My 2018 Mathematics A To Z: Jokes


For today’s entry, Iva Sallay, of Find The Factors, gave me an irresistible topic. I did not resist.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Jokes.

What’s purple and commutes?
An Abelian grape.

Whatever else you say about mathematics we are human. We tell jokes. I will tell some here. You may not understand the words in them. That’s all right. From the Abelian grape there, you gather this is some manner of wordplay. A pun, particularly. It’s built on a technical term. “Abelian groups” come from (not high school) Algebra. In an Abelian group, the group multiplication commutes. That is, if ‘a’ and ‘b’ are any things in the group, then their product “ab” is the same as “ba’. That is, the group works like ordinary addition on numbers does. We say “Abelian” in honor of Niels Henrik Abel, who taught us some fascinating stuff about polynomials. Puns are a common kind of humor. So common, they’re almost base. Even a good pun earns less laughter than groans.

But mathematicians make many puns. A typical page of mathematics jokes has a whole section of puns. “What’s yellow and equivalent to the Axiom of Choice? Zorn’s Lemon.” “What’s nonorientable and lives in the sea?” “Möbius Dick.” “One day Jesus said to his disciples, `The Kingdom of Heaven is like 3x2 + 8x – 9′. Thomas looked very confused and asked peter, `What does the teacher mean?’ Peter replied, `Don’t worry. It’s just another one of his parabolas’.” And there are many jokes built on how it is impossible to tell the difference between the sounds of “π” and “pie”.

It shouldn’t surprise that mathematicians make so many puns. Mathematics trains people to know definitions. To think about precisely what we mean. Puns ignore definitions. They build nonsense out of the ways that sounds interact. Mathematicians practice how to make things interact, even if they don’t know or care what the underlying things are. If you’ve gotten used to proving things about aba^{-1}b^{-1} , without knowing what ‘a’ or ‘b’ are, it’s difficult to avoid turning “poles on the half-plane” (which matters in some mathematical physics) to a story about Polish people on an aircraft.

Popeye's lousy tutor: 'Today I am going to test you at mental multiplication. Quick, how much is 6 1/2 times 656? Quick!' Popeye: '4,264.' 'Right!' 'Blow me down! Anybody what can guess like that don't need no edjacation!'
Elzie Segar’s Thimble Theater from the 14th of September, 1929. Rerun on ComicsKingdom on the 26th of February, 2016. That’s Bernice, the magical Whiffle Hen, as the strange birdlike creature in the last panel there.

If there’s a flaw to this kind of humor it’s that these jokes may sound juvenile. One of the first things that strikes kids as funny is that a thing might have several meanings. Or might sound like another thing. “Why do mathematicians like parks? Because of all the natural logs!”

Jokes can be built tightly around definitions. “What do you get if you cross a mosquito with a mountain climber? Nothing; you can’t cross a vector with a scalar.” “There are 10 kinds of people in the world, those who understand binary mathematics and those who don’t.” “Life is complex; it has real and imaginary parts.”

Paige: 'I keep forgetting ... what's the cosine of 60 degrees?' Jason: 'Well, let's see. If I recall correctly ... 1 - (pi/3)^2/2! + (pi/3)^4/4! - (pi/3)^6/6! + (pi/3)^8/8! - (pi/3)^10/10! + (pi/3)^12/12! - (and this goes on a while, up to (pi/3)^32/32! - ... )' Paige: 'In case you've forgotten, I'm not paying you by the hour.' Jason: '1/2'.
Bill Amend’s FoxTrot Classics for the 23rd of May, 2018. It originally ran the 29th of May, 1996.

There are more sophisticated jokes. Many of them are self-deprecating. “A mathematician is a device for turning coffee into theorems.” “An introvert mathematician looks at her shoes while talking to you. An extrovert mathematician looks at your shoes.” “A mathematics professor is someone who talks in someone else’s sleep”. “Two people are adrift in a hot air balloon. Finally they see someone and shout down, `Where are we?’ The person looks up, and studies them, watching the balloon drift away. Finally, when they are barely in shouting range, the person on the ground shouts back, `You are in a balloon!’ The first passenger curses their luck at running across a mathematician. `How do you know that was a mathematician?’ `Because her answer took a long time, was perfectly correct, and absolutely useless!”’ These have the form of being about mathematicians. But they’re not really. It would be the same joke to say “a poet is a device for turning coffee into couplets”, the sleep-talker anyone who teachers, or have the hot-air balloonists discover a lawyer or a consultant.

Some of these jokes get more specific, with mathematics harder to extract from the story. The tale of the nervous flyer who, before going to the conference, sends a postcard that she has a proof of the Riemann hypothesis. She arrives and admits she has no such thing, of course. But she sends that word ahead of every conference. She knows if she died in a plane crash after that, she’d be famous forever, and God would never give her that. (I wonder if Ian Randal Strock’s little joke of a story about Pierre de Fermat was an adaptation of this joke.) You could recast the joke for physicists uniting gravity and quantum mechanics. But I can’t imagine a way to make this joke about an ISO 9000 consultant.

'If it's a hunnert miles to th' city an' a train is travelin' thurty miles an hour is due t'arrive at 5:00 pm --- what time does th' train leave Hootin' Holler, Jughaid?' 'I dunno, Miz Prunelly, but you better go now jest t'be on th' safe side!!'
John Rose’s Barney Google and Snuffy Smith for the 12th of February, 2016.

A dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

This joke is very mathematical. I know of no important results actually based on spherical cows. But the attitude that tries to make spheres of cows comes from observing mathematicians. To describe any real-world process is to make a model of that thing. A model is a simplification of the real thing. You suppose that things behave more predictably than the real thing. You trust the error made by this supposition is small enough for your needs. A cow is complicated, all those pointy ends and weird contours. A sphere is easy. And, besides, cows are funny. “Spherical cow” is a funny string of sounds, at least in English.

The spherical cows approach parodying the work mathematicians do. Many mathematical jokes are burlesques of deductive logic. Or not even burlesques. Charles Dodgson, known to humans as Lewis Carroll, wrote this in Symbolic Logic:

“No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
∴ This party of tourists need not run.”

[ Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premises to be true, I should be quite clear that I needn’t run — and I should walk!

And you will reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum! Ha! I must think that over a bit!” ]

The punch line is diffused by the text being so educational. And by being written in the 19th century, when it was bad form to excise any word from any writing. But you can recognize the joke, and why it should be a joke.

Not every mathematical-reasoning joke features some manner of cattle. Some are legitimate:

Claim. There are no uninteresting whole numbers.
Proof. Suppose there is a smalled uninteresting whole number. Call it N. That N is uninteresting is an interesting fact. Therefore N is not an uninteresting whole number.

Three mathematicians step up to the bar. The bartender asks, “you all want a beer?” The first mathematician says, “I don’t know.” The second mathematician says, “I don’t know.” The third says, “Yes”.

Some mock reasoning uses nonsense methods to get a true conclusion. It’s the fun of watching Mister Magoo walk unharmed through a construction site to find the department store exchange counter:

5095 / 1019 = 5095 / 1019 = 505 / 101 = 55 / 11 = 5

This one includes the thrill of division by zero.

The Venn Diagram of Grocery Shopping. Overlap 'have teenagers', 'haven't grocery shopped in two weeks', and 'grocery shopping on an empty stomach' and you get 'will need to go back in two days', 'bought entire bakery aisle', and 'bought two of everything'. Where they all overlap, 'need to take out second mortgage'.
Terri Libenson’s Pajama Diaries for the 16th of November, 2016. I was never one for buying too much of the bakery aisle, myself, but then I also haven’t got teenagers. And I did go through so much of my life figuring there was no reason I shouldn’t eat another bagel again.

Venn Diagrams are not by themselves jokes (most of the time). But they are a great structure for jokes. And easy to draw, which is great for us who want to be funny but don’t feel sure about their drafting abilities.

And then there are personality jokes. Mathematics encourages people to think obsessively. Obsessive people are often funny people. Alexander Grothendieck was one of the candidates for “greatest 20th century mathematician”. His reputation is that he worked so well on abstract problems that he was incompetent at practical ones. The story goes that he was demonstrating something about prime numbers and his audience begged him to speak about a specific number, that they could follow an example. And that he grumbled a bit and, finally, said, “57”. It’s not a prime number. But if you speak of “Grothendieck’s prime”, many will recognize what you mean, and grin.

There are more outstanding, preposterous personalities. Paul Erdös was prolific, and a restless traveller. The stories go that he would show up at some poor mathematician’s door and stay with them several months. And then co-author a paper with the elevator operator. (Erdös is also credited as the originator of the “coffee into theorems” quip above.) John von Neumann was supposedly presented with this problem:

Two trains are on the same track, 60 miles apart, heading toward each other, each travelling 30 miles per hour. A fly travels 60 miles per hour, leaving one engine flying toward the other. When it reaches the other engine it turns around immediately and flies back to the other engine. This is repeated until the two trains crash. How far does the fly travel before the crash?

The first, hard way to do this is to realize how far the fly travels is a series. The fly starts at, let’s say, the left engine and flies to the right. Add to that the distance from the right to the left train now. Then left to the right again. Right to left. This is a bunch of calculations. Most people give up on that and realize the problem is easier. The trains will crash in one hour. The fly travels 60 miles per hour for an hour. It’ll fly 60 miles total. John von Neumann, say witnesses, had the answer instantly. He recognized the trick? “I summed the series.”

Henry is frustrated with his arithmetic, until he goes to the pool hall and counts off numbers on those score chips.
Don Trachte’s Henry for the 6th of September, 2015.

The personalities can be known more remotely, from a handful of facts about who they were or what they did. “Cantor did it diagonally.” Georg Cantor is famous for great thinking about infinitely large sets. His “diagonal proof” shows the set of real numbers must be larger than the set of rational numbers. “Fermat tried to do it in the margin but couldn’t fit it in.” “Galois did it on the night before.” (Évariste Galois wrote out important pieces of group theory the night before a duel. It went badly for him. French politics of the 1830s.) Every field has its celebrities. Mathematicians learn just enough about theirs to know a couple of jokes.

Anthropomorphic 3/5: 'Honey, what's wrong?' Anthropomorphic 1/4: 'Sour son is leaving the faith! He said he's converting to decimals!'
Scott Hilburn’s The Argyle Sweater for the 9th of May, 2018. I like the shout-out to Archimedes in the background art, too. Archimedes, though, didn’t use fractions in the way we’d recognize them. He’d write out a number as a combination of ratios of some reference number. So he might estimate the length of something being as to the length of something else as 19 is to 7, or something like that. This seems like a longwinded and cumbersome way to write out numbers, or much of anything, and makes one appreciate his indefatigability as much as his insight.

The jokes can attach to a generic mathematician personality. “How can you possibly visualize something that happens in a 12-dimensional space?” “Easy, first visualize it in an N-dimensional space, and then let N go to 12.” Three statisticians go hunting. They spot a deer. One shoots, missing it on the left. The second shoots, missing it on the right. The third leaps up, shouting, “We’ve hit it!” An engineer and a mathematician are sleeping in a hotel room when the fire alarm goes off. The engineer ties the bedsheets into a rope and shimmies out of the room. The mathematician looks at this, unties the bedsheets, sets them back on the bed, declares, “this is a problem already solved” and goes back to sleep. (Engineers and mathematicians pair up a lot in mathematics jokes. I assume in engineering jokes too, but that the engineers make wrong assumptions about who the joke is on. If there’s a third person in the party, she’s a physicist.)

Do I have a favorite mathematics joke? I suppose I must. There are jokes I like better than others, and there are — I assume — finitely many different mathematics jokes. So I must have a favorite. What is it? I don’t know. It must vary with the day and my mood and the last thing I thought about. I know a bit of doggerel keeps popping into my head, unbidden. Let me close by giving it to you.

Integral z-squared dz
From 1 to the cube root of 3
   Times the cosine
   Of three π over nine
Equals log of the cube root of e.

This may not strike you as very funny. I’m not sure it strikes me as very funny. But it keeps showing up, all the time. That has to add up.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. Also, now and then, I talk about comic strips here. You might like that too.

My 2018 Mathematics A To Z: Infinite Monkey Theorem


Dina Yagodich gave me the topic for today. She keeps up a YouTube channel with a variety of interesting videos. And she did me a favor. I’ve been thinking a long while to write a major post about this theorem. Its subject turns up so often. I’d wanted to have a good essay about it. I hope this might be one.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Infinite Monkey Theorem.

Some mathematics escapes mathematicians and joins culture. This is one such. The monkeys are part of why. They’re funny and intelligent and sad and stupid and deft and clumsy, and they can sit at a keyboard almost look in place. They’re so like humans, except that we empathize with them. To imagine lots of monkeys, and putting them to some silly task, is compelling.

Monkey Typewriter Theory: An immortal monkey pounding on a typewriter will eventually reproduce the text of 'Hamlet'. Baby Keyboard Theory: Left alone, a baby pounding on a computer keyboard will eventually order 32 cases of bathroom caulk from an online retailer.
Paul Trapp’s Thatababy for the 13th of February, 2014.

The metaphor traces back to a 1913 article by the mathematical physicist Émile Borel which I have not read. Searching the web I find much more comment about it than I find links to a translation of the text. And only one copy of the original, in French. And that page wants €10 for it. So I can tell you what everybody says was in Borel’s original text, but can’t verify it. The paper’s title is “Statistical Mechanics and Irreversibility”. From this I surmise that Borel discussed one of the great paradoxes of statistical mechanics. If we open a bottle of one gas in an airtight room, it disperses through the room. Why doesn’t every molecule of gas just happen, by chance, to end up back where it started? It does seem that if we waited long enough, it should. It’s unlikely it would happen on any one day, but give it enough days …

But let me turn to many web sites that are surely not all copying Wikipedia on this. Borel asked us to imagine a million monkeys typing ten hours a day. He posited it was possible but extremely unlikely that they would exactly replicate all the books of the richest libraries of the world. But that would be more likely than the atmosphere in a room un-mixing like that. Fair enough, but we’re not listening anymore. We’re thinking of monkeys. Borel’s is a fantastic image. It would see some adaptation in the years. Physicist Arthur Eddington, in 1928, made it an army of monkeys, with their goal being the writing all the books in the British Museum. By 1960 Bob Newhart had an infinite number of monkeys and typewriters, and a goal of all the great books. Stating the premise gets a laugh I doubt the setup would today. I’m curious whether Newhart brought the idea to the mass audience. (Google NGrams for “monkeys at typewriters” suggest that phrase was unwritten, in books, before about 1965.) We may owe Bob Newhart thanks for a lot of monkeys-at-typewriters jokes.

Kid: 'Mom, Dad, I want to go bungee jumping this summer!' Dad: 'A thousand monkeys working a thousand typewriters would have a better chance of randomly typing the complete works of William Shakespeare over the summer than you have of bungee jumping.' (Awksard pause.) Kid: 'What's a typewriter?' Dad: 'A thousand monkeys randomly TEXTING!'
Bill Hinds’s Cleats rerun for the 1st of July, 2018.

Newhart has a monkey hit on a line from Hamlet. I don’t know if it was Newhart that set the monkeys after Shakespeare particularly, rather than some other great work of writing. Shakespeare does seem to be the most common goal now. Sometimes the number of monkeys diminishes, to a thousand or even to one. Some people move the monkeys off of typewriters and onto computers. Some take the cowardly measure of putting the monkeys at “keyboards”. The word is ambiguous enough to allow for typewriters, computers, and maybe a Megenthaler Linotype. The monkeys now work 24 hours a day. This will be a comment someday about how bad we allowed pre-revolutionary capitalism to get.

