Reading the Comics, June 3, 2020: Subjective Opinions Edition


Thanks for being here for the last week before my All-2020 Mathematics A to Z starts. By the time this posts I should have decided on the A-topic, but I’m still up for B or C topics, if you’d be so kind as to suggest things.

Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.

Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.

Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?

Kid: 'I hate essay assignments. They're so open to subjective grading. It would be a lot simpler if answers would just be right or wrong.' Frazz: 'You're in luck. I understand there's a math test coming up.' Kid: 'What's the lucky part?'
Jef Mallett’s Frazz for the 2nd of June, 2020. Other essays featuring something discussed in Frazz appear at this link.

And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.

Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.

(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.

On the blackboard are written 7 + 0 = 7, 7 - 0 = 7, and 7 x 0 = 0. Wavehead: 'So the takeaway ehre is, if I'm the number 7, avoid multiplication at all costs.'
Mark Anderson’s Andertoons for the 3rd of June, 2020. When I have an essay that features something mentioned in Andertoons the essay’s put up at this link.

Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.

Dan Collins’s Looks Good on Paper for the 3rd has a cook grilling a “Möbius Strip Steak”. It’s a good joke for putting on a mathematics instructor’s door.

Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.


And this wraps up half of the past week’s mathematically-themed comic strips. I hope to have the rest in a Reading the Comics post at this link in a few days. Thanks for reading.

Reading the Comics, December 28, 2018: More Christmas Break Edition


I apologize for running quite so late. Comic Strip Master Command tried to make it easy for me, by issuing few comic strips that had any mathematical content to speak of. I was just busier than all that, and even now, I can’t say quite how. Well, living, I suppose. But I’ve done plenty of things now and can settle back to the usual, if anyone knows just what that was.

Also I am drawing down on the number of cancelled, in-eternal-reruns comic strips on my daily feed. So that should reduce the number of times I feature a comic strip and realize I’ve described it four times already and haven’t got anything new to say. It’s hard for me, since most of these comics have some charms, or at least pleasant weirdness. But clearly just making a note to myself that I’ve said everything there is to say about Randolph Itch, 2 am, isn’t enough. I’m sorry, Randolph.

Bill Holbrook’s On The Fastrack for the 28th is an example of the cartoonist’s habit of drawing metaphors literally. Dethany does ask the auditor Fi about “accepting his numbers”. In this context the numbers aren’t intersting as numbers. They’re interesting as representations for a narrative. If the numbers are consistent with a believable story? If it’s more believable that they represent a truth than that they’re a hoax? We call that “accepting the numbers”, but what we’re accepting is the story they’re given as evidence for.

Man giving a presentation; it's a string of digits on the whiteboard. A giant thumb swipes across it, and the numbers all fall off, leaving the man disheartened. Last panel, the man walks out, dejected. Dethany: 'Did you accept his numbers?' Fi: 'I swiped left.'
Bill Holbrook’s On The Fastrack for the 28th of December, 2018. Essays discussing topics raised by On The Fastrack should all appear at this link.

Auditing, and any critical thinking about numbers, involves some subtle uses of Bayesian probability. We’re working out the probability that this story is something we should believe. Each piece of evidence makes us think this probability is greater or lesser. With experience and skill one learns of patterns which suggest the story is false. Benford’s Law, for example, is often useful. Honestly-taken samples show tendencies, for example, in what leading digits appear. A discrepancy between what’s expected and what appears, if it can’t be explained, can be a sign of forgery.

B.C.: 'How many grains of sand are on this beach?' Peter: 'Five hundred and sixty gillion.' B.C.: 'You're a genius, Peter.'
Johnny Hart’s Back To BC rerun for the 27th of December, 2018. From the dating it apparently originally ran the 1st of July, 1961. Essays mentioning B.C., both 1961 vintage and 2019 current, should be at this link.

Johnny Hart’s Back To BC rerun for the 27th is built on estimating the grains of sand on a beach. This is, as fits the setting, a very old query. Archimedes wrote The Sand Reckoner which estimated how many grains of sand could fit in the universe. Estimating the number of grains of sand on a beach, or in a universe, is a fun mathematical problem. Perhaps not a practical one, not directly. The answer is after all “lots”, and there is no way to verify the number.

