So I’m hosting the 141th Playful Math Education Blog Carnival


I mentioned yesterday Iva Sallay’s hosting of the 140th Playful Math Education Blog Carnival. This is a collection of pieces of educational, recreational, or otherwise just delightful mathematics posts. I’d said I hoped I might have the energy to host one again this year and, you know? Denise Gaskins, who organizes this monthly event, took me up on the offer.

So, if you write, or read, or are just aware of a good mathematics or mathematics-related blog, please, leave me a comment! I’ll need all the help I can get finding things worth sharing. Anything that you’ve learned from, or that’s delighted you, is worth it. It’ll teach and delight other people too.

And if you have a blog and would like to try out hosting it, please do! There are always months available, and it’s a neat different sort of blogging challenge.

The Playful Math Education Blog Carnival #136


Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.

Playful Math Education Blog Carnival banner, showing a coati dressed in bright maroon ringmaster's jacket and top hat, with multiplication and division signs sitting behind atop animal-training podiums; a greyscale photograph audience is in the far background.
Banner art again by Thomas K Dye, creator of Newshounds, Infinity Refugees, Something Happens, and his current comic strip, Projection Edge. You can follow him on Patreon and read his comic strip nine months ahead of its worldwide publication. The banner art was commissioned several weeks ago when I expected I would be in a more playful mood this week.

In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.

Kreig’s writing drew the attention of another kind contributor to my harvesting. Symmetry and Multiplying Negative Numbers explores one of those confusing things about negative numbers: how can a negative number times a negative number be positive? One way to understand this is to represent arithmetic operations as geometric operations. Particularly, we can see negation as a reflection.

That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …

This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.

Fermat’s Two-Squares Theorem, discussed here, is not the famous one about a^n + b^2 = c^n . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.

What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.

Iva Sallay also brought up Jenna Laib’s Making Meaning with Arrays: More Preschooler Division which similarly sees numerical truths revealed through geometric reasoning. Here, particularly, by the problem of baking muffins and thinking through how to divide them up. A key piece here, for a particular child’s learning, was being able to pick up and move things around. Often in shifting between arithmetic and geometry we suppose that we can rearrange things without effort. As adults it’s easy to forget that this is an abstraction that we need to learn.

Sharing of food, in this case cookies, appears in Helena Osana’s Mathematical thinking begins in the early years with dialogue and real-world exploration. Mathematic, Osana notes, is primarily about thinking. An important part in mathematics education is working out how the thinking children most like to do can also find mathematics.

I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?

Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.

Speaking of the game, David Coffey’s Creating Positive Change in Math Class was written in anticipation of the standardized tests meant to prove out mathematics education. Coffey gets to thinking about how to frame teaching to more focus on why students should have a skill, and how they can develop it. How to get students to feel involved in their work. Even how to get students to do homework more reliably. Coffey’s scheduled to present at the Michigan Council of Teachers of Mathematics conference in Grand Rapids this July. This if all starts going well. And this is another post I know of thanks to Goldenoj.

These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.

Mathematical study starts small, though. Often it starts with games. There are many good ones, not least Iva Sallay’s Find the Factors puzzles.

Besides that, Dads Worksheets has provided a set of Math Word Search Puzzles. It’s a new series from people who create worksheets for many grade levels and many aspects of mathematics. They’re on Twitter as @dadsworksheets.

Mr Wu, of the Singapore Math Tuition blog, has also begun a new series of recreational mathematics puzzles. He lays out the plans for this, puzzles aimed at children around eight to ten years old. One of the early ones is the Stickers Math Question. A more recent one is The Secret of the Sweets (Sweet Distribution Problem). Mr Wu can be found on Twitter as @mathtuition88.

Denise Gaskins, on Twitter as @letsplaymath, and indefatigable coordinator for this carnival, offers the chance to Play Math with Your Kids for Free. This is an e-book sampler of mathematics gameplay.

I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.

The Guys and Good Health blog presented Happy Pi Day on the 14th, with — in a move meant to endear the blog to me — several comic strips. This includes one from Grant Snider, who draws lovely strips. I’m sad that his Incidental Comics has left GoComics.com, so I can’t feature it often during my Reading the Comics roundups anymore.

Virtual Brush Box, meanwhile, offers To Celebrate Pi Day, 10 Examples of Numbers and 10 Examples of Math Involved with Horses which delights me by looking at π, and mathematics, as they’re useful in horse-related activities. This may be the only blog post written specifically for me and my sister, and I am so happy that there is the one.

There’s a bit more, a bit of delight. It was my greatest surprise in looking for posts for this month. That is poetry. I mean this literally.

Whimsy-Mimsy wrote on Pi Day a haiku.

D Avery, on Shift N Shake, wrote the longer Another Slice of Pi Day, the third year of their composing poems observing the day.

Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.

Blind On The Light Side similarly wrote Pi poems, including a Pi-Ku, for March 12, 2020. These poems don’t reach long enough to deal with the zero-syllable line, but we can forgive someone not wanting to go on that long.