The cultural legacy of monkeys-at-keyboards might well itself be infinite. It turns up in comic strips every few weeks at least. Television shows, usually writing for a comic beat, mention it. Computer nerds doing humor can’t resist the idea. Here’s a video of a 1979 Apple ][ program titled THE INFINITE NO. OF MONKEYS, which used this idea to show programming tricks. And it’s a great philosophical test case. If a random process puts together a play we find interesting, has it created art? No deliberate process creates a sunset, but we can find in it beauty and meaning. Why not words? There’s likely a book to write about the infinite monkeys in pop culture. Though the quotations of original materials would start to blend together.

But the big question. Have the monkeys got a chance? In a break from every probability question ever, the answer is: it depends on what the question precisely is. Occasional real-world experiments-cum-art-projects suggest that actual monkeys are worse typists than you’d think. They do more of bashing the keys with a stone before urinating on it, a reminder of how slight is the difference between humans and our fellow primates. So we turn to abstract monkeys who behave more predictably, and run experiments that need no ethical oversight.

Toby: 'So this English writer is like a genius, right? And he's the greatest playwright ever. And I want to be just like him! Cause what he does, see, is he gets infinite monkeys on typewriters and just lets 'em go nuts, so eventually they write ALL of Shakespeare's plays!' Brother: 'Cool! And what kind of monkey is an 'infinite'?' Toby: 'Beats me, but I hope I don't have to buy many of them.' Dad: 'Toby, are you *sure* ywou completely pay attention when your teachers are talking?' Toby: 'What? Yes! Why?'

Greg Cravens’ The Buckets for the 30th of March, 2014.

So we must think what we mean by Shakespeare’s Plays. Arguably the play is a specific performance of actors in a set venue doing things. This is a bit much to expect of even a skilled abstract monkey. So let us switch to the book of a play. This has a more clear representation. It’s a string of characters. Mostly letters, some punctuation. Good chance there’s numerals in there. It’s probably a lot of characters. So the text to match is some specific, long string of characters in a particular order.

And what do we mean by a monkey at the keyboard? Well, we mean some process that picks characters randomly from the allowed set. When I see something is picked “randomly” I want to know what the distribution rule is. Like, are Q’s exactly as probable as E’s? As &’s? As %’s? How likely it is a particular string will get typed is easiest to answer if we suppose a “uniform” distribution. This means that every character is equally likely. We can quibble about capital and lowercase letters. My sense is most people frame the problem supposing case-insensitivity. That the monkey is doing fine to type “whaT beArD weRe i BEsT tO pLAy It iN?”. Or we could set the monkey at an old typesetter’s station, with separate keys for capital and lowercase letters. Some will even forgive the monkeys punctuating terribly. Make your choices. It affects the numbers, but not the point.

Literary Calendar. Several jokes, including: Saturday 7pm: an infinite number of chimpanzees discuss their multi-volume 'Treasury of Western Literature with no Typos' at the Museum of Natural History. Nit picking to follow.
Richard Thompson’s Richard’s Poor Almanac rerun for the 7th of November, 2016.

I’ll suppose there are 91 characters to pick from, as a Linotype keyboard had. So the monkey has capitals and lowercase and common punctuation to get right. Let your monkey pick one character. What is the chance it hit the first character of one of Shakespeare’s plays? Well, the chance is 1 in 91 that you’ve hit the first character of one specific play. There’s several dozen plays your monkey might be typing, though. I bet some of them even start with the same character, so giving an exact answer is tedious. If all we want monkey-typed Shakespeare plays, we’re being fussy if we want The Tempest typed up first and Cymbeline last. If we want a more tractable problem, it’s easier to insist on a set order.

So suppose we do have a set order. Then there’s a one-in-91 chance the first character matches the first character of the desired text. A one-in-91 chance the second character typed matches the second character of the desired text. A one-in-91 chance the third character typed matches the third character of the desired text. And so on, for the whole length of the play’s text. Getting one character right doesn’t make it more or less likely the next one is right. So the chance of getting a whole play correct is \frac{1}{91} raised to the power of however many characters are in the first script. Call it 800,000 for argument’s sake. More characters, if you put two spaces between sentences. The prospects of getting this all correct is … dismal.

I mean, there’s some cause for hope. Spelling was much less fixed in Shakespeare’s time. There are acceptable variations for many of his words. It’d be silly to rule out a possible script that (say) wrote “look’d” or “look’t”, rather than “looked”. Still, that’s a slender thread.

Proverb Busters: testing the validity of old sayings. Doctor: 'A hundred monkeys at a hundred typewriters. Over time, will one of them eventually write a Shakepeare play?' Winky: 'Nope. Just the script for Grown-Ups 3'. Doctor: 'Another proverb busted.'
Tim Rickard’s Brewster Rockit for the 1st of April, 2014.

But there is more reason to hope. Chances are the first monkey will botch the first character. But what if they get the first character of the text right on the second character struck? Or on the third character struck? It’s all right if there’s some garbage before the text comes up. Many writers have trouble starting and build from a first paragraph meant to be thrown away. After every wrong letter is a new chance to type the perfect thing, reassurance for us all.

Since the monkey does type, hypothetically, forever … well, so each character has a probability of only \left(\frac{1}{91}\right)^{800,000} (or whatever) of starting the lucky sequence. The monkey will have 91^{800,000} chances to start. More chances than that.

And we don’t have only one monkey. We have a thousand monkeys. At least. A million monkeys. Maybe infinitely many monkeys. Each one, we trust, is working independently, owing to the monkeys’ strong sense of academic integrity. There are 91^{800,000} monkeys working on the project. And more than that. Each one takes their chance.

Melvin: 'Hold on now --- replacement? Who could you find to do all the tasks only Melvin can perform?' Rita: 'A macaque, in fact. Listen, if an infinite number of monkeys can write all the great works, I'm confident that one will more than cover for you.'
John Zakour and Scott Roberts’s Working Daze for the 29th of May, 2018.

There are dizzying possibilities here. There’s the chance some monkey will get it all exactly right first time out. More. Think of a row of monkeys. What’s the chance the first thing the first monkey in the row types is the first character of the play? What’s the chance the first thing the second monkey in the row types is the second character of the play? The chance the first thing the third monkey in the row types is the third character in the play? What’s the chance a long enough row of monkeys happen to hit the right buttons so the whole play appears in one massive simultaneous stroke of the keys? Not any worse than the chance your one monkey will type this all out. Monkeys at keyboards are ergodic. It’s as good to have a few monkeys working a long while as to have many monkeys working a short while. The Mythical Man-Month is, for this project, mistaken.

That solves it then, doesn’t it? A monkey, or a team of monkeys, has a nonzero probability of typing out all Shakespeare’s plays. Or the works of Dickens. Or of Jorge Luis Borges. Whatever you like. Given infinitely many chances at it, they will, someday, succeed.

Except.

A thousand monkeys at a thousand typewriters ... will eventually write 'Hamlet'. A thousand cats at a thousand typewriters ... will tell you go to write your own danged 'Hamlet'.
Doug Savage’s Savage Chickens for the 14th of August, 2018.

What is the chance that the monkeys screw up? They get the works of Shakespeare just right, but for a flaw. The monkeys’ Midsummer Night’s Dream insists on having the fearsome lion played by “Smaug the joiner” instead. This would send the play-within-the-play in novel directions. The result, though interesting, would not be Shakespeare. There’s a nonzero chance they’ll write the play that way. And so, given infinitely many chances, they will.

What’s the chance that they always will? That they just miss every single chance to write “Snug”. It comes out “Smaug” every time?

Eddie: 'You know the old saying about putting an infinite number of monkeys at an infinite number of typewriters, and eventually they'll accidentally write Shakespeare's plays?' Toby: 'I guess.' Eddie: 'My English teacher says that nothing about our class should worry those monkeys ONE BIT!'
Greg Cravens’s The Buckets for the 6th of October, 2018.

We can say. Call the probability that they make this Snug-to-Smaug typo any given time p . That’s a number from 0 to 1. 0 corresponds to not making this mistake; 1 to certainly making it. The chance they get it right is 1 - p . The chance they make this mistake twice is smaller than p . The chance that they get it right at least once in two tries is closer to 1 than 1 - p is. The chance that, given three tries, they make the mistake every time is even smaller still. The chance that they get it right at least once is even closer to 1.

You see where this is going. Every extra try makes the chance they got it wrong every time smaller. Every extra try makes the chance they get it right at least once bigger. And now we can let some analysis come into play.

So give me a positive number. I don’t know your number, so I’ll call it ε. It’s how unlikely you want something to be before you say it won’t happen. Whatever your ε was, I can give you a number M . If the monkeys have taken more than M tries, the chance they get it wrong every single time is smaller than your ε. The chance they get it right at least once is bigger than 1 – ε. Let the monkeys have infinitely many tries. The chance the monkey gets it wrong every single time is smaller than any positive number. So the chance the monkey gets it wrong every single time is zero. It … can’t happen, right? The chance they get it right at least once is closer to 1 than to any other number. So it must be 1. So it must be certain. Right?

Poncho, the dog, looking over his owner's laptop: 'They say if you let an infinite number of cats walk on an infinite number of keyboards, they'll eventually type all the great works of Shakespeare.' The cat walks across the laptop, connecting to their owner's bank site and entering the correct password. Poncho: 'I'll take it.'
Paul Gilligan’s Pooch Cafe for the 17th of September, 2018.

But let me give you this. Detach a monkey from typewriter duty. This one has a coin to toss. It tosses fairly, with the coin having a 50% chance of coming up tails and 50% chance of coming up heads each time. The monkey tosses the coin infinitely many times. What is the chance the coin comes up tails every single one of these infinitely many times? The chance is zero, obviously. At least you can show the chance is smaller than any positive number. So, zero.

Yet … what power enforces that? What forces the monkey to eventually have a coin come up heads? It’s … nothing. Each toss is a fair toss. Each toss is independent of its predecessors. But there is no force that causes the monkey, after a hundred million billion trillion tosses of “tails”, to then toss “heads”. It’s the gambler’s fallacy to think there is one. The hundred million billion trillionth-plus-one toss is as likely to come up tails as the first toss is. It’s impossible that the monkey should toss tails infinitely many times. But there’s no reason it can’t happen. It’s also impossible that the monkeys still on the typewriters should get Shakespeare wrong every single time. But there’s no reason that can’t happen.

It’s unsettling. Well, probability is unsettling. If you don’t find it disturbing you haven’t thought long enough about it. Infinities, too, are unsettling so.

Researcher overseeing a room of monkeys: 'Shakespeare would be OK, but I'd prefer they come up with a good research grant proposal.'
John Deering’s Strange Brew for the 20th of February, 2014.

Formally, mathematicians interpret this — if not explain it — by saying the set of things that can happen is a “probability space”. The likelihood of something happening is what fraction of the probability space matches something happening. (I’m skipping a lot of background to say something that simple. Do not use this at your thesis defense without that background.) This sort of “impossible” event has “measure zero”. So its probability of happening is zero. Measure turns up in analysis, in understanding how calculus works. It complicates a bunch of otherwise-obvious ideas about continuity and stuff. It turns out to apply to probability questions too. Imagine the space of all the things that could possibly happen as being the real number line. Pick one number from that number line. What is the chance you have picked exactly the number -24.11390550338228506633488? I’ll go ahead and say you didn’t. It’s not that you couldn’t. It’s not impossible. It’s just that the chance that this happened, out of the infinity of possible outcomes, is zero.

The infinite monkeys give us this strange set of affairs. Some things have a probability of zero of happening, which does not rule out that they can. Some things have a probability of one of happening, which does not mean they must. I do not know what conclusion Borel ultimately drew about the reversibility problem. I expect his opinion to be that we have a clear answer, and unsettlingly great room for that answer to be incomplete.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. The next essay should come Friday and will, I hope, be shorter.

Someone Else’s Homework: A Postscript


My friend aced the mathematics final. Not due to my intervention, I’d say; my friend only remembered one question on the exam being much like anything we had discussed recently. Though it was very like one of those, a question about the probability of putting together a committee with none, one, two, or more than two members of particular subgroups. And that one we didn’t even work through; I just confirmed my friend’s guess about what calculation to do. Which is good since that particular calculation is a tedious one that I didn’t want to do. No, my friend aced it by working steadily through the whole term. And yes, asking me for tutoring a couple times, but that’s all right. Small, steady work adds up, in mathematics as with so much else.

Meanwhile may I draw your attention over to my humor blog where last night I posted a bit of silliness about number divisibility. Because I can’t help myself, it does include a “quick” test for whether a number could be divided by 21. It’s in the same spirit as tests for whether a number can be divided by 3 or 9 (add the digits add see whether that sum’s divisible by 3 or 9) or 11 (add or subtract digits, in alternate form, and see whether that sum is divisible by 11). The process I give is correct, which is not to say that anyone would ever use it. Even if they did they’d be better off testing for divisibility by both 3 and 7. And I don’t think I’d use an add-the-digits scheme for 7 either.

Why Shouldn’t We Talk About Mathematics In The Deli Line?


You maybe saw this picture going around your social media a couple days ago. I did, but I’m connected to a lot of mathematics people who were naturally interested. Everyone who did see it was speculating about what the story behind it was. Thanks to the CBC, now we know.

So it’s the most obvious if least excitingly gossip-worthy explanation: this Middletown, Connecticut deli is close to the Wesleyan mathematics department’s office and at least one mathematician was too engrossed talking about the subject to actually place an order. We’ve all been stuck behind people like that. It’s enough to make you wonder whether the Cole slaw there is actually that good. (Don’t know, I haven’t been there, not sure I can dispatch my agent in Groton to check just for this.) The sign’s basically a loving joke, which is a relief. Could be any group of people who won’t stop talking about a thing they enjoy, really. And who have a good joking relationship with the deli-owner.

The CBC’s interview gets into whether mathematicians have a sense of humor. I certainly think we do. I think the habit of forming proofs builds a habit of making a wild assumption and seeing where that gets you, often to a contradiction. And it’s hard not to see that the same skills that will let you go from, say, “suppose every angle can be trisected” to a nonsensical conclusion will also let you suppose something whimsical and get to a silly result.

Dr Anna Haensch, who made the sign kind-of famous-ish, gave as an example of a quick little mathematician’s joke going to the board and declaring “let L be a group”. I should say that’s not a riotously funny mathematician’s joke, not the say (like) talking about things purple and commutative are. It’s just a little passing quip, like if you showed a map of New Jersey and labelled the big city just across the Hudson River as “East Hoboken” or something.

But L would be a slightly disjoint name for a group. Not wrong, just odd, unless the context of the problem gave us good reason for the name. Names of stuff are labels, and so are arbitrary and may be anything. But we use them to carry information. If we know something is a group then we know something about the way it behaves. So if in a dense mass of symbols we see that something is given one of the standard names for groups — G, H, maybe G or H with a subscript or a ‘ or * on top of it — we know that, however lost we might be, we know this thing is a group and we know it should have these properties.

It’s a bit of doing what science fiction fans term “incluing”. That’s giving someone the necessary backstory without drawing attention to the fact we’re doing it. To avoid G or H would be like avoiding “Jane [or John] Doe” as the name for a specific but unidentified person. You can do it, but it seems off.

