 6th of April, 2017. I’m not familiar enough with the evolution of the Family Circus style to say whether this is a rerun, a newly-drawn strip, or an old strip with a new caption. I suppose there is a certain timelessness to it, at least once we get into the era when states sported lotteries again.

Bill Bettwy’s Take It From The Tinkersons for the 6th is part of the universe of students resisting class. I can understand the motivation problem in caring about numbers of apples that satisfy some condition. In the role of distinct objects whose number can be counted or deduced cards are as good as apples. In the role of things to gamble on, cards open up a lot of probability questions. Counting cards is even about how the probability of future events changes as information about the system changes. There’s a lot worth learning there. I wouldn’t try teaching it to elementary school students.

Bill Bettwy’s Take It From The Tinkersons for the 6th of April, 2017. That tree in the third panel is a transplant from a Slylock Fox six-differences panel. They’ve been trying to rebuild the population of trees that are sometimes three triangles and sometimes four triangles tall.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 6th uses mathematics as the stuff know-it-alls know. At least I suppose it is; Doctor Know It All speaks of “the pathagorean principle”. I’m assuming that’s meant to be the Pythagorean theorem, although the talk about “in any right triangle the area … ” skews things. You can get to stuf about areas of triangles from the Pythagorean theorem. One of the shorter proofs of it depends on the areas of the squares of the three sides of a right triangle. But it’s not what people typically think of right away. But he wouldn’t be the first know-it-all to start blathering on the assumption that people aren’t really listening. It’s common enough to suppose someone who speaks confidently and at length must know something.

Dave Whamond’s Reality Check for the 6th is a welcome return to anthropomorphic-numerals humor. Been a while.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th builds on the form of a classic puzzle, about a sequence indexed to the squares of a chessboard. The story being riffed on is a bit of mathematical legend. The King offered the inventor of chess any reward. The inventor asked for one grain of wheat for the first square, two grains for the second square, four grains for the third square, eight grains for the fourth square, and so on, through all 64 squares. An extravagant reward, but surely one within the king’s power to grant, right? And of course not: by the 64th doubling the amount of wheat involved is so enormous it’s impossibly great wealth.

The father’s offer is meant to evoke that. But he phrases it in a deceptive way, “one penny for the first square, two for the second, and so on”. That “and so on” is the key. Listing a sequence and ending “and so on” is incomplete. The sequence can go in absolutely any direction after the given examples and not be inconsistent. There is no way to pick a single extrapolation as the only logical choice.

We do it anyway, though. Even mathematicians say “and so on”. This is because we usually stick to a couple popular extrapolations. We suppose things follow a couple common patterns. They’re polynomials. Or they’re exponentials. Or they’re sine waves. If they’re polynomials, they’re lower-order polynomials. Things like that. Most of the time we’re not trying to trick our fellow mathematicians. Or we know we’re modeling things with some physical base and we have reason to expect some particular type of function.

In this case, the $1.27 total is consistent with getting two cents for every chess square after the first. There are infinitely many other patterns that would work, and the kid would have been wise to ask for what precisely “and so on” meant before choosing.

Berkeley Breathed’s Bloom County 2017 for the 7th is the climax of a little story in which Oliver Wendell Holmes has been annoying people by shoving scientific explanations of things into their otherwise pleasant days. It’s a habit some scientifically-minded folks have, and it’s an annoying one. Many of us outgrow it. Anyway, this strip is about the curious evidence suggesting that the universe is not just expanding, but accelerating its expansion. There are mathematical models which allow this to happen. When developing General Relativity, Albert Einstein included a Cosmological Constant for little reason besides that without it, his model would suggest the universe was of a finite age and had expanded from an infinitesimally small origin. He had grown up without anyone knowing of any evidence that the size of the universe was a thing that could change.

Anyway, the Cosmological Constant is a puzzle. We can find values that seem to match what we observe, but we don’t know of a good reason it should be there. We sciencey types like to have models that match data, but we appreciate more knowing why the models look like that and not anything else. So it’s a good problem some of the cosmologists have been working on. But we’ve been here before. A great deal of physics, especially in the 20th Century, has been driven by looking for reasons behind what look like arbitrary points in a successful model. If Oliver were better-versed in the history of science — something scientifically minded people are often weak on, myself included — he’d be less easily taunted by Opus.

Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 7th thinks that we forgot they ran this same strip back on the 17th of March. I spotted it, though. Nyah.

Reading the Comics, February 6, 2016: Lottery Edition

As mentioned, the lottery was a big thing a couple of weeks ago. So there were a couple of lottery-themed comics recently. Let me group them together. Comic strips tend to be anti-lottery. It’s as though people trying to make a living drawing comics for newspapers are skeptical of wild long-shot dreams.