As a last note, I have joined Mathstodon, the Mastodon instance with a mathematics theme. You can follow my shy writings there as @nebusj@mathstodon.xyz, or follow a modest number of people talking, largely, about mathematics. Mathstodon is a mathematically-themed microblogging site. On WordPress, I do figure to keep reading the comics for their mathematics topics. And sometime this year, when I feel I have the energy, I hope to do another A to Z, my little glossary project.

And this is what I have to offer. I hope the carnival has brought you some things of interest, and some things of delight. And, if I may, please consider this Grant Snider cartoon, Hope.

Life Through A Mathematician’s Eyes is scheduled to host the 137th installment of the Playful Math Education Blog Carnival, at the end of April. I look forward to seeing it. Good luck to us all.

Getting Ready for Pi Day, and also the Playful Math Blog Carnival


So the first bit of news: I’m hosting the Playful Math Education Blog Carnival later this month. This is a roaming blog link party, sharing blogs that delight or educate, or ideally both, about mathematics. As mentioned the other day Iva Sallay of Find the Factors hosted the 135th of these. My entry, the 136th, I plan to post sometime the last week of March.

And I’ll need help! If you’ve run across a web site, YouTube video, blog post, or essay that discusses something mathematical in a way that makes you grin, please let me know, and let me share it with the carnival audience.


This Saturday is March 14th, which we’ve been celebrating as Pi Day. I remain skeptical that it makes a big difference in people’s view of mathematics or in their education. But an afternoon spent talking about mathematics with everyone agreeing that, for today, we won’t complain about how hard it always was or how impossible we always found it, is pleasant. And that’s a good thing. I don’t know how much activity there’ll be for it, since the 14th is a weekend day this year. And the Covid-19 problem has got all the schools in my state closed through to April, so any calendar relevance is shattered.

But I have some things in the archive anyway. Last year I gathered Six Or Arguably Four Things For Pi Day, a collection of short essays about ways to calculate π well or poorly, and about some of the properties we’re pretty sure that π has, even if we can’t prove it. Also this fascinating physics problem that yields the digits of π.

And the middle of March often brings out Comic Strip Master Command. It looks like I’ve had at least five straight Pi Day editions of Reading the Comics, although most of them cover strips from more than just the 14th of March. From the past:

What will 2020 offer? There’s no guessing about anything in 2020 anymore, really. But when I get to look at the Pi Day comic strips for 2020 my essay on them should appear at this link. Thanks ever for reading. And for letting me know about sites that would be good for this month’s Carnival.

Find the Factors hosted the Playful Math Education Blog Carnival this month


I apologize that obligations have kept me from writing some things that I mean to. So let me just point you to Iva Sallay, whose Find the Factor recreational math puzzle page hosted the 135th Playful Math Education Blog Carnival this past month. The Blog Carnival is a fun roaming thing that I’ve hosted once, and do hope to host again. It’s a curated collection of other mathematics sites that are fun or interesting or hopefully both together.

A Weird Kind Of Ruler


I ran across something neat. It’s something I’ve seen before, but the new element is that I have a name for it. This is the Golomb Ruler. It’s a ruler made with as few marks as possible. The marks are supposed to be arranged so that the greatest possible number of different distances can be made, by measuring between selected pairs of points.

So, like, in a regularly spaced ruler, you have a lot of ways to measure a distance of 1 unit of length. Only one fewer way to measure a distance of 2 units. One fewer still ways to measure a distance of 3 units and so on. Convenient but wasteful of marks. A Golomb ruler might, say, put marks only where the regularly spaced ruler has the units 1, 2, and 4. Then by choosing the correct pairs you can measure a distance of 1, 2, 3, or 4 units.

There’s applications of the Golomb ruler, stuff in information theory and sensor design and stuff. Also logistics. Never mind those. They present a neat little puzzle: can you find, for a given number of marks, the best possible arrangement of them into a ruler? That would be the arrangement that allows the greatest number of different lengths. Or perhaps the one that allows the longest string of whole-number differences. Your definition of best-possible determines what the answer is.

As a number theory problem it won’t surprise you to know there’s not a general answer. If I’m reading accurately most of the known best arrangements — the ones that allow the greatest number of differences — were proven by testing out cases. The 24-mark arrangement needed a test of 555,529,785,505,835,800 different rulers. MathWorld’s page on this tells me that optimal mark placement isn’t known for 25 or more marks. It also says that the 25-mark ruler’s optimal arrangement was published in 2008. So it isn’t just Wikipedia where someone will write an article, and then someone else will throw a new heap of words onto it, and nobody will read to see if the whole thing still makes sense. Wikipedia meanwhile lists optimal configurations for up to 27 points, demonstrated by 2014.

And as this suggests, you aren’t going to discover an optimal arrangement for some number of marks yourself. Unless you should be the first person to figure out an algorithm to do it. It’s not even known how complex an algorithm has to be. It’s suspected that it has to be NP-hard, though. But, while you won’t discover anything new to mathematics in pondering this, you can still have the fun of working out arrangements yourself, at least for a handful of points. There are numbers of points with more than one optimal arrangement.