Finally, What I Learned Doing Theorem Thursdays


Here’s the index to the stuff I posted on them.

The biggest thing I learned from my Theorem Thursdays project was: don’t do this for Thursdays. The appeal is obvious. If things were a little different I’d have no problem with Thursdays. But besides being a slightly-read pop-mathematics blogger I’m also a slightly-read humor blogger. And I try to have a major piece, about seven hundred words that are more than simply commentary on how a comic strip’s gone wrong, ready for Thursday evenings my time.

That’s all my doing. It’s a relic of my thinking that the humor blog should run at least a bit like a professional syndicated columnist’s, with a fixed deadline for bigger pieces. While I should be writing more ahead of deadline than this, what I would do is get to Wednesday realizing I have two major things to write in a day. I’d have an idea for one of them, the mathematics thing, since I would pick a topic the previous Thursday. And once I’ve picked an idea the rest is easy. (Part of the process of picking is realizing whether there’s any way to make seven hundred words about something.) But that’s a lot of work for something that’s supposed to be recreational. Plus Wednesdays are, two weeks a month, a pinball league night.

So Thursday is right out, unless I get better about having first drafts of stuff done Monday night. So Thursday is right out. This has problems for future appearances of the gimmick. The alliterative pull is strong. The only remotely compelling alternative is Theorems on the Threes, maybe one the 3rd, 13th, and 23rd of the month. That leaves the 30th and 31st unaccounted for, and room for a good squabble about whether they count in an “on the threes” scheme.

There’s a lot of good stuff to say about the project otherwise. The biggest is that I had fun with it. The Theorem Thursday pieces sprawled into for-me extreme lengths, two to three thousand words. I had space to be chatty and silly and autobiographic in ways that even the A To Z projects don’t allow. Somehow those essays didn’t get nearly as long, possibly because I was writing three of them a week. I didn’t actually write fewer things in July than I did in, say, May. But it was fewer kinds of things; postings were mostly Theorem Thursdays and Reading the Comics posts. Still, overall readership didn’t drop and people seemed to quite like what I did write. It may be fewer but longer-form essays are the way I should go.

Also I found that people like stranger stuff. There’s an understandable temptation in doing pop-mathematics to look for topics that are automatically more accessible. People are afraid enough of mathematics. They have good reason to be terrified of some topic even mathematics majors don’t encounter until their fourth year. So there’s a drive to simpler topics, or topics that have fewer prerequisites, and that’s why every mathematics blogger has an essay about how the square root of two is irrational and how there’s different sizes to infinitely large sets. And that’s produced some excellent writing about topics like those, which are great topics. They have got the power to inspire awe without requiring any warming up. That’s special.

But it also means they’re hard to write anything new or compelling about if you’re like me, and in somewhere like the second hundred billion of mathematics bloggers. I can’t write anything better than what’s already gone about that. Liouville’s Theorem? That’s something I can be a good writer about. With that, I can have a blog personality. It’s like having a real personality but less work.

As I did with the Leap Day 2016 A To Z project, I threw the topics open to requests. I didn’t get many. Possibly the form gave too much freedom. Picking something to match a letter, as in the A to Z, gives a useful structure for choosing something specific. Pick a theorem from anywhere in mathematics? Something from algebra class? Something mentioned in a news report about a major breakthrough the reporter doesn’t understand but had an interesting picture? Something that you overheard the name of once without any context? How should people know what the scope of it is, before they’ve even seen a sample? And possibly people don’t actually remember the names of theorems unless they stay in mathematics or mathematics-related fields. Those folks hardly need explained theorems with names they remember. This is a hard problem to imagine people having, but it’s something I must consider.

So this is what I take away from the two-month project. There’s a lot of fun digging into the higher-level mathematics stuff. There’s an interest in it, even if it means I write longer and therefore fewer pieces. Take requests, but have a structure for taking them that makes it easy to tell what requests should look like. Definitely don’t commit to doing big things for Thursday, not without a better scheme for getting the humor blog pieces done. Free up some time Wednesday and don’t put up an awful score on Demolition Man like I did last time again. Seriously, I had a better score on The Simpsons Pinball Party than I did on Demolition Man and while you personally might not find this amusing there’s at least two people really into pinball who know how hilarious that is. (The games have wildly different point scorings. This like having a basketball score be lower than a hockey score.) That isn’t so important to mathematics blogging but it’s a good lesson to remember anyway.

Reading the Comics, December 30, 2015: Seeing Out The Year Edition


There’s just enough comic strips with mathematical themes that I feel comfortable doing a last Reading the Comics post for 2015. And as maybe fits that slow week between Christmas and New Year’s, there’s not a lot of deep stuff to write about. But there is a Jumble puzzle.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips gives us someone so wrapped up in measuring data as to not notice the obvious. The obvious, though, isn’t always right. This is why statistics is a deep and useful field. It’s why measurement is a powerful tool. Careful measurement and statistical tools give us ways to not fool ourselves. But it takes a lot of sampling, a lot of study, to give those tools power. It can be easy to get lost in the problems of gathering data. Plus numbers have this hypnotic power over human minds. I understand Lard’s problem.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 27th of December messes with a kid’s head about the way we know 1 + 1 equals 2. The classic Principia Mathematica construction builds it out of pure logic. We come up with an idea that we call “one”, and another that we call “plus one”, and an idea we call “two”. If we don’t do anything weird with “equals”, then it follows that “one plus one equals two” must be true. But does the logic mean anything to the real world? Or might we be setting up a game with no relation to anything observable? The punchy way I learned this question was “one cup of popcorn added to one cup of water doesn’t give you two cups of soggy popcorn”. So why should the logical rules that say “one plus one equals two” tell us anything we might want to know about how many apples one has?

Words: LIHWE, CAQUK, COYKJE, TALAFO. Unscramble to - - - O O, O O - - -, - - - - - O, - - O - O -, and solve the puzzle: 'The math teacher liked teaching addition and subtraction - - - - - - -'.
David L Hoyt and Jeff Knurek’s Jumble for the 28th of December, 2015. The link will probably expire in late January 2016.

David L Hoyt and Jeff Knurek’s Jumble for the 28th of December features a mathematics teacher. That’s enough to include here. (You might have an easier time getting the third and fourth words if you reason what the surprise-answer word must be. You can use that to reverse-engineer what letters have to be in the circles.)

Richard Thompson’s Richard’s Poor Almanac for the 28th of December repeats the Platonic Fir Christmas Tree joke. It’s in color this time. Does the color add to the perfection of the tree, or take away from it? I don’t know how to judge.

A butterfly tells another 'you *should* feel guilty --- the flutter of your wings ended up causing a hurricane that claimed thousands of lives!'
Rina Piccolo filling in for Hilary Price on Rhymes With Orange for the 29th of December, 2015. It’s a small thing but I always like the dog looking up in the title panel for Cartoonist Showcase weeks.

Hilary Price’s Rhymes With Orange for the 29th of December gives its panel over to Rina Piccolo. Price often has guest-cartoonist weeks, which is a generous use of her space. Piccolo already has one and a sixth strips — she’s one of the Six Chix cartoonists, and also draws the charming Tina’s Groove — but what the heck. Anyway, this is a comic strip about the butterfly effect. That’s the strangeness by which a deterministic system can still be unpredictable. This counter-intuitive conclusion dates back to the 1890s, when Henri Poincaré was trying to solve the big planetary mechanics question. That question is: is the solar system stable? Is the Earth going to remain in about its present orbit indefinitely far into the future? Or might the accumulated perturbations from Jupiter and the lesser planets someday pitch it out of the solar system? Or, less likely, into the Sun? And the sad truth is, the best we can say is we can’t tell.

In Brian Anderson’s Dog Eat Doug for the 30th of December, Sophie ponders some deep questions. Most of them are purely philosophical questions and outside my competence. “What are numbers?” is also a philosophical question, but it feels like something a mathematician ought to have a position on. I’m not sure I can offer a good one, though. Numbers seem to be to be these things which we imagine. They have some properties and that obey certain rules when we combine them with other numbers. The most familiar of these numbers and properties correspond with some intuition many animals have about discrete objects. Many times over we’ve expanded the idea of what kinds of things might be numbers without losing the sense of how numbers can interact, somehow. And those expansions have generally been useful. They strangely match things we would like to know about the real world. And we can discover truths about these numbers and these relations that don’t seem to be obviously built into the definitions. It’s almost as if the numbers were real objects with the capacity to surprise and to hold secrets.

Why should that be? The lazy answer is that if we came up with a construct that didn’t tell us anything interesting about the real world, we wouldn’t bother studying it. A truly irrelevant concept would be a couple forgotten papers tucked away in an unread journal. But that is missing the point. It’s like answering “why is there something rather than nothing” with “because if there were nothing we wouldn’t be here to ask the question”. That doesn’t satisfy. Why should it be possible to take some ideas about quantity that ravens, raccoons, and chimpanzees have, then abstract some concepts like “counting” and “addition” and “multiplication” from that, and then modify those concepts, and finally have the modification be anything we can see reflected in the real world? There is a mystery here. I can’t fault Sophie for not having an answer.

Making A Joke Of Entropy


This entered into my awareness a few weeks back. Of course I’ve lost where I got it from. But the headline is of natural interest to me. Kristy Condon’s “Researchers establish the world’s first mathematical theory of humor” describes the results of an interesting paper studying the phenomenon of funny words.

The original paper is by Chris Westbury, Cyrus Shaoul, Gail Moroschan, and Michael Ramscar, titled “Telling the world’s least funny jokes: On the quantification of humor as entropy”. It appeared in The Journal of Memory and Language. The thing studied was whether it’s possible to predict how funny people are likely to find a made-up non-word.

As anyone who tries to be funny knows, some words just are funnier than others. Or at least they sound funnier. (This brings us into the problem of whether something is actually funny or whether we just think it is.) Westbury, Shaoul, Moroschan, and Ramscar try testing whether a common measure of how unpredictable something is — the entropy, a cornerstone of information theory — can tell us how funny a word might be.

We’ve encountered entropy in these parts before. I used it in that series earlier this year about how interesting a basketball tournament was. Entropy, in this context, is low if something is predictable. It gets higher the more unpredictable the thing being studied is. You see this at work in auto-completion: if you have typed in ‘th’, it’s likely your next letter is going to be an ‘e’. This reflects the low entropy of ‘the’ as an english word. It’s rather unlikely the next letter will be ‘j’, because English has few contexts that need ‘thj’ to be written out. So it will suggest words that start ‘the’ (and ‘tha’, and maybe even ‘thi’), while giving ‘thj’ and ‘thq’ and ‘thd’ a pass.

Westbury, Shaoul, Moroschan, and Ramscar found that the entropy of a word, how unlikely that collection of letters is to appear in an English word, matches quite well how funny people unfamiliar with it find it. This fits well with one of the more respectable theories of comedy, Arthur Schopenhauer’s theory that humor comes about from violating expectations. That matches well with unpredictability.

Of course it isn’t just entropy that makes words funny. Anyone trying to be funny learns that soon enough, since a string of perfect nonsense is also boring. But this is one of the things that can be measured and that does have an influence.

(I doubt there is any one explanation for why things are funny. My sense is that there are many different kinds of humor, not all of them perfectly compatible. It would be bizarre if any one thing could explain them all. But explanations for pieces of them are plausible enough.)

Anyway, I recommend looking at the Kristy Condon report. It explains the paper and the research in some more detail. And if you feel up to reading an academic paper, try Westbury, Shaoul, Moroschan, and Ramscar’s report. I thought it readable, even though so much of it is outside my field. And if all else fails there’s a list of two hundred made-up words used in field tests for funniness. Some of them look pretty solid to me.

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, October 29, 2015: Spherical Squirrel Edition


John Zakour and Scott Roberts’s Maria’s Day is going to Sunday-only publication. A shame, but I understand Zakour and Roberts choosing to focus their energies on better-paying venues. That those venues are “writing science fiction novels” says terrifying things about the economic logic of web comics.

This installment, from the 23rd, is a variation on the joke about the lawyer, or accountant, or consultant, or economist, who carefully asks “what do you want the answer to be?” before giving it. Sports are a rich mine of numbers, though. Mostly they’re statistics, and we might wonder: why does anyone care about sports statistics? Once the score of a game is done counted, what else matters? A sociologist and a sports historian are probably needed to give true, credible answers. My suspicion is that it amounts to money, as it ever does. If one wants to gamble on the outcomes of sporting events, one has to have a good understanding of what is likely to happen, and how likely it is to happen. And I suppose if one wants to manage a sporting event, one wants to spend money and time and other resources to best effect. That requires data, and that we see in numbers. And there are so many things that can be counted in any athletic event, aren’t there? All those numbers carry with them a hypnotic pull.

In Darrin Bell’s Candorville for the 24th of October, Lemont mourns how he’s forgotten how to do long division. It’s an easy thing to forget. For one, we have calculators, as Clyde points out. For another, long division ultimately requires we guess at and then try to improve an answer. It can’t be reduced to an operation that will never require back-tracking and trying some part of it again. That back-tracking — say, trying to put 28 into the number seven times, and finding it actually goes at least eight times — feels like a mistake. It feels like the sort of thing a real mathematician would never do.

And that’s completely wrong. Trying an answer, and finding it’s not quite right, and improving on it is perfectly sound mathematics. Arguably it’s the whole field of numerical mathematics. Perhaps students would find long division less haunting if they were assured that it is fine to get a wrong-but-close answer as long as you make it better.

John Graziano’s Ripley’s Believe It or Not for the 25th of October talks about the Rubik’s Cube, and all the ways it can be configured. I grant it sounds like 43,252,003,274,489,856,000 is a bit high a count of possible combinations. But it is about what I hear from proper mathematics texts, the ones that talk about group theory, so let’s let it pass.

The Rubik’s Cube gets talked about in group theory, the study of things that work kind of like arithmetic. In this case, turning one of the faces — well, one of the thirds of a face — clockwise or counterclockwise by 90 degrees, so the whole thing stays a cube, works like adding or subtracting one, modulo 4. That is, we pretend the only numbers are 0, 1, 2, and 3, and the numbers wrap around. 3 plus 1 is 0; 3 plus 2 is 1. 1 minus 2 is 3; 1 minus 3 is 2. There are several separate rotations that can be done, each turning a third of each face of the cube. That each face of the cube starts a different color means it’s easy to see how these different rotations interact and create different color patterns. And rotations look easy to understand. We can at least imagine rotating most anything. In the Rubik’s Cube we can look at a lot of abstract mathematics in a handheld and friendly-looking package. It’s a neat thing.

Scott Hilburn’s The Argyle Sweater for the 26th of October is really a physics joke. But it uses (gibberish) mathematics as the signifier of “a fully thought-out theory” and that’s good enough for me. Also the talk of a “big boing” made me giggle and I hope it does you too.

Izzy Ehnes’s The Best Medicine Cartoon makes, I believe, its debut for Reading the Comics posts with its entry for the 26th. It’s also the anthropomorphic-numerals joke for the week.

Frank Page’s Bob the Squirrel is struggling under his winter fur this week. On the 27th Bob tries to work out the Newtonian forces involved in rolling about in his condition. And this gives me the chance to share a traditional mathematicians joke and a cliche punchline.