T Lewis and Michael Fry’s Over The Hedge started a lottery storyline the 1st of February. Verne, the turtle, repeats the tired joke that the lottery is a tax on people bad at mathematics. Enormous jackpots, like the $1,500,000,000 payout of a couple weeks back, break one leg of the anti-lottery argument. If the expected payout is large enough then the expectation value of playing can become positive. The expectation value is one of those statistics terms that almost tells you what it is just by the name. It’s what you would expect as the average result if you could repeat some experiment arbitrarily many times. If the payout is 1.5 billion, and the chance of winning one in 250 million, then the expected value of the payout is six dollars. If a ticket costs less than six dollars, then — if you could play over and over, hundreds of millions of times — you’d expect to come out ahead each time you play.

If you could. Of course, you can’t play the lottery hundreds of millions of times. You can play a couple of times at most. (Even if you join a pool at work and buy, oh, a thousand tickets. That’s still barely better than playing twice.) And the payout may be less than the full jackpot; multiple winners are common things in the most enormous jackpots. Still, if you’re pondering whether it’s sensible to spend two dollars on a billion-dollar lottery jackpot? You’re being fussy. You’ll spend at least that much on something more foolish and transitory — the lottery ticket can at least be used as a bookmark — I’ll bet.

Jef Mallett’s Frazz for the 4th of February picks up the anti-lottery crusade. Caulfield does pin down that lotteries work because people figure they have a better chance of winning than they truly do. Nobody buys a ticket because they figure it’s worth losing a dollar or two. It’s because they figure the chance is worth a little money.

Ken Cursoe’s Tiny Sepuku for the 4th of February consults the Chinese Zodiac Monkey for help on finding lucky numbers. There’s not really any finding them. Lotteries work hard to keep the winning numbers as unpredictable as possible. I have heard the lore that numbers up to 31 are picked by more people — they’re numbers that can be birthdays — so that multiple winners on the same drawing are more likely. I don’t know that this is true, though. I suspect that I could feel comfortable even with a four-way split of one and a half billions of dollars. Five-way would be out of the question, of course. Better to tear up the ticket than take that undignified split.

In Rick Detorie’s One Big Happy for the 3rd of February, 2016. The link will probably expire in early March.

In Rick Detorie’s One Big Happy for the 3rd of February features Ruthie tossing off a confusing pile of numbers on the way to declaring herself bad at mathematics. It’s always the way.

Breaking up a whole number like 4 into different sums of whole numbers is a mathematics problem also. Splitting up 4 into, say, ‘2 plus 1 plus 1’, is a ‘partition’ of the number. I’m not sure of important results that follow this sort of integer partition directly. But splitting up sets of things different ways runs through a lot of mathematics. Integer partitions are the ones you can do in elementary school.

Percy Crosby’s Skippy for the 3rd of February — I believe it originally ran December 1928 — is a Roman numerals joke. The mathematical content may be low, but what the heck. It’s kind of timely. The Super Bowl, set for today, has been the most prominent use of Roman numerals we have anymore since the Star Trek movies stopped using them a quarter-century ago.

Bill Amend’s FoxTrot for the 7th of February seems to be in agreement. And yes, I’m disappointed the Super Bowl is giving up on Roman numerals, much the way I’m disappointed they’re using a standardized and quite boring logo for each year. Part of the glory of past Super Bowls is seeing old graphic design eras preserved like fossils.

Brian Gordon’s Fowl Language for the 5th of February shows a duck trying to explain incredibly huge numbers to his kid. It’s hard. You need to appreciate mathematics some to start appreciating real vastness. I’m not sure anyone can really have a feel for a number like 300 sextillion, the character’s estimate for the number of stars there are. You can make rationalizations for what numbers that big are like, but I suspect the mind shies back from staring directly at it.

Infinity, and the many different sizes of infinity, might be easier to work with. One doesn’t need to imagine infinitely many things to work out the properties of infinitely large sets. You could do as well with a neatly drawn rectangle and some other, bigger, rectangles. But if you want to talk about the number 300,000,000,000,000,000,000,000 then you do want to think of something true about that number which isn’t also true about eight or about nine hundred million. But geology teaches us to ponder Deep Time. Astronomy trains us to imagine incredibly vast distances. Why not spend some time pondering huge numbers?

And with all that said, I’d like to make one more call for any requests for my winter 2016 Mathematics A To Z glossary. There are quite a few attractive letters left unclaimed; a word or short term could be yours!

Reading the Comics, November 18, 2015: All Caught Up Edition

Yes, I feel a bit bad that I didn’t have anything posted yesterday. I’d had a nice every-other-day streak going for a couple weeks there. But I had honestly expected more mathematically themed comic strips, and there just weren’t enough in my box by the end of the 17th. So I didn’t have anything to schedule for a post the 18th. The 18th came through, though, and now I’ve got enough to talk about. And that before I get to reading today’s comics. So, please, enjoy.