(Golomb here is Solomon W Golomb, a mathematician and electrical engineer with a long history in information theory and also recreational mathematics problems. There are several parties who independently invented the problem. But Golomb actually did work with rulers, so at least they aren’t incorrectly named.)

My 2019 Mathematics A To Z: Relatively Prime


I have another subject nominated by goldenoj today. And it even lets me get into number theory, the field of mathematics questions that everybody understands and nobody can prove.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Relatively Prime.

I was once a young grad student working as a teaching assistant and unaware of the principles of student privacy. Near the end of semesters I would e-mail students their grades. This so they could correct any mistakes and know what they’d have to get on the finals. I was learning Perl, which was an acceptable pastime in the 1990s. So I wrote scripts that would take my spreadsheet of grades and turn it into e-mails that were automatically sent. And then I got all fancy.

It seemed boring to send out completely identical form letters, even if any individual would see it once. Maybe twice if they got me for another class. So I started writing variants of the boilerplate sentences. My goal was that every student would get a mass-produced yet unique e-mail. To best the chances of this I had to make sure of something about all these variant sentences and paragraphs.

So you see the trick. I needed a set of relatively prime numbers. That way, it would be the greatest possible number of students before I had a completely repeated text. We know what prime numbers are. They’re the numbers that, in your field, have exactly two factors. In the counting numbers the primes are numbers like 2, 3, 5, 7 and so on. In the Gaussian integers, these are numbers like 3 and 7 and 3 - 2\imath . But not 2 or 5. We can look to primes among the polynomials. Among polynomials with rational coefficients, x^2 + x + 1 is prime. So is 2x^2 + 14x + 1 . x^2 - 4 is not.

The idea of relative primes appears wherever primes appears. We can say without contradiction that 4 and 9 are relative primes, among the whole numbers. Though neither’s prime, in the whole numbers, neither has a prime factor in common. This is an obvious way to look at it. We can use that definition for any field that has a concept of primes. There are others, though. We can say two things are relatively prime if there’s a linear combination of them that adds to the identity element. You get a linear combination by multiplying each of the things by a scalar and adding these together. Multiply 4 by -2 and 9 by 1 and add them and look what you get. Or, if the least common multiple of a set of elements is equal to their product, then the elements are relatively prime. Some make sense only for the whole numbers. Imagine the first quadrant of a plane, marked in Cartesian coordinates. Draw the line segment connecting the point at (0, 0) and the point with coordinates (m, n). If that line segment touches no dots between (0, 0) and (m, n), then the whole numbers m and n are relatively prime.

We start looking at relative primes as pairs of things. We can be interested in larger sets of relative primes, though. My little e-mail generator, for example, wouldn’t work so well if any pair of sentence replacements were not relatively prime. So, like, the set of numbers 2, 6, 9 is relatively prime; all three numbers share no prime factors. But neither the pair 2, 6 and the pair 6, 9 are not relatively prime. 2, 9 is, at least there’s that. I forget how many replaceable sentences were in my form e-mails. I’m sure I did the cowardly thing, coming up with a prime number of alternate ways to phrase as many sentences as possible. As an undergraduate I covered the student government for four years’ worth of meetings. I learned a lot of ways to say the same thing.

Which is all right, but are relative primes important? Relative primes turn up all over the place in number theory, and in corners of group theory. There are some thing that are easier to calculate in modulo arithmetic if we have relatively prime numbers to work with. I know when I see modulo arithmetic I expect encryption schemes to follow close behind. Here I admit I’m ignorant whether these imply things which make encryption schemes easier or harder.

Some of the results are neat, certainly. Suppose that the function f is a polynomial. Then, if its first derivative f’ is relatively prime to f, it turns out f has no repeated roots. And vice-versa: if f has no repeated roots, then it and its first derivative are relatively prime. You remember repeated roots. They’re factors like (x - 2)^2 , that foiled your attempt to test a couple points and figure roughly where a polynomial crossed the x-axis.

I mentioned that primeness depends on the field. This is true of relative primeness. Polynomials really show this off. (Here I’m using an example explained in a 2007 Ask Dr Math essay.) Is the polynomial 3x + 6 relatively prime to 3x^2 + 12 ?

It is, if we are interested in polynomials with integer coefficients. There’s no linear combination of 3x + 6 and 3x^2 + 12 which gets us to 1. Go ahead and try.

It is not, if we are interested in polynomials with rational coefficients. Multiply 3x + 6 by \frac{1}{12}\left(1 - \frac{1}{2}x\right) and multiply 3x^2 + 12 by \frac{1}{24} . Then add those up.

Tell me what polynomials you want to deal with today and I will tell you which answer is right.

This may all seem cute if, perhaps, petty. A bunch of anonymous theorems dotting the center third of an abstract algebra text will inspire that. The most important relative-primes thing I know of is the abc conjecture, posed in the mid-80s by Joseph Oesterlé and David Masser. Start with three counting numbers, a, b, and c. Require that a + b = c.