But it can still be indirectly practical. To work with enormous but finite numbers of things is hard. We do well working with small numbers like ‘six’ and ‘fourteen’ and some of us are even good at around ‘thirty’. We don’t have a good intuition for how a number like 480,000,000,000,000,000 should work. And that’s important; if we try adding six and fourteen and get thirty, we realize there’s something not quite right before we’ve done too much more work. With enormous numbers we can go on not noticing the mistake’s there. We need to find ways to understand these inconvenient numbers using the skills and intuitions we already have. Aristotle had to develop new terminology for numbers to get the Ancient Greek numerals system to handle the problem coherently. Peter’s invention of a gillion is — I’ll go ahead and say — a sly reenactment of that.

Neil: 'Hey, a Rubik's Cube! I used to be really good at those! But you can only solve them five times. Then you have to buy another one.' Leticia: 'Five times? Er ... why's that, Neil?' Neil: 'After that, the little square stickers aren't sticky anymore.'
Mark Pett’s Lucky Cow rerun for the 27th of December, 2018. So I am dropping the strip from my routine reading. But the various appearances of Lucky Cow should remain at this link.

Mark Pett’s Lucky Cow rerun for the 27th I do intend to make this enjoyable but cancelled strip’s last appearance here. It’s a Rubik’s Cube joke. It’s one about using a solution outside the rules of the problem. And as marginal as this one, I couldn’t quite bring myself to write a paragraph about the Todd the Dinosaur strip of the 29th, which also features the Rubik’s Cube.

An anthropomorphic numeral 8 talks to the doctor about its weight and its eating. The doctor performs surgery, cutting off a loop and turning the figure into a 9. It smiles and waves, and the new 9 goes off to join its friends 2, 0, and 1.
Ryan Pagelow’s Buni for the 28th of December, 2018. Essays which mention Buni should be gathered at this link.

Ryan Pagelow’s Buni for the 28th I’ll list as the anthropomorphic-numerals joke for the week, since it did turn out to be that slow a week here. I’m a bit curious what the now-9 is figuring to do next year. I suppose that one’s easy; it’s going to be going from 3 to 4 in a couple years that’s a real problem.


The various Reading the Comics posts should all be at this link. I like to think I’ll be back to having a post this coming Sunday, and maybe a second one next week if there are enough comic strips near enough to on-topic. Thanks for reading.

Reading the Comics, December 30, 2017: Looking To 2018 Edition


The last full week of 2017 was also a slow one for mathematically-themed comic strips. You can tell by how many bits of marginally relevant stuff I include. In this case, it also includes a couple that just mention the current or the upcoming year. So you’ve been warned.

Mac King and Bill King’s Magic in a Minute activity for the 24th is a logic puzzle. I’m not sure there’s deep mathematics to it, but it’s some fun to reason out.

John Graziano’s Ripley’s Believe It Or Not for the 24th mentions the bit of recreational group theory that normal people know, the Rubik’s Cube. The group theory comes in from rotations: you can take rows or columns on the cube and turn them, a quarter or a half or a three-quarters turn. Which rows you turn, and which ways you turn them, form a group. So it’s a toy that inspires deep questions. Who wouldn’t like to know in how few moves a cube could be solved? We know there are at least some puzzles that take 18 moves to solve. (You can calculate the number of different cube arrangements there are, and how many arrangements you could make by shuffling a cube around with 17 moves. There’s more possible arrangements than there are ones you can get to in 17 moves; therefore, there must be at least one arrangement that takes 18 moves to solve.) A 2010 computer-assisted proof by Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge showed that at most 20 face turns are needed for every possible cube to be solved. I don’t know if there’s been any success figuring out whether 19 or even 18 is necessarily enough.

Griffith: 'Here we are, Zippy, back in the land of our childhood.' Zippy: 'Are we still in the ninth grade?' Griffith: 'Kind of ... although I still can't remember a thing about algebra.' Zippy: 'So many spitballs and paper airplanes ago!!' Griffith: 'Why did I act up so much in school, Zippy? Was it a Freudian thing?' Zippy: 'It was a cry for kelp.' Griffith: 'Don't you mean a cry for help? I don't think kelp was even a word I knew back in the 50s.' Zippy: 'Seaweed is the fifth dimension!'
Bill Griffith’s Zippy the Pinhead for the 26th of December, 2017. This is not as strongly a memoir or autobiographical strip as Griffith will sometimes do, which is a shame. Those are always captivating. I have fun reading Zippy the Pinhead and understand why people wouldn’t. But the memoir strips I recommend even to people who don’t care for the usual fare.