Scott Adams’s Dilbert Classics for the 16th of November (originally published the 21st of September, 1992) features Dilbert discovering Bell’s Theorem. Bell’s Theorem is an important piece of our understanding of quantum mechanics. It’s a theorem that excites people who first hear about it. It implies quantum mechanics can’t explain reality unless it can allow information to be transmitted between interacting particles faster than light. And quantum mechanics does explain reality. The thing is, and the thing that casual readers don’t understand, is that there’s no way to use this to send a signal. Imagine that I took two cards, one an ace and one an eight, seal them in envelopes, and gave them to astronauts. The astronauts each travel to ten light-years away from me in opposite directions. (They took extreme offense at something I said and didn’t like one another anyway.) Then one of them opens her envelope, finding that she’s got the eight. Then instantly, even though they’re twenty light-years apart, she knows the other astronaut has an ace in her envelope. But there is no way the astronauts can use this to send information to one another, which is what people want Bell’s Theorem to tell us. (My example is not legitimate quantum mechanics and do not try to use it to pass your thesis defense. It just shows why Bell’s Theorem does not give us a way to send information we care about faster than light.) The next day Dilbert’s Garbageman, the Smartest Man in the World, mentions Dilbert’s added something to Bell’s Theorem. It’s the same thing everybody figuring they can use quantum entanglement to communicate adds to the idea.

Tom Thaves’ Frank and Ernest for the 16th of November riffs on the idea of a lottery as a “tax on people who are bad at math”. Longtime readers here know that I have mixed feelings about that, and not just because I’m wary of cliché. If the jackpot is high enough, you can reach the point where the expectation value of the prize is positive. That is, you would expect to make money if you played the game under the same conditions often enough. But that chance is still vanishingly small. Even playing a million times would not make it likely you would more earn money than you spent. I’m not dogmatic enough to say what your decision should be, at least if the prize is big enough. (And that’s not considering the value placed on the fun of playing. One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket. But many people do enjoy it and I can’t bring myself to say they’re all wrong about feeling enjoyment.)

And it happens that on the 18th Brant Parker and Johnny Hart’s Wizard of Id Classics (originally run the 20th of November, 1965) did a lottery joke. That one is about a lottery one shouldn’t play, except that the King keeps track of who refuses to buy a ticket. I know when we’re in a genre.

Peter Mann’s The Quixote Syndrome for the 16th of November explores something I had never known but that at least the web seems to think is true. Apparently in 1958 Samuel Beckett knew the 12-year-old André Roussimoff. People of my age cohort have any idea who that is when they hear Roussimoff became pro wrestling star André the Giant. And Beckett drove the kid to school. Mann — taking, I think, a break from his usual adaptations of classic literature — speculates on what they might have talked about. His guess: Beckett attempting to ease one of his fears through careful study and mathematical treatment. The problem is goofily funny. But the treatment is the sort of mathematics everyone understands needing and understands using.

John Deering’s Strange Brew for the 17th of November tells a rounding up joke. Scott Hilburn’s The Argyle Sweater told it back in August. I suspect the joke is just in the air. Most jokes were formed between 1922 and 1978 anyway, and we’re just shuffling around the remains of that fruitful era.

Tony Cochrane’s Agnes for the 18th of November tells a resisting-the-word-problem joke. I admit expecting better from Cochrane. But casting arithmetic problems into word problems is fraught with peril. It isn’t enough to avoid obsolete references. (If we accept trains as obsolete. I’m from the United States Northeast, where subways and even commuter trains are viable things.) The problem also has to ask something the problem-solver can imagine wanting to know. It may not matter whether the question asks how far apart two trains, two cars, or two airplanes are, if the student can’t see their distance as anything but trivia. We may need better practice in writing stories if we’re to write story problems.

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.

Reading the Comics, July 4, 2015: Symbolic Curiosities Edition

Comic Strip Master Command was pretty kind to me this week, and didn’t overload me with too many comics when my computer problems were the most time-demanding. You’ve seen how bad that is by how long it’s taken me to get to answering people’s comments. But they have kept publishing mathematical comic strips, and so I’m ready for another review. This time around a couple of the strips talk about the symbols of mathematics, so that’s enough of a hook for my titling needs.

Henry Scarpelli and Craig Boldman’s Archie for the 30th of June, 2015, although that’s a rerun.

Henry Scarpelli and Craig Boldman’s Archie (June 30, rerun) is about living with long odds. People react to very improbable events in strange ways. Moose is being maybe more consistent than normal for folks in figuring that if he’s going to be lucky enough to win a contest then he’s just lucky enough to be hit by a meteor too. (It feels like a lottery to me, although I guess Moose has to be too young to enter a lottery.) And I’m amused by the logic of someone’s behavior becoming funny because it is logically consistent.

Dave Blazek’s Loose Parts (June 30) shows the offices of Math, Inc. (I believe this is actually the Chicago division, not the main headquarters.) This is also a strip I could easily see happening in the real world. It’s not different in principle from clocks which put some arithmetic expression up for the hours, or those calendars which make a math puzzle out of the date.