The story goes that a dairy farmer knew he could be milking his cows better. He could surely get more milk, and faster, if only the operations of his farm were arranged better. So he hired a mathematician, to find the optimal way to configure everything. The mathematician toured every part of the pastures, the milking barn, the cows, everything relevant. And then the mathematician set to work devising a plan for the most efficient possible cow-milking operation. The mathematician declared, “First, assume a spherical cow.”

The punch line has become a traditional joke in the mathematics and science fields. As a joke it comments on the folkloric disconnection between mathematicians and practicality. It also comments on the absurd assumptions that mathematicians and scientists will make for the sake of producing a model, and for getting an answer.

The joke within the joke is that it’s actually fine to make absurd assumptions. We do it all the time. All models are simplifications of the real world, tossing away things that may be important to the people involved but that just complicate the work we mean to do. We may assume cows are spherical because that reflects, in a not too complicated way, that while they might choose to get near one another they will also, given the chance, leave one another some space. We may pretend a fluid has no viscosity, because we are interested in cases where the viscosity does not affect the behavior much. We may pretend people are fully aware of the costs, risks, and benefits of any action they wish to take, at least when they are trying to decide which route to take to work today.

That an assumption is ridiculous does not mean the work built on it is ridiculous. We must defend why we expect those assumptions to make our work practical without introducing too much error. We must test whether the conclusions drawn from the assumption reflect what we wanted to model reasonably well. We can still learn something from a spherical cow. Or a spherical squirrel, if that’s the case.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 28th of October is a binary numbers joke. It’s the other way to tell the joke about there being 10 kinds of people in the world. (I notice that joke made in the comments on Gocomics.com. That was inevitable.)

Eric the Circle for the 29th of October, this one by “Gilly” again, jokes about mathematics being treated as if quite subject to law. The truth of mathematical facts isn’t subject to law, of course. But the use of mathematics is. It’s obvious, for example, in the setting of educational standards. What things a member of society must know to be a functioning part of it are, western civilization has decided, a subject governments may speak about. Thus what mathematics everyone should know is a subject of legislation, or at least legislation in the attenuated form of regulated standards.

But mathematics is subject to parliament (or congress, or the diet, or what have you) in subtler ways. Mathematics is how we measure debt, that great force holding society together. And measurement again has been (at least in western civilization) a matter for governments. We accept the principle that a government may establish a fundamental unit of weight or fundamental unit of distance. So too may it decide what is a unit of currency, and into how many pieces the unit may be divided. And from this it can decide how to calculate with that currency: if the “proper” price of a thing would be, say, five-ninths of the smallest available bit of currency, then what should the buyer give the seller?

Who cares, you might ask, and fairly enough. I can’t get worked up about the risk that I might overpay four-ninths of a penny for something, nor feel bad that I might cheat a merchant out of five-ninths of a penny. But consider: when Arabic numerals first made their way to the west they were viewed with suspicion. Everyone at the market or the moneylenders’ knew how Roman numerals worked, and could follow addition and subtraction with ease. Multiplication was harder, but it could be followed. Division was a diaster and I wouldn’t swear that anyone has ever successfully divided using Roman numerals, but at least everything else was nice and familiar.

But then suddenly there was this influx of new symbols, only one of them something that had ever been a number before. One of them at least looked like the letter O, but it was supposed to represent a missing quantity. And every calculation on this was some strange gibberish where one unfamiliar symbol plus another unfamiliar symbol turned into yet another unfamiliar symbol or maybe even two symbols. Sure, the merchant or the moneylender said it was easier, once you learned the system. But they were also the only ones who understood the system, and the ones who would profit by making “errors” that could not be detected.

Thus we see governments, even in worldly, trade-friendly city-states like Venice, prohibiting the use of Arabic numerals. Roman numerals may be inferior by every measure, but they were familiar. They stood at least until enough generations passed that the average person could feel “1 + 1 = 2” contained no trickery.

If one sees in this parallels to the problem of reforming mathematics education, all I can offer is that people are absurd, and we must love the absurdness of them.

One last note, so I can get this essay above two thousand words somehow. In the 1910s Alfred North Whitehead and Bertrand Russell published the awesome and menacing Principia Mathematica. This was a project to build arithmetic, and all mathematics, on sound logical grounds utterly divorced from the great but fallible resource of human intuition. They did probably as well as human beings possibly could. They used a bewildering array of symbols and such a high level of abstraction that a needy science fiction movie could put up any random page of the text and pass it off as Ancient High Martian.

But they were mathematicians and philosophers, and so could not avoid a few wry jokes, and one of them comes in Volume II, around page 86 (it’ll depend on the edition you use). There, in Proposition 110.643, Whitehead and Russell establish “1 + 1 = 2” and remark, “the above proposition is occasionally useful”. They note at least three uses in their text alone. (Of course this took so long because they were building a lot of machinery before getting to mere work like this.)

Back in my days as a graduate student I thought it would be funny to put up a mock political flyer, demanding people say “NO ON PROP *110.643”. I was wrong. But the joke is strong enough if you don’t go to the trouble of making up the sign. I didn’t make up the sign anyway.

And to murder my own weak joke: arguably “1 + 1 = 2” is established much earlier, around page 380 of the first volume, in proposition *54.43. The thing is, that proposition warns that “it will follow, when mathematical addition has been defined”, which it hasn’t been at that point. But if you want to say it’s Proposition *54.43 instead go ahead; it will not get you any better laugh.

If you’d like to see either proof rendered as non-head-crushingly as possible, the Metamath Proof Explorer shows the reasoning for Proposition *54.43 as well as that for *110.643. And it contains hyperlinks so that you can try to understand the exact chain of reasoning which comes to that point. Good luck. I come from a mathematical heritage that looks at the Principia Mathematica and steps backward, quickly, before it has the chance to notice us and attack.

Reading the Comics, October 5, 2015: Boxes and Hyperboxes Edition


I’ve got more mathematically-themed comic strips than this to write about, but this should do for one day’s postings. Motley did give me the puzzle of figuring out whether the character’s description of a process could be made sensible, which is a bit of extra fun. Boxes and cubes come up in three of the comics, too.

John McPherson’s Close to Home for the 3rd of October drops in the abacus as a backup for the bank’s computers. It’s a cute enough idea. Deep down, I admit, I’m not sure that an abacus would be needed for most of the work a teller has to do during a temporary computer outage, though. Most of the calculations to do would be working out whether there’s enough money in the account to allow a given withdrawal. That’s database-checking, really. Also I’m not sure that’s a model of abacus that’s actually been made, but if I understood what was wanted, then in some ways wasn’t the artwork successful?

Larry Wright’s Motley Classics for the 3rd of October is a rerun from the same day in 1987. Debbie gives the terribly complicated instructions on how to calculate a tip. I’m not sure how tip-calculating got to the pop culture position of “most complicated thing people do with mathematics that isn’t taxes”. Probably that it is a fairly universal need for mathematics that isn’t taxes (and so seasonally bound) explains it. I think she’s describing a valid algorithm, though, if we make some assumptions about her pronouns.

Suppose we start with the price P. Double that and move the decimal one place over, to the left I suppose, and we have 0.20 times P. Suppose that this is the first answer. If we divide this first answer by four, then, this second answer will be 0.05 times P. And subtracting the second answer from the first is, indeed, 0.15 times P, or fifteen percent of the original price. While correct, though, it’s still a lousy algorithm. Too many steps, too much division, and subtraction is a challenge. Taking one-tenth the price plus half a tenth would be numerically identical and less challenging. Taking one-sixth the price would be a division, yes, but get you to near enough fifteen percent with only one move.

Mark Pett’s Lucky Cow for the 4th of October, another rerun, shows off one of the silly semantic-equation games that mathematics majors sometimes play. Forgive them. There’s a similar argument which proves that half a ham sandwich is greater than God. It all amounts to playing on arguments which might (not always!) be correct in form but have things with silly meanings plugged into them.

Stephan Pastis’s Pearls Before Swine for the 4th of October gives Pig the chance to panic. It’s another strip about the difference between what “positive” and “negative” mean in inference testing, and so in medical testing, versus the connotations of “good” and “`bad” they have. I’ve explained this before, in other Reading the Comics essays, so I’ll spare the whole thing. But in short, “positive” in this case means “these test results are so far away from normal values that it strains plausibility to think it’s normal”. “Negative” means “these test results are not so far away from normal values as to strain plausibility to think it’s normal”.

Geoff Grogan’s Jetpack Jr for the 5th of October draws a hypercube as the box little alien Jetpack Junior arrived in. Well, these are some of the common representations of how a four-dimensional cube would look in our three-dimensional space (and that, rendered on a two-dimensional screen). The difficult-to-conceptualize part is that in the cube, seen in the middle third of the strip, every one of the red lines is the same length, and is perpendicular to all its neighbors. The triptych of shapes are all the same four-dimensional cube, too, just rotated along different axes by different amounts.

All my old links to play with hypercube rotations seem to have expired or turn out to be Java applets. Here’s a page that offers a couple of pictures, though. It has a link to an iOS app that should let people play with rotating a four-dimensional hypercube. Might enjoy it. I think this is the first time Jetpack Jr as such has got around here. It used to run as Plastic Babyheads from Outer Space, with a silly overarching story about aliens with plastic baby heads, ah, invading. I don’t think that made the Reading the Comics roster, though, unless some of the aliens mentioned pi, which they might have done.

Charles Brubaker’s Ask A Cat for the 5th of October I think is another debuting strip around here. It’s about the problem of Schrödinger’s Cat, a thought-experiment designed to show we don’t really understand what the conventional mathematical models of quantum mechanics mean. In at least some views, the mathematics of quantum mechanics suggests we could have an apparently ridiculous result: something big, like a cat, that we expect should work like a classical-physics entity, behaving instead like a quantum-mechanical entity, with no definable state. The problem has been with us for eighty years and isn’t well-answered, but that happens. Zeno’s paradoxes have been with us three thousand years and are still showing us things we don’t quite understand about divisibility and continuity.

Anthony Smith’s Learn to Speak Cat for the 5th of October is a completely different cat comic strip that I think is making a debut here. This is more a matter of silly symbolic manipulation than anything serious, though.

Tom Toles’s Randolph Itch, 2 am from the 5th of October is a rerun from 1999. And it shows a soap-bubble cube. Soap bubbles allow for some neat mathematics. They act like animate computers working out the way to enclose a given volume with the least surface area. A web site written by Dr Michael Hutchings at the University of California/Berkeley describes some of the mathematical work involved. Surprising to me is that it was only in the 1970s that the “double bubble conjecture” was proven. That’s a question about how to cover a given volume using two bubbles. The answer is what you might get from playing with soap bubble wands, but it took about a century of working on to prove. Granting, mathematicians did other things with their time, so it wasn’t uninterrupted soap-bubble work. Hutchings includes some review of the field as it existed in the early 2000s, and lists three open problems. The first of them is one that’s understandable even without knowing more mathematical lingo than what R3 is. (And folks who’re hanging around here know that by now.) Also it has pictures of soap bubbles, which are good for a lazy Friday morning.

Reading the Comics, September 6, 2015: September 6, 2015 Edition


Well, we had another of those days where Comic Strip Master Command ordered everybody to do mathematics jokes. I’ll survive.

Henry is frustrated with his arithmetic, until he goes to the pool hall and counts off numbers on those score chips.
Don Trachte’s Henry for the 6th of September, 2015.

Don Trachte’s Henry is a reminder that arithmetic, like so many things, is easier to learn when you’re comfortable with the context. Personally I’ve never understood why some of the discs on pool scoring racks are different colors but imagine it relates to scoring values, somehow. I’ve encountered multiple people who assume I must be good at pool, since it’s all geometry, and what isn’t just geometry is physics. I’ve disappointed them all so far.

Tony Rubino and Gary Markstein’s Daddy’s Home uses arithmetic as an example of joy-crushing school drudgery. It could’ve as easily been asking the capital of Montana.

Scott Adams’s Dilbert Classics, a rerun from the 29th of June, 1992, has Dilbert make a breakthrough in knot theory. The fundamental principle is correct: there are many knots that one could use for tying shoelaces, just as there are many knots that could be used for tying ties. Discovering new ones is a good ways for knot theorists to get a bit of harmless publicity. Nobody needs them. From a knot-theory perspetive it also doesn’t matter if you lace the shoe’s holes crosswise or ladder-style. There are surely other ways to lace the holes, too, but nobody needs them either.

Maria Scrivan’s Half Full uses a blackboard full of mathematical symbols and name-drops Common Core. Fifty years ago this same joke was published, somewhere, with “Now solve it using the New Math” as punchline. Thirty years from now it will run again, with “Now solve it using the (insert name here)” as punchline. Some things are eternal truths.

T Lewis and Michael Fry’s Over The Hedge presents one of those Cretan paradox-style logic problems. Anyway, I choose to read it as such. I’m tickled by it.

And to close things out, both Leigh Rubin’s Rubes and Mikael Wulff and Anders Morgenthaler’s WuMo did riffs on the story of Newton and the falling apple. Is this truly mathematically-themed? Well, it’s tied to the legend of calculus’s origin, so that’s near enough for me.

Reading the Comics, July 24, 2015: All The Popular Topics Are Here Edition


This week all the mathematically-themed comic strips seem to have come from Gocomics.com. Since that gives pretty stable URLs I don’t feel like I can include images of those comics. So I’m afraid it’s a bunch of text this time. I like to think you enjoy reading the text, though.

Mark Anderson’s Andertoons seemed to make its required appearance here with the July 20th strip. And the kid’s right about parentheses being very important in mathematics and “just” extra information in ordinary language. Parentheses as a way of grouping together terms appear as early as the 16th century, according to Florian Cajori. But the symbols wouldn’t become common for a couple of centuries. Cajori speculates that the use of parentheses in normal rhetoric may have slowed mathematicians’ acceptance of them. Vinculums — lines placed over a group of terms — and colons before and after the group seem to have been more popular. Leonhard Euler would use parentheses a good bit, and that settled things. Besides all his other brilliances, Euler was brilliant at setting notation. There are still other common ways of aggregating terms. But most of them are straight brackets or curled braces, which are almost the smallest possible changes from parentheses you can make.

Though his place was secure, Mark Anderson got in another strip the next day. This one’s based on the dangers of extrapolating mindlessly. One trouble with extrapolation is that if we just want to match the data we have then there are literally infinitely many possible extrapolations, each equally valid. But most of them are obvious garbage. If the high temperature the last few days was 78, 79, 80, and 81 degrees Fahrenheit, it may be literally true that we could extrapolate that to a high of 120,618 degrees tomorrow, but we’d be daft to believe it. If we understand the factors likely to affect our data we can judge what extrapolations are plausible and what ones aren’t. As ever, sanity checking, verifying that our results could be correct, is critical.

Bill Amend’s FoxTrot Classics (July 20) continues Jason’s attempts at baking without knowing the unstated assumptions of baking. See above comments about sanity checking. At least he’s ruling out the obviously silly rotation angle. (The strip originally ran the 22nd of July, 2004. You can see it in color, there, if you want to see things like that.) Some commenters have gotten quite worked up about Jason saying “degrees Kelvin” when he need only say “Kelvin”. I can’t join them. Besides the phenomenal harmlessness of saying “degrees Kelvin”, it wouldn’t quite flow for Jason to think “350 degrees” short for “350 Kelvin” instead of “350 degrees Kelvin”.