There is a product of the unique prime factors of a, b, and c. That is, let’s say a is 36. This is 2 times 2 times 3 times 3. Let’s say b is 5. This is prime. c is 41; it’s prime. Their unique prime factors are 2, 3, 5, and 41; the product of all these is 1,230.

The conjecture deals with this product of unique prime factors for this relatively prime triplet. Almost always, c is going to be smaller than this unique prime factors product. The conjecture says that there will be, for every positive real number \epsilon , at most finitely many cases where c is larger than this product raised to the power 1 + \epsilon . I do not know why raising this product to this power is so important. I assume it rules out some case where this product raised to the first power would be too easy a condition.

Apart from that 1 + \epsilon bit, though, this is a classic sort of number theory conjecture. Like, it involves some technical terms, but nothing too involved. You could almost explain it at a party and expect to be understood, and to get some people writing down numbers, testing out specific cases. Nobody will go away solving the problem, but they’ll have some good exercise and that’s worthwhile.

And it has consequences. We do not know whether the abc conjecture is true. We do know that if it is true, then a bunch of other things follow. The one that a non-mathematician would appreciate is that Fermat’s Last Theorem would be provable by an alterante route. The abc conjecture would only prove the cases for Fermat’s Last Theorem for powers greater than 5. But that’s all right. We can separately work out the cases for the third, fourth, and fifth powers, and then cover everything else at once. (That we know Fermat’s Last Theorem is true doesn’t let us conclude the abc conjecture is true, unfortunately.)

There are other implications. Some are about problems that seem like fun to play with. If the abc conjecture is true, then for every integer A, there are finitely many values of n for which n! + A is a perfect square. Some are of specialist interest: Lang’s conjecture, about elliptic curves, would be true. This is a lower bound for the height of non-torsion rational points. I’d stick to the n! + A stuff at a party. A host of conjectures about Diophantine equations — (high school) algebra problems where only integers may be solutions — become theorems. Also coming true: the Fermat-Catalan conjecture. This is a neat problem; it claims that the equation

a^m + b^n = c^k

where a, b, and c are relatively prime, and m, n, and k are positive integers satisfying the constraint

\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1

has only finitely many solutions with distinct triplets \left(a^m, b^n, c^k\right) . The inequality about reciprocals of m, n, and k is needed so we don’t have boring solutions like 2^2 + 3^3 = 31^1 clogging us up. The bit about distinct triplets is so we don’t clog things up with a or b being 1 and then technically every possible m or n giving us a “different” set. To date we know something like ten solutions, one of them having a equal to 1.

Another implication is Pillai’s Conjecture. This one asks whether every positive integer occurs only finitely many times as the difference between perfect powers. Perfect powers are, like 32 (two to the fifth power) or 81 (three to the fourth power) or such.

So as often happens when we stumble into a number theory thing, the idea of relative primes is easy. And there are deep implications to them. But those in turn give us things that seem like fun arithmetic puzzles.


This closes out the A to Z essays for this week. Tomorrow and Saturday I hope to bring some attention to essays from past years. And next week I figure to open for topics for the end of the alphabet, the promising letters U through Z. This and the rest of the 2019 essays should appear at this link, as should the letter S next Tuesday. And all of the A to Z essays ought to be at this link. Thank you for reading.

My 2019 Mathematics A To Z: Koenigsberg Bridge Problem


Today’s A To Z term was nominated by Bunny Hugger. I’m glad to write about it. The problem is foundational to both graph theory and topology.

I’m more fluent in graph theory, and my writing will reflect that. But its critical insight involves looking at spaces and ignoring things like distance and area and angle. It is amazing that one can discard so much of geometry and still have anything to consider. What we do learn then applies to very many problems.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Königsberg Bridge Problem.

Once upon a time there was a city named Königsberg. It no longer is. It is Kaliningrad now. It’s no longer in that odd non-contiguous chunk of Prussia facing the Baltic Sea. It’s now in that odd non-contiguous chunk of Russia facing the Baltic Sea.

I put it this way because what the city evokes, to mathematicians, is a story. I do not have specific reason to think the story untrue. But it is a good story, and as I think more about history I grow more skeptical of good stories. A good story teaches, though not always the thing it means to convey.

The story is this. The city is on two sides of the Pregel river, now the Pregolya River. Two large islands are in the river. For several centuries these four land masses were connected by a total of seven bridges. And we are told that people in the city would enjoy free time with an idle puzzle. Was there a way to walk all seven bridges one and only one time? If no one did something fowl like taking a boat to cross the river, or not going the whole way across a bridge, anyway? There were enough bridges, though, and enough possible ways to cross them, that trying out every option was hopeless.

Then came Leonhard Euler. Who is himself a preposterous number of stories. Pick any major field of mathematics; there is an Euler’s Theorem at its center. Or an Euler’s Formula. Euler’s Method. Euler’s Function. Likely he brought great new light to it.