Bill Griffith’s Zippy the Pinhead for the 26th just mentions algebra as a thing that Griffith can’t really remember, even in one of his frequent nostalgic fugues. I don’t know that Zippy’s line about the fifth dimension is meant to refer to geometry. It might refer to the band, but that would be a bit odd. Yes, I know, Zippy the Pinhead always speaks oddly, but in these nostalgic fugue strips he usually provides some narrative counterpoint.

Larry Wright’s Motley Classics for the 26th originally ran in 1986. I mention this because it makes the odd dialogue of getting “a new math program” a touch less odd. I confess I’m not sure what the kid even got. An educational game? Something for numerical computing? The coal-fired, gear-driven version of Mathematica that existed in the 1980s? It’s a mystery, it is.

Ryan Pagelow’s Buni for the 27th is really a calendar joke. It seems to qualify as an anthropomorphic numerals joke, though. It’s not a rare sentiment either.

Jef Mallett’s Frazz for the 29th is similarly a calendar joke. It does play on 2017 being a prime number, a fact that doesn’t really mean much besides reassuring us that it’s not a leap year. I’m not sure just what’s meant by saying it won’t repeat for another 2017 years, at least that wouldn’t be just as true for (say) 2015 or 2019. But as Frazz points out, we do cling to anything that floats in times like these.

Reading the Comics, December 30, 2016: New Year’s Eve Week Edition


So last week, for schedule reasons, I skipped the Christmas Eve strips and promised to get to them this week. There weren’t any Christmas Eve mathematically-themed comic strips. Figures. This week, I need to skip New Year’s Eve comic strips for similar schedule reasons. If there are any, I’ll talk about them next week.

Lorie Ransom’s The Daily Drawing for the 28th is a geometry wordplay joke for this installment. Two of them, when you read the caption.

John Graziano’s Ripley’s Believe It or Not for the 28th presents the quite believable claim that Professor Dwight Barkley created a formula to estimate how long it takes a child to ask “are we there yet?” I am skeptical the equation given means all that much. But it’s normal mathematician-type behavior to try modelling stuff. That will usually start with thinking of what one wants to represent, and what things about it could be measured, and how one expects these things might affect one another. There’s usually several plausible-sounding models and one has to select the one or ones that seem likely to be interesting. They have to be simple enough to calculate, but still interesting. They need to have consequences that aren’t obvious. And then there’s the challenge of validating the model. Does its description match the thing we’re interested in well enough to be useful? Or at least instructive?

Len Borozinski’s Speechless for the 28th name-drops Albert Einstein and the theory of relativity. Marginal mathematical content, but it’s a slow week.

John Allison’s Bad Machinery for the 29th mentions higher dimensions. More dimensions. In particular it names ‘ana’ and ‘kata’ as “the weird extra dimensions”. Ana and kata are a pair of directions coined by the mathematician Charles Howard Hinton to give us a way of talking about directions in hyperspace. They echo the up/down, left/right, in/out pairs. I don’t know that any mathematicians besides Rudy Rucker actually use these words, though, and that in his science fiction. I may not read enough four-dimensional geometry to know the working lingo. Hinton also coined the “tesseract”, which has escaped from being a mathematician’s specialist term into something normal people might recognize. Mostly because of Madeline L’Engle, I suppose, but that counts.

Samson’s Dark Side of the Horse for the 29th is Dark Side of the Horse‘s entry this essay. It’s a fun bit of play on counting, especially as a way to get to sleep.

John Graziano’s Ripley’s Believe It or Not for the 29th mentions a little numbers and numerals project. Or at least representations of numbers. Finding other orders for numbers can be fun, and it’s a nice little pastime. I don’t know there’s an important point to this sort of project. But it can be fun to accomplish. Beautiful, even.

Mark Anderson’s Andertoons for the 30th relieves us by having a Mark Anderson strip for this essay. And makes for a good Roman numerals gag.

Ryan Pagelow’s Buni for the 30th can be counted as an anthropomorphic-numerals joke. I know it’s more of a “ugh 2016 was the worst year” joke, but it parses either way.

John Atkinson’s Wrong Hands for the 30th is an Albert Einstein joke. It’s cute as it is, though.