Nate Frakes’s Break of Day (July 21) is the pure number wordplay strip for this roundup. This might be my favorite of this bunch, mostly because I can imagine the way it would be staged as a bit on The Muppet Show or a similar energetic and silly show. John Atkinson’s Wrong Hands for July 23 is this roundup’s general mathematics wordplay strip. And Mark Parisi’s Off The Mark for July 22nd is the mathematics-literalist strip for this roundup.

Ruben Bolling’s Tom The Dancing Bug (July 23, rerun) is nominally an economics strip. Its premise is that since rational people do what maximizes their reward for the risk involved, then pointing out clearly how the risks and possible losses have changed will change their behavior. Underlying this are assumptions from probability and statistics. The core is the expectation value. That’s an average of what you might gain, or lose, from the different outcomes of something. That average is weighted by the probability of each outcome. A strictly rational person who hadn’t ruled anything in or out would be expected to do the thing with the highest expected gain, or the smallest expected loss. That people do not do things this way vexes folks who have not known many people.

Reading the Comics, July 19, 2015: Rerun Comics Edition


I’m stepping my blog back away from the daily posting schedule. It’s fun, but it’s also exhausting. Sometimes, Comic Strip Master Command helps out. It slowed the rate of mathematically-themed comics just enough.

By this post’s title I don’t mean that my post is a rerun. But several of the comics mentioned happen to be. One of the good — maybe best — things about the appearance of comics on Gocomics.com and ComicsKingdom is that comic strips that have ended, such as Randolph Itch, 2 am or (alas) Cul de Sac can still appear without taking up space. And long-running comic strips such as Luann can have earlier strips be seen to a new audience, again without doing any harm to the newest generation of cartoonists. So, there’s that.

Greg Evans’s Luann Againn (July 13, originally run July 13, 1987) makes a joke of Tiffany not understanding the odds of a contest. That’s amusing enough. Estimating the probability of something happening does require estimating how many things are possible, though, and how likely they are relative to one another. Supposing that every entry in a sweepstakes is equally likely to win seems fair enough. Estimating the number of sweepstakes entries is another problem.

Tom Toles’s Randolph Itch, 2 am (July 13, rerun from July 29, 2002) tells a silly little pirates-and-algebra joke. I like this one for the silliness and the artwork. The only sad thing is there wasn’t a natural way to work equations for a circle into it, so there’d be a use for “r”.

Continue reading “Reading the Comics, July 19, 2015: Rerun Comics Edition”

The Alice in Wonderland Sesquicentennial


I did not realize it was the 150th anniversary of the publication of Alice in Wonderland, which is probably the best-liked piece of writing by any mathematician. At least it’s the only one I can think of that’s clearly inspired a Betty Boop cartoon. I’ve had cause to talk about Carroll’s writing about logic and some other topics in the past. (One was just a day short of three years ago, by chance.)

As mentioned in his tweet John Allen Paulos reviewed a book entirely about Lewis Carroll/Charles Dodgson’s mathematical and logic writing. I was unaware of the book before, but am interested now.

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


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

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

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

Life at the quantum level: one subatomic particle suspects the other of being unfaithful because both know he could be in two places at once.
Niklas Eriksson’s Carpe Diem for the 20th of June, 2015.

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

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

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

Little Iodine does badly in arithmetic in class. But she's very good at counting the calories, and the cost, of what her teacher eats.
Jimmy Hatlo’s Little Iodine for the 18th of April, 1954, and rerun the 18th of June, 2015.

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

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

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

Reading the Comics, June 11, 2015: Bonus Education Edition


The coming US summer vacation suggests Comic Strip Master Command will slow down production of mathematics-themed comic strips. But they haven’t quite yet. And this week I also found a couple comics that, while not about mathematics, amused me enough that I want to include them anyway. So those bonus strips I’ll run at the end of my regular business here.

Bill Hinds’s Tank McNamara (June 6) does a pi pun. The pithon mathematical-snake idea is fun enough and I’d be interested in a character design. I think the strip’s unjustifiably snotty about tattoos. But comic strips have a strange tendency to get snotty about other forms of art.

A friend happened to mention one problem with tattoos that require straight lines or regular shapes is that human skin has a non-flat Gaussian curvature. Yes, that’s how the friend talks. Gaussian curvature is, well, a measure of how curved a surface is. That sounds obvious enough, but there are surprises: a circular cylinder, such as the label of a can, has the same curvature as a flat sheet of paper. You can see that by how easy it is to wrap a sheet of paper around a can. But a ball hasn’t, and you see that by how you can’t neatly wrap a sheet of paper around a ball without crumpling or tearing the paper. Human skin is kind of cylindrical in many places, but not perfectly so, and it changes as the body moves. So any design that looks good on paper requires some artistic imagination to adapt to the skin.

Bill Amend’s FoxTrot (June 7) sets Jason and Marcus working on their summer tans. It’s a good strip for adding to the cover of a trigonometry test as part of the cheat-sheet.

Dana Simpson’s Phoebe and her Unicorn (June 8) makes what I think is its first appearance in my Reading the Comics series. The strip, as a web comic, had been named Heavenly Nostrils. Then it got the vanishingly rare chance to run as a syndicated newspaper comic strip. And newspaper comics page editors don’t find the word “nostril” too inherently funny to pass up. Thus the more marketable name. After that interesting background I’m sad to say Simpson delivers a bog-standard “kids not understanding fractions” joke. I can’t say much about that.

Ruben Bolling’s Super Fun-Pak Comix (June 10, rerun) is an installment of everyone’s favorite literary device model of infinite probabilities. A Million Monkeys At A Million Typewriters subverts the model. A monkey thinking about the text destroys the randomness that it depends upon. This one’s my favorite of the mathematics strips this time around.

And Dan Thompson’s traditional Brevity appearance is the June 11th strip, an Anthropomorphic Numerals joke combining a traditional schoolyard gag with a pun I didn’t notice the first time I read the panel.


And now here’s a couple strips that aren’t mathematical but that I just liked too much to ignore. Also this lets Mark Anderson’s Andertoons get back on my page. The June 10th strip is a funny bit of grammar play.

Percy Crosby’s Skippy (June 6, rerun from sometime in 1928) tickles me for its point about what you get at the top and the bottom of the class. Although tutorials and office hours and extracurricular help, and automated teaching tools, do customize things a bit, teaching is ultimately a performance given to an audience. Some will be perfectly in tune with the performance, and some won’t. Audiences are like that.

What Is 13 Times 7?


AbyssBrain, author of the Mathemagical Site blog on WordPress, commented on that 2-plus-2-equals-5 post a couple days ago with a link to an Abbot and Costello Show sketch, in which Lou Costello proves to the landlord that 13 times 7 equals 28. And better than that, he does it three different ways. I didn’t want something fun as that to languish in the comments, so please, enjoy it here on the front page.

I have always liked comedy sketches about complicated chains of mock reasoning so this sort of thing is designed just for me.

Reading the Comics, February 24, 2014: Getting Caught Up Edition


And now, I think, I’ve got caught up on the mathematics-themed comics that appeared at Comics Kingdom and at Gocomics.com over the past week and a half. I’m sorry to say today’s entries don’t get to be about as rich a set of topics as the previous bunch’s, but on the other hand, there’s a couple Comics Kingdom strips that I feel comfortable using as images, so there’s that. And come to think of it, none of them involve the setup of a teacher asking a student in class a word problem, so that’s different.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (February 21) tells the old joke about how much of fractions someone understands. To me the canonical version of the joke was a Sydney Harris panel in which one teacher complains that five-thirds of the class doesn’t understand a word she says about fractions, but it’s all the same gag. I’m a touch amused that three and five turn up in this version of the joke too. That probably reflects writing necessity — especially for this B.C. the numbers have to be a pair that obviously doesn’t give you one-half — and that, somehow, odd numbers seem to read as funnier than even ones.

Bud Fisher’s Mutt and Jeff (February 21) decimates one of the old work-rate problems, this one about how long it takes a group of people to eat a pot roast. It was surely an old joke even when this comic first appeared (and I can’t tell you when it was; Gocomics.com’s reruns have been a mixed bunch of 1940s and 1950s ones, but they don’t say when the original run date was), but the spread across five panels treats the joke well as it’s able to be presented as a fuller stage-ready sketch. Modern comic strips value an efficiently told, minimalist joke, but pacing and minor punch lines (“some men don’t eat as fast as others”) add their charm to a comic.

Continue reading “Reading the Comics, February 24, 2014: Getting Caught Up Edition”

Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition


So, the mathematics comics ran away from me a little bit, and I didn’t have the chance to write up a proper post on Thursday or Friday. So I’m writing what I probably would have got to on Friday had time allowed, and there’ll be another in this sequence sooner than usual. I hope you’ll understand.

The title for this entry is basically thanks to Zach Weinersmith, because his comics over the past week gave me reasons to talk about Georg Cantor and Bernard Riemann. These were two of the many extremely sharp, extremely perceptive German mathematicians of the 19th Century who put solid, rigorously logical foundations under the work of centuries of mathematics, only to discover that this implied new and very difficult questions about mathematics. Some of them are good material for jokes.

Eric and Bill Teitelbaum’s Bottomliners panel (February 14) builds a joke around everything in some set of medical tests coming back negative, as well as the bank account. “Negative”, the word, has connotations that are … well, negative, which may inspire the question why is it a medical test coming back “negative” corresponds with what is usually good news, nothing being wrong? As best I can make out the terminology derives from statistics. The diagnosis of any condition amounts to measuring some property (or properties), and working out whether it’s plausible that the measurements could reflect the body’s normal processes, or whether they’re such that there just has to be some special cause. A “negative” result amounts to saying that we are not forced to suppose something is causing these measurements; that is, we don’t have a strong reason to think something is wrong. And so in this context a “negative” result is the one we ordinarily hope for.

Continue reading “Reading the Comics, February 20, 2015: 19th-Century German Mathematicians Edition”

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


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

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

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

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

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

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

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

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

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

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

Reading The Comics, October 20, 2014: No Images This Edition


Since I started including Comics Kingdom strips in my roundups of mathematically-themed strips I’ve been including images of those, because I’m none too confident that Comics Kingdom’s pages are accessible to normal readers after some time has passed. Gocomics.com has — as far as I’m aware, and as far as anyone has told me — no such problems, so I haven’t bothered doing more than linking to them. So this is the first roundup in a long while I remember that has only Gocomics strips, with nothing from Comics Kingdom. It’s also the first roundup for which I’m fairly sure I’ve done one of these strips before.

Guy Endore-Kaiser and Rodd Perry and Dan Thompson’s Brevity (October 15, but a rerun) is an entry in the anthropomorphic-numbers line of mathematics comics, and I believe it’s one that I’ve already mentioned in the past. This particular strip is a rerun; in modern times the apparently indefatigable Dan Thompson has added this strip to the estimated fourteen he does by himself. In any event it stands out in the anthropomorphic-numbers subgenre for featuring non-integers that aren’t pi.

Ralph Hagen’s The Barn (October 16) ponders how aliens might communicate with Earthlings, and like pretty much everyone who’s considered the question mathematics is supposed to be the way they’d do it. It’s easy to see why mathematics is plausible as a universal language: a mathematical truth should be true anywhere that deductive logic holds, and it’s difficult to conceive of a universe existing in which it could not hold true. I have somewhere around here a mention of a late-19th-century proposal to try contacting Martians by planting trees in Siberia which, in bloom, would show a proof of the Pythagorean theorem.

In modern times we tend to think of contact with aliens being done by radio more likely (or at least some modulated-light signal), which makes a signal like a series of pulses counting out prime numbers sound likely. It’s easy to see why prime numbers should be interesting too: any species that has understood multiplication has almost certainly noticed them, and you can send enough prime numbers in a short time to make clear that there is a deliberate signal being sent. For comparison, perfect numbers — whose factors add up to the original number — are also almost surely noticed by any species that understands multiplication, but the first several of those are 6, 28, 496, and 8,128; by the time 8,128 pulses of anything have been sent the whole point of the message has been lost.

And yet finding prime numbers is still not really quite universal. You or I might see prime numbers as key, but why not triangular numbers, like the sequence 1, 3, 6, 10, 15? Why not square or cube numbers? The only good answer is, well, we have to pick something, so to start communicating let’s hope we find something that everyone will be able to recognize. But there’s an arbitrariness that can’t be fully shed from the process.

John Zakour and Scott Roberts’s Maria’s Day (October 17) reminds us of the value of having a tutor for mathematics problems — if you’re having trouble in class, go to one — and of paying them appropriately.

Steve Melcher’s That Is Priceless (October 17) puts comic captions to classic paintings and so presented Jusepe de Ribera’s 1630 Euclid, Letting Me Copy His Math Homework. I confess I have a broad-based ignorance of art history, but if I’m using search engines correctly the correct title was actually … Euclid. Hm. It seems like Melcher usually has to work harder at these things. Well, I admit it doesn’t quite match my mental picture of Euclid, but that would have mostly involved some guy in a toga wielding a compass. Ribera seems to have had a series of Greek Mathematician pictures from about 1630, including Pythagoras and Archimedes, with similar poses that I’ll take as stylized representations of the great thinkers.

Mark Anderson’s Andertoons (October 18) plays around statistical ideas that include expectation values and the gambler’s fallacy, but it’s a good puzzle: has the doctor done the procedure hundreds of times without a problem because he’s better than average at it, or because he’s been lucky? For an alternate formation, baseball offers a fine question: Ted Williams is the most recent Major League Baseball player to have a season batting average over .400, getting a hit in at least two-fifths of his at-bats over the course of the season. Was he actually good enough to get a hit that often, though, or did he just get lucky? Consider that a .250 hitter — with a 25 percent chance of a hit at any at-bat — could quite plausibly get hits in three out of his four chances in one game, or for that matter even two or three games. Why not a whole season?

Well, because at some point it becomes ridiculous, rather the way we would suspect something was up if a tossed coin came up tails thirty times in a row. Yes, possibly it’s just luck, but there’s good reason to suspect this coin doesn’t have a fifty percent chance of coming up heads, or that the hitter is likely to do better than one hit for every four at-bats, or, to the original comic, that the doctor is just better at getting through the procedure without complications.

Ryan North’s quasi-clip-art Dinosaur Comics (October 20) thrilled the part of me that secretly wanted to study language instead by discussing “light verb constructions”, a grammatical touch I hadn’t paid attention to before. The strip is dubbed “Compressed Thesis Comics”, though, from the notion that the Tyrannosaurus Rex is inspired to study “computationally” what forms of light verb construction are more and what are less acceptable. The impulse is almost perfect thesis project, really: notice a thing and wonder how to quantify it. A good piece of this thesis would probably be just working out how to measure acceptability of a particular verb construction. I imagine the linguistics community has a rough idea how to measure these or else T Rex is taking on way too big a project for a thesis, since that’d be an obvious point for the thesis to crash against.

Well, I still like the punch line.

Reading the Comics, October 14, 2014: Not Talking About Fourier Transforms Edition


I know that it’s disappointing to everyone, given that one of the comic strips in today’s roundup of mathematically-themed such gives me such a good excuse to explain what Fourier Transforms are and why they’re interesting and well worth the time learning. But I’m not going to do that today. There’s enough other things to think about and besides you probably aren’t going to need Fourier Transforms in class for a couple more weeks yet. For today, though, no, I’ll go on to other things instead. Sorry to disappoint.