And in 1736 he solved the Königsberg Bridge Problem. The answer was to look at what would have to be true for a solution to exist. He noticed something so obvious it required genius not to dismiss it. It seems too simple to be useful. In a successful walk you enter each land mass (river bank or island) the same number of times you leave it. So if you cross each bridge exactly once, you use an even number of bridges per land mass. The exceptions are that you must start at one land mass, and end at a land mass. Maybe a different one. How you get there doesn’t count for the problem. How you leave doesn’t either. So the land mass you start from may have an odd number of bridges. So may the one you end on. So there are up to two land masses that may have an odd number of bridges.

Once this is observed, it’s easy to tell that Königsberg’s Bridges did not match that. All four land masses in Königsberg have an odd number of bridges. And so we could stop looking. It’s impossible to walk the seven bridges exactly once each in a tour, not without cheating.

Graph theoreticians, like the topologists of my prologue, now consider this foundational to their field. To look at a geographic problem and not concern oneself with areas and surfaces and shapes? To worry only about how sets connect? This guides graph theory in how to think about networks.

The city exists, as do the islands, and the bridges existed as described. So does Euler’s solution. And his reasoning is sound. The reasoning is ingenious, too. Everything hard about the problem evaporates. So what do I doubt about this fine story?

Well, I don’t know that this bridge problem was something the people of Königsberg thought about. At least not in the way it’s presented, this idle problem everyone who visited the river wondered about without trying very hard to solve. The only people I ever hear discussing this are mathematicians. And mathematicians are as fond of good stories as anyone else, and accept that when the reality is messy and ambiguous and confused. I’m not alone in having doubts. The Mathematics Association of America’s web page about the problem concedes it is “according to lore” that the people of the city had this problem.

Teo Paoletti, author of that web page, says Danzig mayor Carl Leonhard Gottlieb Ehler wrote Euler, asking for a solution. This falls short of proving that the bridges were a common subject of speculation. It does show at least that Ehler thought it worth pondering. Euler apparently did not think it was even mathematics. Not that he thought it was hard; he simply thought it didn’t depend on mathematical principles. It took only reason. But he did find something interesting: why was it not mathematics? Paoletti quotes Euler as writing:

This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it.

I am reminded of a mathematical joke. It’s about the professor who always went on at great length about any topic, however slight. I have no idea why this should stick with me. Finally one day the professor admitted of something, “This problem is not interesting.” The students barely had time to feel relief. The professor went on: “But the reasons why it is not interesting are very interesting. So let us explore that.”

The Königsberg Bridge Problem is in the first chapter of every graph theory book ever. And it is a good graph theory problem. It may not be fair to say it created graph theory, though. Euler seems to have treated this as a little side bit of business, unrelated to his real mathematics. Graph theory as we know it — as a genre — formed in the 19th century. So did topology. In hindsight we can see how studying these bridges brought us good questions to ask, and ways to solve them. But for something like a century after Euler published this, it was just the clever solution to a recreational mathematics puzzle. It was as important as finding knight’s tours of chessboards.

That we take it as the introduction to graph theory, and maybe topology, tells us something. It is an easy problem to pose. Its solution is clever, but not obscure. It takes no long chains of complex reasoning. Many people approach mathematics problems with fear. By telling this story, we promise mathematics that feels as secure as a stroll along the riverfront. This promise is good through about chapter three, section four, where there are four definitions on one page and the notation summons obscure demons of LaTeX.

Still. Look at what the story of the bridges tells us. We notice something curious about our environment. The problem seems mathematical, or at least geographic. The problem is of no consequence. But it lingers in the mind. The obvious approaches to solving it won’t work. But think of the problem differently. The problem becomes simple. And better than simple. It guides one to new insights. In a century it gives birth to two fields of mathematics. In two centuries these are significant fields. They’re things even non-mathematicians have heard of. It’s almost a mathematician’s fantasy of insight and accomplishment.

But this does happen. The world suggests no end of little mathematics problems. Sometimes they are wonderful. Richard Feynman’s memoirs tell of his imagination being captured by a plate spinning in the air. Solving that helped him resolve a problem in developing Quantum Electrodynamics. There are more mundane problems. One of my professors in grad school remembered tossing and catching a tennis racket and realizing he didn’t know why sometimes it flipped over and sometimes didn’t. His specialty was in dynamical systems, and he could work out the mechanics of what a tennis racket should do, and when. And I know that within me is the ability to work out when a pile of books becomes too tall to stand on its own. I just need to work up to it.

The story of the Königsberg Bridge Problem is about this. Even if nobody but the mayor of Danzig pondered how to cross the bridges, and he only got an answer because he infected Euler with the need to know? It is a story of an important piece of mathematics. Good stories will tell us things that are true, which are not necessarily the things that happen in them.


Thanks for reading this. All of the Fall 2019 A To Z posts ought to be at this link. On Thursday I should publish my ‘L’ post. All of my past A To Z essays should be available at this link, And tomorrow I hope to finish off the comic strips worth just quick mentions from last week. See you then.