Glen McCoy and Gary McCoy’s The Flying McCoys (October 9) jokes about how one can go through life without ever using algebra. I imagine other departments get this, too, like, “I made it through my whole life without knowing anything about US History!” or “And did any of that time I spent learning Art do anything for me?” I admit a bias here: I like learning stuff even if it isn’t useful because I find it fun to learn stuff. I don’t insist that you share in finding that fun, but I am going to look at you weird if you feel some sense of triumph about not learning stuff.

Tom Thaves’s Frank and Ernest (October 10) does a gag about theoretical physics, and string theory, which is that field where physics merges almost imperceptibly into mathematics and philosophy. The rough idea of string theory is that it’d be nice to understand why the particles we actually observe exist, as opposed to things that we could imagine existing that that don’t seem to — like, why couldn’t there be something that’s just like an electron, but two times as heavy? Why couldn’t there be something with the mass of a proton but three-quarters the electric charge? — by supposing that what we see are the different natural modes of behavior of some more basic construct, these strings. A natural mode is, well, what something will do if it’s got a bunch of energy and is left to do what it will with it.

Probably the most familiar kind of natural mode is how if you strike a glass or a fork or such it’ll vibrate, if we’re lucky at a tone we can hear, and if we’re really lucky, at one that sounds good. Things can have more than one natural mode. String theory hopes to explain all the different kinds of particles, and the different ways in which they interact, as being different modes of a hopefully small and reasonable variety of “strings”. It’s a controversial theory because it’s been very hard to find experiments that proves, or soundly rules out, a particular model of it as representation of reality, and the models require invoking exotic things like more dimensions of space than we notice. This could reflect string theory being an intriguing but ultimately non-physical model of the world; it could reflect that we just haven’t found the right way to go about proving it yet.

Charles Schulz’s Peanuts (October 10, originally run October 13, 1967) has Sally press Charlie Brown into helping her with her times tables. She does a fair bit if guessing, which isn’t by itself a bad approach. For one, if you don’t know the exact answer, but you can pin down a lower and and upper bound, you’re doing work that might be all you really need and you’re doing work that may give you a hint how to get what you really want. And for that matter, guessing at a solution can be the first step to finding one. One of my favorite areas of mathematics, Monte Carlo methods, finds solutions to complicated problems by starting with a wild guess and making incremental refinements. It’s not guaranteed to work, but when it does, it gets extremely good solutions and with a remarkable ease. Granted this, doesn’t really help the times tables much.

On the 11th (originally run October 14, 1967), Sally incidentally shows the hard part of refining guesses about a solution; there has to be some way of telling whether you’re getting warmer. In your typical problem for a Monte Carlo approach, for example, you have some objective function — say, the distance travelled by something going along a path, or the total energy of a system — and can measure whether an attempted change is improving your solution — say, minimizing your distance or reducing the potential energy — or is making it worse. Typically, you take any refinement that makes the provisional answer better, and reject most, but not all, refinements that make the provisional answer worse.

That said, “Overly-Eight” is one of my favorite made-up numbers. A “Quillion” is also a pretty good one.

Jeff Mallet’s Frazz (October 12) isn’t explicitly about mathematics, but it’s about mathematics. “Why do I have to show my work? I got the right answer?” There are good responses on two levels, the first of which is practical, and which blends into the second: if you give me-the-instructor the wrong answer then I can hopefully work out why you got it wrong. Did you get it wrong because you made a minor but ultimately meaningless slip in your calculations, or did you get it wrong because you misunderstood the problem and did not know what kind of calculation to do? Error comes in many forms; some are boring — wrote the wrong number down at the start and never noticed, missed a carry — some are revealing — doesn’t know the order of operations, doesn’t know how the chain rule applies in differentiation — and some are majestic.

These last are the great ones, the errors that I love seeing, even though they’re the hardest to give a fair grade to. Sometimes a student will go off on a tack that doesn’t look anything like what we did in class, or could have reasonably seen in the textbook, but that shows some strange and possibly mad burst of creative energy. Usually this is rubbish and reflects the student flailing around, but, sometimes the student is on to something, might be trying an approach that, all right, doesn’t work here, but which if it were cleaned of its logical flaws might be a new and different way to work out the problem.

And that blends to the second reason: finding answers is nice enough and if you’re good at that, I’m glad, but is it all that important? We have calculators, after all. What’s interesting, and what is really worth learning in mathematics, is how to find answers: what approaches can efficiently be used on this problem, and how do you select one, and how do you do it to get a correct answer? That’s what’s really worth learning, and what is being looked for when the instruction is to show your work. Caulfield had the right answer, great, but is it because he knew a good way to work out the problem, or is it because he noticed the answer was left on the blackboard from the earlier class when this one started, or is it because he guessed and got lucky, or is it because he thought of a clever new way to solve the problem? If he did have a clever new way to do the problem, shouldn’t other people get to see it? Coming up with clever new ways to find answers is the sort of thing that gets you mathematical immortality as a pioneer of some approach that gets mysteriously named for somebody else.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 14) makes fun of tenure, the process by which people with a long track record of skill, talent, and drive are rewarded with no longer having to fear being laid off or fired except for cause. (Though I should sometime write about Fourier Transforms, as they’re rather neat.)

'Albert, stop daydreaming and eat your soup', which is alphabet soup, apparently, and where you could find E = m c c if you looked just right.
Margaret Shulock’s Six Chix comic for the 14th of October, 2014: Albert Einstein is evoked alongside the origins of his famous equation about c and m and soup and stuff.

Margaret Shulock’s turn at Six Chix (October 14) (the comic strip is shared among six women because … we couldn’t have six different comic strips written and drawn by women all at the same time, I guess?) evokes the classic image of Albert Einstein, the genius, and drawing his famous equation out of the ordinary stuff of daily life. (I snark a little; Shulock is also the writer for Apartment 3-G, to the extent that things can be said to be written in Apartment 3-G.)

Reading the Comics, September 28, 2014: Punning On A Sunday Edition


I honestly don’t intend this blog to become nothing but talk about the comic strips, but then something like this Sunday happens where Comic Strip Master Command decided to send out math joke priority orders and what am I to do? And here I had a wonderful bit about the natural logarithm of 2 that I meant to start writing sometime soon. Anyway, for whatever reason, there’s a lot of punning going on this time around; I don’t pretend to explain that.

Jason Poland’s Robbie and Bobby (September 25) puns off of a “meth lab explosion” in a joke that I’ve seen passed around Twitter and the like but not in a comic strip, possibly because I don’t tend to read web comics until they get absorbed into the Gocomics.com collective.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (September 26) shows how an infinity pool offers the chance to finally, finally, do a one-point perspective drawing just like the art instruction book says.

Bill Watterson’s Calvin and Hobbes (September 27, rerun) wrapped up the latest round of Calvin not learning arithmetic with a gag about needing to know the difference between the numbers of things and the values of things. It also surely helps the confusion that the (United States) dime is a tiny coin, much smaller in size than the penny or nickel that it far out-values. I’m glad I don’t have to teach coin values to kids.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 27) mentions Lagrange points. These are mathematically (and physically) very interesting because they come about from what might be the first interesting physics problem. If you have two objects in the universe, attracting one another gravitationally, then you can describe their behavior perfectly and using just freshman or even high school calculus. For that matter, describing their behavior is practically what Isaac Newton invented his calculus to do.

Add in a third body, though, and you’ve suddenly created a problem that just can’t be done by freshman calculus, or really, done perfectly by anything but really exotic methods. You’re left with approximations, analytic or numerical. (Karl Fritiof Sundman proved in 1912 that one could create an infinite series solution, but it’s not a usable solution. To get a desired accuracy requires so many terms and so much calculation that you’re better off not using it. This almost sounds like the classical joke about mathematicians, coming up with solutions that are perfect but unusable. It is the most extreme case of a possible-but-not-practical solution I’m aware of, if stories I’ve heard about its convergence rate are accurate. I haven’t tried to follow the technique myself.)

But just because you can’t solve every problem of a type doesn’t mean you can’t solve some of them, and the ones you do solve might be useful anyway. Joseph-Louis Lagrange did that, studying the problem of one large body — like a sun, or a planet — and one middle-sized body — a planet, or a moon — and one tiny body — like an asteroid, or a satellite. If the middle-sized body is orbiting the large body in a nice circular orbit, then, there are five special points, dubbed the Lagrange points. A satellite that’s at one of those points (with the right speed) will keep on orbiting at the same rotational speed that the middle body takes around the large body; that is, the system will turn as if the large, middle, and tiny bodies were fixed in place, relative to each other.

Two of these spots, dubbed numbers 4 and 5, are stable: if your tiny body is not quite in the right location that’s all right, because it’ll stay nearby, much in the same way that if you roll a ball into a pit it’ll stay in the pit. But three of these spots, numbers 1, 2, and 3, are unstable: if your tiny body is not quite on those spots, it’ll fall away, in much the same way if you set a ball on the peak of the roof it’ll roll off one way or another.

When Lagrange noticed these points there wasn’t any particular reason to think of them as anything but a neat mathematical construct. But the points do exist, and they can be stable even if the medium body doesn’t have a perfectly circular orbit, or even if there are other planets in the universe, which throws off the nice simple calculations yet. Something like 1700 asteroids are known to exist in the number 4 and 5 Lagrange points for the Sun and Jupiter, and there are a handful known for Saturn and Neptune, and apparently at least five known for Mars. For Earth apparently there’s just the one known to exist, catchily named 2010 TK7, discovered in October 2010, although I’d be surprised if that were the only one. They’re just small.

Professor Peter Peddle has the crazy idea of studying boxing scientifically and preparing strategy accordingly.
Elliot Caplin and John Cullen Murphy’s Big Ben Bolt, from the 23rd of August, 1953 (rerun the 28th of September, 2014).

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt (September 28, originally run August 23, 1953) has been on the Sunday strips now running a tale about a mathematics professor, Peter Peddle, who’s threatening to revolutionize Big Ben Bolt’s boxing world by reducing it to mathematical abstraction; past Sunday strips have even shown the rather stereotypically meek-looking professor overwhelming much larger boxers. The mathematics described here is nonsense, of course, but it’d be asking a bit of the comic strip writers to have a plausible mathematical description of the perfect boxer, after all.

But it’s hard for me anyway to not notice that the professor’s approach is really hard to gainsay. The past generation of baseball, particularly, has been revolutionized by a very mathematical, very rigorous bit of study, looking at questions like how many pitches can a pitcher actually throw before he loses control, and where a batter is likely to hit based on past performance (of this batter and of batters in general), and how likely is this player to have a better or a worse season if he’s signed on for another year, and how likely is it he’ll have a better enough season than some cheaper or more promising player? Baseball is extremely well structured to ask these kinds of questions, with football almost as good for it — else there wouldn’t be fantasy football leagues — and while I am ignorant of modern boxing, I would be surprised if a lot of modern boxing strategy weren’t being studied in Professor Peddle’s spirit.

Eric the Circle (September 28), this one by Griffinetsabine, goes to the Shapes Singles Bar for a geometry pun.

Bill Amend’s FoxTrot (September 28) (and not a rerun; the strip is new runs on Sundays) jumps on the Internet Instructional Video bandwagon that I’m sure exists somewhere, with child prodigy Jason Fox having the idea that he could make mathematics instruction popular enough to earn millions of dollars. His instincts are probably right, too: instructional videos that feature someone who looks cheerful and to be having fun and maybe a little crazy — well, let’s say eccentric — are probably the ones that will be most watched, at least. It’s fun to see people who are enjoying themselves, and the odder they act the better up to a point. I kind of hate to point out, though, that Jason Fox in the comic strip is supposed to be ten years old, implying that (this year, anyway) he was born nine years after Bob Ross died. I know that nothing ever really goes away anymore, but, would this be a pop culture reference that makes sense to Jason?

Tom Thaves’s Frank and Ernest (September 28) sets up the idea of Euclid as a playwright, offering a string of geometry puns.

Jef Mallet’s Frazz (September 28) wonders about why trains show up so often in story problems. I’m not sure that they do, actually — haven’t planes and cars taken their place here, too? — although the reasons aren’t that obscure. Questions about the distance between things changing over time let you test a good bit of arithmetic and algebra while being naturally about stuff it’s reasonable to imagine wanting to know. What more does the homework-assigner want?

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 28) pops back up again with the prospect of blowing one’s mind, and it is legitimately one of those amazing things, that e^{i \pi} = -1 . It is a remarkable relationship between a string of numbers each of which are mind-blowing in their ways — negative 1, and pi, and the base of the natural logarithms e, and dear old i (which, multiplied by itself, is equal to negative 1) — and here they are all bundled together in one, quite true, relationship. I do have to wonder, though, whether anyone who would in a social situation like this understand being told “e raised to the i times pi power equals negative one”, without the framing of “we’re talking now about exponentials raised to imaginary powers”, wouldn’t have already encountered this and had some of the mind-blowing potential worn off.

Reading The Comics, September 24, 2014: Explained In Class Edition


I’m a fan of early 20th century humorist Robert Benchley. You might not be yourself, but it’s rather likely that among the humorists you do like are a good number of people who are fans of his. He’s one of the people who shaped the modern American written-humor voice, and as such his writing hasn’t dated, the way that, for example, a 1920s comic strip will often seem to come from a completely different theory of what humor might be. Among Benchley’s better-remembered quotes, and one of those striking insights into humanity, not to mention the best productivity tip I’ve ever encountered, was something he dubbed the Benchley Principle: “Anyone can do any amount of work, provided it isn’t the work he is supposed to be doing at the moment.” One of the comics in today’s roundup of mathematics-themed comics brought the Benchley Principle to mind, and I mean to get to how it did and why.

Eric The Circle (by ‘Griffinetsabine’ this time) (September 18) steps again into the concerns of anthropomorphized shapes. It’s also got a charming-to-me mention of the trapezium, the geometric shape that’s going to give my mathematics blog whatever immortality it shall have.

Bill Watterson’s Calvin and Hobbes (September 20, rerun) dodged on me: I thought after the strip from the 19th that there’d be a fresh round of explanations of arithmetic, this time including imaginary numbers like “eleventeen” and “thirty-twelve” and the like. Not so. After some explanation of addition by Calvin’s Dad,
Spaceman Spiff would take up the task on the 22nd of smashing together Mysterio planets 6 and 5, which takes a little time to really get started, and finally sees the successful collision of the worlds. Let this serve as a reminder: translating a problem to a real-world application can be a fine way to understand what is wanted, but you have to make sure that in the translation you preserve the result you wanted from the calculation.

Joe has memorized the odds for various poker hands. Four times four, not so much.
Rick Detorie’s One Big Happy for the 21st of September, 2014. I confess ignorance as to whether these odds are accurate.

It’s Rick DeTorie’s One Big Happy (September 21) which brought the Benchley Principle to my mind. Here, Joe is shown to know extremely well the odds of poker hands, but to have no chance at having learned the multiplication table. It seems like something akin to Benchley’s Principle is at work here: Joe memorizing the times tables might be socially approved, but it isn’t what he wants to do, and that’s that. But inspiring the desire to know something is probably the one great challenge facing everyone who means to teach, isn’t it?

Jonathan Lemon’s Rabbits Against Magic (September 21) features a Möbius strip joke that I imagine was a good deal of fun to draw. The Möbius strip is one of those concepts that really catches the imagination, since it seems to defy intuition that something should have only the one side. I’m a little surprise that topology isn’t better-popularized, as it seems like it should be fairly accessible — you don’t need equations to get some surprising results, and you can draw pictures — but maybe I just don’t understand the field well enough to understand what’s difficult about bringing it to a mass audience.

Hector D. Cantu and Carlos Castellanos’s Baldo (September 23) tells a joke about percentages and students’ self-confidence about how good they are with “numbers”. In strict logic, yes, the number of people who say they are and who say they aren’t good at numbers should add up to something under 100 percent. But people don’t tend to be logically perfect, and are quite vulnerable to the way questions are framed, so the scenario is probably more plausible in the real world than the writer intended.

Steve Moore’s In The Bleachers (September 23) falls back on the most famous of all equations as representative of “something it takes a lot of intelligence to understand”.

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


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

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

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

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

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

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

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

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

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

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

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

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

Reading the Comics, September 8, 2014: What Is The Problem Edition


Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.

Edison Lee works out the shape of the universe, and as ever in this sort of thing, he forgot to carry a number.
I’d have thought the universe to be at least three-dimensional.

John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.

There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.

Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.

Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.

Jeff Harris’s Shortcuts children’s activity panel (September 9) is a page of stuff about “Geometry”, and it’s got some nice facts (some mathematical, some historical), and a fair bunch of puzzles about the field.

Morrie Turner’s Wee Pals (September 7, perhaps a rerun; Turner died several months ago, though I don’t know how far ahead of publication he was working) features a word problem in terms of jellybeans that underlines the danger of unwarranted assumptions in this sort of problem-phrasing.

Moose has trouble working out 15 percent of $8.95; Jughead explains why.
How far back is this rerun from if Moose got lunch for two for $8.95?

Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.

Mark Parisi’s Off The Mark (September 8) is another entry into the world of anthropomorphized numbers, so you can probably imagine just what π has to say here.

Reading the Comics, August 29, 2014: Recurring Jokes Edition


Well, I did say we were getting to the end of summer. It’s taken only a couple days to get a fresh batch of enough mathematics-themed comics to include here, although the majority of them are about mathematics in ways that we’ve seen before, sometimes many times. I suppose that’s fair; it’s hard to keep thinking of wholly original mathematics jokes, after all. When you’ve had one killer gag about “537”, it’s tough to move on to “539” and have it still feel fresh.

Tom Toles’s Randolph Itch, 2 am (August 27, rerun) presents Randolph suffering the nightmare of contracting a case of entropy. Entropy might be the 19th-century mathematical concept that’s most achieved popular recognition: everyone knows it as some kind of measure of how disorganized things are, and that it’s going to ever increase, and if pressed there’s maybe something about milk being stirred into coffee that’s linked with it. The mathematical definition of entropy is tied to the probability one will find whatever one is looking at in a given state. Work out the probability of finding a system in a particular state — having particles in these positions, with these speeds, maybe these bits of magnetism, whatever — and multiply that by the logarithm of that probability. Work out that product for all the possible ways the system could possibly be configured, however likely or however improbable, just so long as they’re not impossible states. Then add together all those products over all possible states. (This is when you become grateful for learning calculus, since that makes it imaginable to do all these multiplications and additions.) That’s the entropy of the system. And it applies to things with stunning universality: it can be meaningfully measured for the stirring of milk into coffee, to heat flowing through an engine, to a body falling apart, to messages sent over the Internet, all the way to the outcomes of sports brackets. It isn’t just body parts falling off.

Stanley's old algebra teacher insists there is yet hope for him.
Randy Glasbergen’s _The Better Half_ For the 28th of August, 2014.

Randy Glasbergen’s The Better Half (August 28) does the old joke about not giving up on algebra someday being useful. Do teachers in other subjects get this? “Don’t worry, someday your knowledge of the Panic of 1819 will be useful to you!” “Never fear, someday they’ll all look up to you for being able to diagram a sentence!” “Keep the faith: you will eventually need to tell someone who only speaks French that the notebook of your uncle is on the table of your aunt!”

Eric the Circle (August 28, by “Gilly” this time) sneaks into my pages again by bringing a famous mathematical symbol into things. I’d like to make a mention of the links between mathematics and music which go back at minimum as far as the Ancient Greeks and the observation that a lyre string twice as long produced the same note one octave lower, but lyres and strings don’t fit the reference Gilly was going for here. Too bad.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 28) is another strip to use a “blackboard full of mathematical symbols” as visual shorthand for “is incredibly smart stuff going on”. The symbols look to me like they at least started out as being meaningful — they’re the kinds of symbols I expect in describing the curvature of space, and which you can find by opening up a book about general relativity — though I’m not sure they actually stay sensible. (It’s not the kind of mathematics I’ve really studied.) However, work in progress tends to be sloppy, the rough sketch of an idea which can hopefully be made sound.

Anthony Blades’s Bewley (August 29) has the characters stare into space pondering the notion that in the vastness of infinity there could be another of them out there. This is basically the same existentially troublesome question of the recurrence of the universe in enough time, something not actually prohibited by the second law of thermodynamics and the way entropy tends to increase with the passing of time, but we have already talked about that.

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


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

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

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

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

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

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

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

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

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

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

Reading the Comics, August 16, 2014: Saturday Morning Breakfast Cereal Edition


Zach Weinersmith’s Saturday Morning Breakfast Cereal is a long-running and well-regarded web comic that I haven’t paid much attention to because I don’t read many web comics. XKCD, Newshounds, and a couple others are about it. I’m not opposed to web comics, mind you, I just don’t get around to following them typically. But Saturday Morning Breakfast Cereal started running on Gocomics.com recently, and Gocomics makes it easy to start adding comics, and I did, and that’s served me well for the mathematical comics collections since it’s been a pretty dry spell. I bet it’s the summer vacation.

Saturday Morning Breakfast Cereal (July 30) seems like a reach for inclusion in mathematical comics since its caption is “Physicists make lousy firemen” and it talks about the action of a fire — and of the “living things” caught in the fire — as processes producing wobbling and increases in disorder. That’s an effort at describing a couple of ideas, the first that the temperature of a thing is connected to the speed at which the molecules making it up are moving, and the second that the famous entropy is a never-decreasing quantity. We get these notions from thermodynamics and particularly the attempt to understand physically important quantities like heat and temperature in terms of particles — which have mass and position and momentum — and their interactions. You could write an entire blog about entropy and probably someone does.

Randy Glasbergen’s Glasbergen Cartoons (August 2) uses the word-problem setup for a strip of “Dog Math” and tries to remind everyone teaching undergraduates the quotient rule that it really could be worse, considering.

Nate Fakes’s Break of Day (August 4) takes us into an anthropomorphized world that isn’t numerals for a change, to play on the idea that skill in arithmetic is evidence of particular intelligence.

Jiggs tries to explain addition to his niece, and learns his brother-in-law is his brother-in-law.
George McManus’s _Bringing Up Father_, originally run the 12th of April, 1949.

George McManus’s Bringing Up Father (August 11, rerun from April 12, 1949) goes to the old motif of using money to explain addition problems. It’s not a bad strategy, of course: in a way, arithmetic is one of the first abstractions one does, in going from the idea that a hundred of something added to a hundred fifty of something will yield two hundred fifty of that thing, and it doesn’t matter what that something is: you’ve abstracted out the ideas of “a hundred plus a hundred fifty”. In algebra we start to think about whether we can add together numbers without knowing what one or both of the numbers are — “x plus y” — and later still we look at adding together things that aren’t necessarily numbers.

And back to Saturday Morning Breakfast Cereal (August 13), which has a physicist type building a model of his “lack of dates” based on random walks and, his colleague objects, “only works if we assume you’re an ideal gas molecule”. But models are often built on assumptions that might, taken literally, be nonsensical, like imagining the universe to have exactly three elements in it, supposing that people never act against their maximal long-term economic gain, or — to summon a traditional mathematics/physics joke — assuming a spherical cow. The point of a model is to capture some interesting behavior, and avoid the complicating factors that can’t be dealt with precisely or which don’t relate to the behavior being studied. Choosing how to simplify is the skill and art that earns mathematicians the big money.

And then for August 16, Saturday Morning Breakfast Cereal does a binary numbers joke. I confess my skepticism that there are any good alternate-base-number jokes, but you might like them.

Reading the Comics, July 24, 2014: Math Is Just Hard Stuff, Right? Edition


Maybe there is no pattern to how Comic Strip Master Command directs the making of mathematics-themed comic strips. It hasn’t quite been a week since I had enough to gather up again. But it’s clearly the summertime anyway; the most common theme this time seems to be just that mathematics is some hard stuff, without digging much into particular subjects. I can work with that.

Pab Sungenis’s The New Adventures of Queen Victoria (July 19) brings in Erwin Schrödinger and his in-strip cat Barfly for a knock-knock joke about proof, with Andrew Wiles’s name dropped probably because he’s the only person who’s gotten to be famous for a mathematical proof. Wiles certainly deserves fame for proving Fermat’s Last Theorem and opening up what I understand to be a useful new field for mathematical research (Fermat’s Last Theorem by itself is nice but unimportant; the tools developed to prove it, though, that’s worthwhile), but remembering only Wiles does slight Richard Taylor, whose help Wiles needed to close a flaw in his proof.

Incidentally I don’t know why the cat is named Barfly. It has the feel to me of a name that was a punchline for one strip and then Sungenis felt stuck with it. As Thomas Dye of the web comic Newshounds said, “Joke names’ll kill you”. (I’m inclined to think that funny names can work, as the Marx Brotehrs, Fred Allen, and Vic and Sade did well with them, but they have to be a less demanding kind of funny.)

John Deering’s Strange Brew (July 19) uses a panel full of mathematical symbols scrawled out as the representation of “this is something really hard being worked out”. I suppose this one could also be filed under “rocket science themed comics”, but it comes from almost the first problem of mathematical physics: if you shoot something straight up, how long will it take to fall back down? The faster the thing starts up, the longer it takes to fall back, until at some speed — the escape velocity — it never comes back. This is because the size of the gravitational attraction between two things decreases as they get farther apart. At or above the escape velocity, the thing has enough speed that all the pulling of gravity, from the planet or moon or whatever you’re escaping from, will not suffice to slow the thing down to a stop and make it fall back down.

The escape velocity depends on the size of the planet or moon or sun or galaxy or whatever you’re escaping from, of course, and how close to the surface (or center) you start from. It also assumes you’re talking about the speed when the thing starts flying away, that is, that the thing doesn’t fire rockets or get a speed boost by flying past another planet or anything like that. And things don’t have to reach the escape velocity to be useful. Nothing that’s in earth orbit has reached the earth’s escape velocity, for example. I suppose that last case is akin to how you can still get some stuff done without getting out of the recliner.

Mel Henze’s Gentle Creatures (July 21) uses mathematics as the standard for proving intelligence exists. I’ve got a vested interest in supporting that proposition, but I can’t bring myself to say more than that it shows a particular kind of intelligence exists. I appreciate the equation of the final panel, though, as it can be pretty well generalized.

To disguise a sports venue it's labelled ``Math Arena'', with ``lectures on the actual odds of beating the casino''.
Bill Holbrook’s _Safe Havens_ for the 22nd of July, 2014.

Bill Holbrook’s Safe Havens (July 22) plays on mathematics’ reputation of being not very much a crowd-pleasing activity. That’s all right, although I think Holbrook makes a mistake by having the arena claim to offer a “lecture on the actual odds of beating the casino”, since the mathematics of gambling is just the sort of mathematics I think would draw a crowd. Probability enjoys a particular sweet spot for popular treatment: many problems don’t require great amounts of background to understand, and have results that are surprising, but which have reasons that are easy to follow and don’t require sophisticated arguments, and are about problems that are easy to imagine or easy to find interesting: cards being drawn, dice being rolled, coincidences being found, or secrets being revealed. I understand Holbrook’s editorial cartoon-type point behind the lecture notice he put up, but the venue would have better scared off audiences if it offered a lecture on, say, “Chromatic polynomials for rigidly achiral graphs: new work on Yamada’s invariant”. I’m not sure I could even explain that title in 1200 words.

Missy Meyer’s Holiday Doodles (July 22) revelas to me that apparently the 22nd of July was “Casual Pi Day”. Yeah, I suppose that passes. I didn’t see much about it in my Twitter feed, but maybe I need some more acquaintances who don’t write dates American-fashion.

Thom Bluemel’s Birdbrains (July 24) again uses mathematics — particularly, Calculus — as not just the marker for intelligence but also as The Thing which will decide whether a kid goes on to success in life. I think the dolphin (I guess it’s a dolphin?) parent is being particularly horrible here, as it’s not as if a “B+” is in any way a grade to be ashamed of, and telling kids it is either drives them to give up on caring about grades, or makes them send whiny e-mails to their instructors about how they need this grade and don’t understand why they can’t just do some make-up work for it. Anyway, it makes the kid miserable, it makes the kid’s teachers or professors miserable, and for crying out loud, it’s a B+.

(I’m also not sure whether a dolphin would consider a career at Sea World success in life, but that’s a separate and very sad issue.)

Reading the Comics, July 18, 2014: Summer Doldrums Edition


Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.

Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.

Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.

Richard Thompson’s Richard’s Poor Almanac (July 7, rerun) brings back our favorite set of infinite monkeys, here, to discuss their ambitious book set at the Museum of Natural History.

Tom Thaves’s Frank and Ernest (July 16) builds on the (true) point that the ancient Greeks had no symbol for zero, and would probably have had a fair number of objections to the concept.

'The day Einstein got the wind knocked out of his sails': Einstein tells his wife he's discovered the theory of relativity.
Joe Martin’s _Mr Boffo_ strip for the 18th of July, 2014.

Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.

Charles Schulz’s Peanuts (July 18) — a rerun, of course, from the 21st of July, 1967 — mentions Sally going to Summer School and learning all about the astronomical details of summertime. Astronomy has always been one of the things driving mathematical discovery, but I admit, thinking mostly this would be a good chance to point out Dr Helmer Aslaksen’s page describing the relationship between the solstices and the times of earliest and latest sunrise (and sunset). It’s not quite as easy as finding when the days are longest and shortest. Dr Aslaksen has a number of fascinating astronomy- and calendar-based pages which I think worth reading, so, I hope you enjoy.

Reading the Comics, July 3, 2014: Wulff and Morgenthaler Edition


Sorry to bring you another page of mathematics comics so soon after the last one, but, I don’t control Comic Strip Master Command. I’m not sure who does, but it’s obviously someone who isn’t paying very close attention to Mary Worth because the current psychic-child/angel-warning-about-pool-safety storyline is really going off the rails. But I can’t think of a way to get that back to mathematical topics, so let me go to safer territories instead.

Mickey's nephew figures this is rocket science.
The Disney Corporation’s _Mickey Mouse_ comic strip rerun on the 28th of June, 2014.

The Disney Corporation’s Mickey Mouse (June 28, rerun) uses the familiar old setup of mathematics stuff — here crossbred with rocket science — as establishment that someone is just way smarter than the rest of the room.

Wulff and Morgenthaler’s Truth Facts — a new strip from the people who do that WuMu which is replacing the strangely endless reruns of Get Fuzzy in your local newspaper (no, I don’t know why Get Fuzzy has been rerunning daily strips since November, and neither do its editors, so far as they’re admitting) — shows a little newspaper sidebar each day. The premise is sure to include a number of mathematics/statistics type jokes and on June 28th they went ahead with the joke that delivers statistics about statistics, so that’s out of the way.

Dave Whamond’s Reality Check (June 29) brings out two of the songs that prominently mention numbers.