Particle Physics Made Hard


A friend was playing with that cute little particle-physics simulator idea I mentioned last week. And encountered a problem. With a little bit of thought, I was able to not solve the problem. But I was able to explain why it was a subtler and more difficult problem than they had realized. These are the moments that make me feel justified calling myself a mathematician.

The proposed simulation was simple enough: imagine a bunch of particles that interact by rules that aren’t necessarily symmetric. Like, the attraction particle A exerts on particle B isn’t the same as what B exerts on A. Or there are multiple species of particles. So (say) red particles are attracted to blue but repelled by green. But green is attracted to red and repelled by blue twice as strongly as red is attracted to blue. Your choice.

Give a mathematician a perfectly good model of something. She’ll have the impulse to try tinkering with it. One reliable way to tinker with it is to change the domain on which it works. If your simulation supposes you have particles moving on the plane, then, what if they were in space instead? Or on the surface of a sphere? Or what if something was strange about the plane? My friend had this idea: what if the particles were moving on the surface of a cube?

And the problem was how to find the shortest distance between two particles on the surface of a cube. The distance matters since most any attraction rule depends on the distance. This may be as simple as “particles more than this distance apart don’t interact in any way”. The obvious approach, or if you prefer the naive approach, is to pretend the cube is a sphere and find distances that way. This doesn’t get it right, not if the two points are on different faces of the cube. If they’re on adjacent faces, ones which share an edge — think the floor and the wall of a room — it seems straightforward enough. My friend got into trouble with points on opposite faces. Think the floor and the ceiling.

This problem was posed (to the public) in January 1905 by Henry Ernest Dudeney. Dudeney was a newspaper columnist with an exhaustive list of mathematical puzzles. A couple of the books collecting them are on Project Gutenberg. The puzzles show their age in spots. Some in language; some in problems that ask to calculate money in pounds-shillings-and-pence. Many of them are chess problems. But many are also still obviously interesting, and worth thinking about. This one, I was able to find, was a variation of The Spider and the Fly, problem 75 in The Canterbury Puzzles:

Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.

(Also I admire Dudeney’s efficient closing off of the snarky, problem-breaking answer someone was sure to give. It suggests experienced thought about how to pose problems.)

What makes this a puzzle, even a paradox, is that the obvious answer is wrong. At least, what seems like the obvious answer is to start at point A, move to one of the surfaces connecting the spider’s and the fly’s starting points, and from that move to the fly’s surface. But, no: you get a shorter answer by using more surfaces. Going on a path that seems like it wanders more gets you a shorter distance. The solution’s presented here, along with some follow-up problems. In this case, the spider’s shortest path uses five of the six surfaces of the room.

The approach to finding this is an ingenious one. Imagine the room as a box, and unfold it into something flat. Then find the shortest distance on that flat surface. Then fold the box back up. It’s a good trick. It turns out to be useful in many problems. Mathematical physicists often have reason to ponder paths of things on flattenable surfaces like this. Sometimes they’re boxes. Sometimes they’re toruses, the shape of a doughnut. This kind of unfolding often makes questions like “what’s the shortest distance between points” easier to solve.

There are wrinkles to the unfolding. Of course there are. How interesting would it be if there weren’t? The wrinkles amount to this. Imagine you start at the corner of the room, and walk up a wall at a 45 degree angle to the horizon. You’ll get to the far corner eventually, if the room has proportions that allow it. All right. But suppose you walked up at an angle of 30 degrees to the horizon? At an angle of 75 degrees? You’ll wind your way around the walls (and maybe floor and ceiling) some number of times, each path you start with. Probably different numbers of times. Some path will be shortest, and that’s fine. But … like, think about the path that goes along the walls and ceiling and floor three times over. The room, unfolded into a flat panel, has only one floor and one ceiling and each wall once. The straight line you might be walking goes right off the page.

And this is the wrinkle. You might need to tile the room. In a column of blocks (like in Dudeney’s solution) every fourth block might be the floor, with, between any two of them, a ceiling. This is fine, and what’s needed. It can be a bit dizzying to imagine such a state of affairs. But if you’ve ever zoomed a map of the globe out far enough that you see Australia six times over then you’ve understood how this works.

I cannot attest that this has helped my friend in the slightest. I am glad that my friend wanted to think about the surface of the cube. The surface of a dodecahedron would be far, far past my ability to help with.

Some More Mathematics I’ve Been Reading, 6 October 2018


I have a couple links I’d not included in the recent Playful Mathematics Education Blog Carnival. Looking at them, I can’t say why.

The top page of this asks, with animated text, whether you want to see something amazing. Forgive its animated text. It does do something amazing. This paper by Javier Cilleruelo, Florian Luca, and Lewis Baxter proves that every positive whole number is the sum of at most three palindromic numbers. The web site, by Mathstodon host Christian Lawson-Perfect, demonstrates it. Enter a number and watch the palindromes appear and add up.