Mel Henze’s Gentle Creatures (June 30) drops in a bit of mathematics technobabble for the sake of sounding all serious and science-y and all that. But “apply the standard Lagrangian model” is a better one than average since Joseph-Louis Lagrange was an astoundingly talented and omnipresent mathematician and physicist. Probably his most useful work is a recasting of Newton’s laws of physics in a form in which you don’t have to worry so much about forces at every moment and can instead look at the kinetic and potential energy of a system. This generally reduces the number of equations one has to work with to describe what’s going on, and that usually means it’s easier to understand them. That said I don’t know a specific “Lagrangian model” that would necessarily be relevant. The most popular “Lagrangian model” I can find talks about the flow of particles in a larger fluid and is popular in studying atmospheric pollutants, though the couple of medical citations stuggest it’s also useful for studying how things get transported by the bloodstream. Anyway, it’s nice to hear somebody besides Einstein get used as a science name.

Mary Beth figures if she works her apple-dividing and giving right she can get al the apple.
John Rose’s _Barney Google and Snuffy Smith_ for the 1st of July, 2014, featuring neither Barney Google nor Snuffy Smith.

John Rose’s Barney Google and Snuffy Smith (July 1) plays with division word problems and percentages and the way people can subvert the intentions of a problem given any chance.

Bill Watterson’s Calvin and Hobbes (July 1, rerun) lets Calvin’s Dad gently blow Calvin’s mind by pointing out that rotational motion means that different spots on the same object are moving at different speeds yet the object stays in one piece. When you think hard enough about it rotation is a very strange phenomenon (I suppose you could say that about any subject, though), and the difference in speeds within a single object is just part of it. Sometime we must talk about the spinning pail of water.

Wulff and Morgenthaler’s WuMo (July 1) — I named this edition after them for some reason, after all — returns to the potential for mischief in how loosely one uses the word “half”.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (July 3) dips into the well of mathematics puns. I admit I had to reread the caption before noticing where the joke was. It’s been a busy week.

Reading the Comics, June 27, 2014: Pretty Easy Edition


I don’t mean to complain, because it really is a lot of fun to do these comic strip roundups, but Comic Strip Master Command has been sending a flood of comics my way. I hope it’s not overwhelming readers, or me. The downside of the great number of mathematics-themed comics this past week has been that they aren’t very deep examples, but, what the heck. Many of them are interesting anyway. As usual I’m including examples of the Comics Kingdom and the Creators.com comics because I’m not yet confident how long those links remain visible to non-subscribers.

A caveman invents the 'science fictiony' number of eleven.
Mike Peters’s _Mother Goose and Grimm_ for the 23rd of June, 2014

Mike Peters’ Mother Goose and Grimm (June 23) presents the cavemen-inventing-stuff pattern and the invention of a “science-fictiony” number. This is amusing, sure, but the dynamic is historically valid: it does seem like the counting numbers (1, 2, 3, and so on) were more or less intuitive, but negative numbers? Rationals? Irrationals? Zero? They required development and some fairly sophisticated reasoning to think of. You get a hint of the suspicion with which the newly-realized numbers were viewed when you think of the connotations of terms like “complex” numbers, or “imaginary” numbers, or even “negative” numbers. For that matter, Arabic numerals required some time for Europeans — who were comfortable with Roman numerals — to feel comfortable with; histories of mathematics will mention how Arabic numerals were viewed with suspicion and sometimes banned as being too easy for merchants or bankers to use to defraud customers who didn’t know what the symbols meant or how to use them.

Thom Bluemel’s Birdbrains (June 23) also takes us to the dawn of time and the invention of the calendar. Calendars are deeply intwined with mathematics, as they typically try to reconcile several things that aren’t quite perfectly reconcilable: the changes of the season, the cycles of the moon, the position of the sun in the sky, the length of the day. But the attempt to do as well as possible, using rules easy enough for normal human beings to understand, is productive.

Mark Pett’s Lucky Cow (June 23, rerun) lets Neil do some accounting the modern old-fashioned way. I trust there are abacus applications out there; somewhere in my pile of links I had a Javascript-based slide rule simulator, after all. I never quite got abacus use myself.

Mark Parisi’s Off The Mark (June 23) shows off one of those little hazards of skywriting and mathematical symbols. I admit the context threw me; I had to look again to read the birds as the less-than sign.

Dilton Doiley and Jughead find wonder in the vast and the numerous.
Henry Scarpelli and Craig Boldman’s _Archie_ rerun on the 24th of June, 2014

Henry Scarpelli and Craig Boldman’s Archie (June 24) has resident nerd Dilton Doiley pondering the vastness of the sky and the number of stars and feel the sense of wonder that inspires. The mind being filled with ever-increasing wonder and awe isn’t a unique sentiment, and thinking hard of very large, very numerous things is one of the paths to that sensation. Jughead has a similar feeling, evidently.

The orangutan who 'probably never had a thought in his life' works out a pretty involved derivative using the product and chain rules.
Mort Walker (Addison)’s _Boner’s Ark_ rerun the 26th of June, 2014 (originally run the 31st of July, 1968).

Mort Walker (“Addison”)’s Boner’s Ark (June 26, originally run July 31, 1968) features once again the motif of “a bit of calculus proves someone is really smart”. The orangutan’s working out of a derivative starts out well, too, using the product rule correctly through the first three lines, a point at which the chain rule and the derivative of the arccotangent function conspire to make things look really complicated. I admit I’m impressed Walker went to the effort to get things right that far in and wonder where he got the derivative worked out. It’s not one of the standard formulas you’d find in every calculus textbook, although you might find it as one of the more involved homework problem for Calculus I.

Mark Pett’s Lucky Cow comes up again (June 26, rerun) sees Neil a little gloomy at the results of a test coming back “negative”, a joke I remember encountering on The Office (US) too. It brings up the question of why, given the connotations of the words, a “positive” test result is usually a bad thing and a “negative” one a good, and it back to the language of statistics. Normally a test — medical, engineering, or otherwise — is really checking to see how often some phenomenon occurs within a given sample. But the phenomenon will normally happen a little bit anyway, even if nothing untoward is happening. It also won’t normally happen at exactly the same rate, even if there’s nothing to worry about. What statistics asks, then, is, “is this phenomenon happening so much in this sample space that it’s not plausible for it to just be coincidence?” And in that context, yeah, everything being normal is the negative result. What happens isn’t suspicious. Of course, Neil has other issues, here.

Chip Dunham’s Overboard (June 26) plays on the fact that “half” does have a real proper meaning, but will get used pretty casually when people aren’t being careful. Or when dinner’s involved.

Percy Crosby’s Skippy (June 26, rerun) must have originally run in March sometime, and it does have Skippy and the other kid arguing about how many months it is until Christmas. Counting intervals like this does invite what’s termed a “fencepost error”, and the kids present it perfectly: do you count the month you’re in if you want to count how many months until something? There isn’t really an absolutely correct answer, though; you and the other party just have to agree on whether you mean, say, the pages on the calendar you’ll go through between today and Christmas, or whether you mean how many more times you’ll pass the 24th of the month until you get to Christmas. You will see this same dynamic in every argument about conventions ever. Two spaces after the end of the sentence.

Moose has a fair idea for how to get through the algebra book in time.
Henry Scarpelli and Craig Boldman’s _Archie_ rerun for the 27th of June, 2014.

In Henry Scarpelli and Craig Boldman’s Archie (June 27, rerun), Moose has a pretty good answer to how to get the whole algebra book read in time. It’d be nice if it quite worked that way.

Mel Henze’s Gentle Creatures (June 27) has the characters working out just what the calculations for a jump into hyperspace would be. I admit I’ve always wondered just what the calculations for that sort of thing are, but that’s a bit silly of me.

Reading the Comics, June 22, 2014: Name-Dropping Stuff Edition


Comic Strip Master Command apparently really is ordering strips to finish their mathematics jokes before the summer vacation sets in, based on how many we’ve gotten in the past week. I confess this set doesn’t give me so much to write about; it’s more a set of mathematics things getting name-dropped. And there’s always something, isn’t there?

Tom Thaves’s Frank and Ernest (June 17) showcases a particularly severe form of math anxiety. I’m sympathetic to people who’re afraid of mathematics, naturally; it’s rotten being denied a big and wonderful and beautiful part of human ingenuity. I don’t know where math anxiety comes from, although I’d imagine a lot of it comes from that mix of doing something you aren’t quite sure you’re doing correctly and being hit too severely with a sense of rejection in the case that you did it wrong. I’d like to think that recreational mathematics puzzles would help overcome that, but I have no evidence that it does, just my hunch that getting to play with numbers and pictures and logic puzzles is good for you.

Russell Myers’ Broom Hilda (June 18) taunts the schoolkid Nerwin with the way we “used to do math with our brains instead of calculators”. One hesitates to know too much about the continuity of Broom Hilda, but I believe she’s over a thousand years old and so when she was Nerwin’s age they didn’t even have Arabic numerals just yet. I’ll assume there’s some way she’d have been in school then. (Also, given how long Broom Hilda‘s been running Nerwin did used to be in classes that did mathematics without calculators.)

Hagar can't count how many beers he had, and so proposes getting a math tutor.
Chris Browne’s Hagar the Horrible for the 19th of June, 2014.

Chris Brown’s Hagar the Horrible (June 19) tries to get itself cut out and put up on the walls of math tutors’ offices. Good luck.

Crankshaft does arithmetic by counting on his fingers, including long division.
Tom Batiuk and Chuck Ayers’s Crankshaft for the 20th of June, 2014.

Tom Batiuk and Chuck Ayers’ Crankshaft (June 20) spent a couple days this week explaining how he just counts on fingers to do his arithmetic. It’s a curious echo of the storyline several years ago revealing Crankshaft suffered from Backstory Illiteracy, in which we suddenly learned he had gone all his life without knowing how to read. I hesitate to agree with him but, yeah, there’s no shame in counting on your fingers if that does all the mathematics you need to do and you get the answers you want reliably. I don’t know what his long division thing is; if it weren’t for Tom Batiuk writing the comic strip I’d call it whimsy.

Keith Knight’s The Knight Life carried on with the story of the personal statistician this week. I think the entry from the 20th is most representative. It’s fine, and fun, to gather all kinds of data about whatever you encounter, but if you aren’t going to study the data and then act on its advice you’re wasting your time. The personal statistician ends up quitting the job.

Steve McGarry’s kid-activity feature KidTown (June 22) promotes the idea of numbers as a thing to notice in the newspapers, and includes a couple of activities, one featuring a maze to be navigated by way of multiples of seven. It also has one of those math tricks where you let someone else pick a number, give him a set of mathematical operations to do, and then you can tell them what the result is. It seems to me working out why that scheme works is a good bit of practice for someone learning algebra, and developing your own mathematics trick that works along this line is further good practice.

Reading the Comics, June 16, 2014: Cleaning Out Before Summer, I Guess, Edition


I had thought the folks at Comic Strip Master Command got most of their mathematics-themed comics cleaned out ahead of the end of the school year (United States time zones) by last week, and then over the course of the weekend they went and published about a hundred million of them, so let me try catching up on that before the long dry spell of summer sets in. (And yet none of them mentioned monkeys writing Shakespeare; go figure.) I’m kind of expecting an all-mathematics-strips series tomorrow morning.

Jason Chatfield’s Ginger Meggs (June 12) puns a bit on negative numbers as also meaning downbeat or pessimistic ones. Negative numbers tend to make people uneasy, when they’re first encountered. It took western mathematics several centuries to be quite fully comfortable with them and that even with the good example of debts serving as a mental model of what negative numbers might mean. Descartes, for example, apparently used four separate quadrants, giving points their positions to the right and up, to the left and up, to the left and down, or to the right and down, from the origin point, rather than deal with negative numbers; and the Fahrenheit temperature scale was pretty much designed around the constraint that Daniel Fahrenheit shouldn’t have to deal with negative numbers in measuring the temperature in his hometown of the Netherlands. I have seen references to Immanuel Kant writing about the theoretical foundation of negative numbers, but not a clear explanation of just what he did, alas. And skepticism of exotic number constructs would last; they’re not called imaginary numbers because people appreciated the imaginative leaps that working with the square roots of negative numbers inspired.

Steve Breen and Mike Thompson’s Grand Avenue (June 12) served notice that, just like last summer, Grandma is going to make sure the kids experience mathematics as a series of chores they have to endure through an otherwise pleasant summer break.

Mike Twohy’s That’s Life (June 12) might be a marginal inclusion here, but it does refer to a lab mouse that’s gone from merely counting food pellets to cost-averaging them. The mathematics abilities of animals are pretty amazing things, certainly, and I’d also be impressed by an animal that was so skilled in abstract mathematics that it was aware “how much does a thing cost?” is a pretty tricky question when you look hard at it.

Jim Scancarelli’s Gasoline Alley (June 13) features a punch line that’s familiar to me — it’s what you get by putting a parrot and the subject of geometry together — although the setup seems clumsy to me. I think that’s because the kid has to bring up geometry out of nowhere in the first panel. Usually the setup as I see it is more along the lines of “what geometric figure is drawn by a parrot that then leaves the room”, which I suppose also brings geometry up out of nowhere to start off, really. I guess the setup feels clumsy to me because I’m trying to imagine the dialogue as following right after the previous day’s, so the flow of the conversation feels odd.

Eric the Circle (June 14), this one signed “andel”, riffs on the popular bit of mathematics trivia that in a randomly selected group of 22 people there’s about a fifty percent chance that some pair of them will share a birthday; that there’s a coincidental use for 22 in estimating π is, believe it or not, something I hadn’t noticed before.

Pab Sungenis’s New Adventures of Queen Victoria (June 14) plays with infinities, and whether the phrase “forever and a day” could actually mean anything, or at least anything more than “forever” does. This requires having a clear idea what you mean by “forever” and, for that matter, by “more”. Normally we compare infinitely large sets by working out whether it’s possible to form pairs which match one element of the first set to one element of the second, and seeing whether elements from either set have to be left out. That sort of work lets us realize that there are just as many prime numbers as there are counting numbers, and just as many counting numbers as there are rational numbers (positive and negative), but that there are more irrational numbers than there are rational numbers. And, yes, “forever and a day” would be the same length of time as “forever”, but I suppose the Innamorati (I tried to find his character’s name, but I can’t, so, Pab Sungenis can come in and correct me) wouldn’t do very well if he promised love for the “power set of forever”, which would be a bigger infinity than “forever”.

Mark Anderson’s Andertoons (June 15) is actually roughly the same joke as the Ginger Meggs from the 12th, students mourning their grades with what’s really a correct and appropriate use of mathematics-mentioning terminology.

Keith Knight’s The Knight Life (June 16) introduces a “personal statistician”, which is probably inspired by the measuring of just everything possible that modern sports has gotten around to doing. But the notion of keeping track of just what one is doing, and how effectively, is old and, at least in principle, sensible. It’s implicit in budgeting (time, money, or other resources) that you are going to study what you do, and what you want to do, and what’s required by what you want to do, and what you can do. And careful tracking of what one’s doing leads to what’s got to be a version of the paradox of Achilles and the tortoise, in which the time (and money) spent on recording the fact of one’s recordings starts to spin out of control. I’m looking forward to that. Don’t read the comments.

Max Garcia’s Sunny Street (June 16) shows what happens when anthropomorphized numerals don’t appear in Scott Hilburn’s The Argyle Sweater for too long a time.