Next bit is an article that relates to my years-long odd interest in pasta making. Mathematicians solve age-old spaghetti mystery reports a group of researchers at MIT — the renowned “Rensselaer Polytechnic Institute of Boston” [*] — studying why dry spaghetti fractures the way it does. Like many great problems, it sounds ridiculous to study at first. Who cares why, basically, you can’t snap a dry spaghetti strand in two equal pieces by bending it at the edges? The problem has familiarity to it and seems to have little else. But then you realize this is a matter of how materials work, and how they break. And realize it’s a great question. It’s easy to understand and subtle to solve.

And then, how about quaternions? Everybody loves quaternions. Well, @SheckyR here links to an article from Thatsmath.com, The Many Modern Uses of Quaternions. It’s some modern uses anyway. The major uses for quaternions are in rotations. They’re rather good at representing rotations. And they’re really good at representing doing several rotations, along different axes, in a row.

The article finishes with (as teased in the tweet above) a report of an electric toothbrush that should keep track of positions inside the user’s head, even as the head rotates. This is intriguing. I say as a person who’s reluctantly started using an electric toothbrush. I’m one of those who brushes, manually, too hard, to the point of damaging my gums. The electric toothbrush makes that harder to do. I’m not sure how an orientation-aware electric toothbrush will improve the situation any, but I’m open-minded.


[*] I went to graduate school at Rensselaer Polytechnic Institute, the “RPI of New York”. The school would be a rival to MIT if RPI had any self-esteem. I’m guessing, as I never went to a school that had self-esteem.

Playful Mathematics Education Blog Carnival #121


Greetings one and all! Come, gather round! Wonder and spectate and — above all else — tell your friends of the Playful Mathematics Blog Carnival! Within is a buffet of delights and treats, fortifications for the mind and fire for the imagination.

121 is a special number. When I was a mere tot, growing in the wilds of suburban central New Jersey, it stood there. It held a spot of privilege in the multiplication tables on the inside front cover of composition books. On the forward diagonal, yet insulated from the borders. It anchors the safe interior. A square number, eleventh of that set in the positive numbers.

Cartoon of several circus tents, with numbered flags above them and balloons featuring arithmetic symbols. The text, in a carnival-poster font, is 'PLAYFUL MATH EDUCATION CARNIVAL'.
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.

The First Tent

The first wonder to consider is Iva Sallay’s Find the Factors blog. She brings each week a sequence of puzzles, all factoring challenges. The result of each, done right, is a scrambling of the multiplication tables; it’s up to you the patron to find the scramble. She further examines each number in turn, finding its factors and its interesting traits. And furthermore, usually, when beginning a new century of digits opens a horserace, to see which of the numbers have the greatest number of factorizations. She furthermore was the host of this Playful Mathematics Education Carnival for August of 2018.

121 is more than just a square. It is the lone square known to be the sum of the first several powers of a prime number: it is 1 + 3 + 3^2 + 3^3 + 3^4 , a fantastic combination. If there is another square that is such a sum of primes, it is unknown to any human — and must be at least 35 digits long.

We look now for a moment at some astounding animals. From the renowned Dr Nic: Introducing Cat Maths cards, activities, games and lessons — a fine collection of feline companions, such toys as will enterain them. A dozen attributes each; twenty-seven value cards. These cats, and these cards, and these activity puzzles, promise games and delights, to teach counting, subtraction, statistics, and inference!

Next and no less incredible is the wooly Mathstodon. Christian Lawson-Perfect hosts this site, an instance of the open-source Twitter-like service Mastodon. Its focus: a place for people interested in mathematicians to write of what they know. To date over 1,300 users have joined, and have shared nearly 25,000 messages. You need not join to read many of these posts — your host here has yet to — but may sample its wares as you like.


The Second Tent

121 is one of only two perfect squares known to be four less than the cube of a whole number. The great Fermat conjectured that 4 and 121 are the only such numbers; no one has found a counter-example. Nor a proof.

Friends, do you know the secret to popularity? There is an astonishing truth behind it. Elias Worth of the MathSection blog explains the Friendship Paradox. This mind-warping phenomenon tells us your friends have more friends than you do. It will change forever how you look at your followers and following accounts.

And now to thoughts of learning. Stepping forward now is Monica Utsey, @Liveonpurpose47 of Chocolate Covered Boy Joy. Her declaration: “I incorporated Montessori Math materials with my right brain learner because he needed literal representations of the work we were doing. It worked and we still use it.” See now for yourself the representations, counting and comparing and all the joys of several aspects of arithmetic.

Take now a moment for your own fun. Blog Carnival patron and organizer Denise Gaskins wishes us to know: “The fun of mathematical coloring isn’t limited to one day. Enjoy these coloring resources all year ’round!” Happy National Coloring Book Day offers the title, and we may keep the spirit of National Coloring Book Day all the year round.

Confident in that? Then take on a challenge. Can you scroll down faster than Christian Lawson-Perfect’s web site can find factors? Prove your speed, prove your endurance, and see if you can overcome this infinite scroll.


The Third Tent

121 is a star number, the fifth of that select set. 121 identical items can be tiled to form a centered hexagon. You may have seen it in the German game of Chinese Checkers, as the board of that has 121 holes.

We come back again to teaching. “Many homeschoolers struggle with teaching their children math. Here are some tips to make it easier”, offers Denise Gaskins. Step forth and benefit from this FAQ: Struggling with Arithmetic, a collection of tips and thoughts and resources to help make arithmetic the more manageable.

Step now over to the arcade, and to the challenge of Pac-Man. This humble circle-inspired polygon must visit the entirety of a maze, and avoid ghosts as he does. Matthew Scroggs of Chalk Dust Magazine here seeks and shows us Optimal Pac-Man. Graph theory tells us there are thirteen billion different paths to take. Which of them is shortest? Which is fastest? Can it be known, and can it help you through the game?

And now a recreation, one to become useful if winter arrives. Think of the mysteries of the snowball rolling down a hill. How does it grow in size? How does it speed up? When does it stop? Rodolfo A Diaz, Diego L Gonzalez, Francisco Marin, and R Martinez satisfy your curiosity with Comparative kinetics of the snowball respect to other dynamical objects. Be warned! This material is best suited for the college-age student of the mathematical snow sciences.


The Fourth Tent

121 is furthermore the sixth of the centered octagonal numbers. 121 of a thing may be set into six concentric octagons of one, then two, then three, then four, then five, and then six of them on a side.

To teach is to learn! And we have here an example of such learning. James Sheldon writing for the American Mathematical Society Graduate Student blog offers Teaching Lessons from a Summer of Taking Mathematics Courses. What secrets has Sheldon to reveal? Come inside and learn what you may.

And now step over to the games area. The game Entanglement wraps you up in knots, challenging you to find the longest knot possible. David Richeson of Division By Zero sees in this A game for budding knot theorists. What is the greatest score that could be had in this game? Can it ever be found? Only Richeson has your answer.

Step now back to the amazing Mathstodon. Gaze in wonder at the account @dudeney_puzzles. Since the September of 2017 it has brought out challenges from Henry Ernest Dudeney’s Amusements in Mathematics. Puzzles given, yes, with answers that follow along. The impatient may find Dudeney’s 1917 book on Project Gutenberg among other places.


The Fifth Tent

Sum the digits of 121; you will find that you have four. Take its prime factors, 11 and 11, and sum their digits; you will find that this is four again. This makes 121 a Smith number. These marvels of the ages were named by Albert Wilansky, in honor of his brother-in-law, a man known to history as Harold Smith, and whose telephone number of 4,937,775 was one such.

Now let us consider terror. What is it to enter a PhD program? Many have attempted it; some have made it through. Mathieu Besançon gives to you a peek behind academia’s curtain. A year in PhD describes some of this life.

And now to an astounding challenge. Imagine an assassin readies your death. Can you protect yourself? At all? Tai-Danae Bradley invites you to consider: Is the Square a Secure Polygon? This question takes you on a tour of geometries familiar and exotic. Learn how mathematicians consider how to walk between places on a torus — and the lessons this has for a square room. The fate of the universe itself may depend on the methods described herein — the techniques used to study it relate to those that study whether a physical system can return to its original state. And then J2kun turned this into code, Visualizing an Assassin Puzzle, for those who dare to program it.

Have you overcome this challenge? Then step into the world of linear algebra, and this delight from the Mathstodon account of Christian Lawson-Perfect. The puzzle is built on the wonders of eigenvectors, those marvels of matrix multiplication. They emerge from multiplication longer or shorter but unchanged in direction. Lawson-Perfect uses whole numbers, represented by Scrabble tiles, and finds a great matrix with a neat eigenvalue. Can you prove that this is true?


The Sixth Tent

Another wonder of the digits of 121. Take them apart, then put them together again. Contorted into the form 112 they represent the same number. 121 is, in the base ten commonly used in the land, a Friedman Number, second of that line. These marvels, in the Arabic, the Roman, or even the Mayan numerals schemes, are named for Erich Friedman, a figure of mystery from the Stetson University.

We draw closer to the end of this carnival’s attractions! To the left I show a tool for those hoping to write mathematics: Donald E Knuth, Tracy Larrabee, and Paul M Roberts’s Mathematical Writing. It’s a compilation of thoughts about how one may write to be understood, or to avoid being misunderstood. Either would be a marvel for the ages.

To the right please see Gregory Taylor’s web comic Any ~Qs. Taylor — @mathtans on Twitter — brings a world of math-tans, personifications of mathematical concepts, together for adventures and wordplay. And if the strip is not to your tastes, Taylor is working on ε Project, a serialized written story with new installments twice a month.

If you will look above you will see the marvels of curved space. On YouTube, Eigenchris hopes to learn differential geometry, and shares what he has learned. While he has a series under way he suggested Episode 15, ‘Geodesics and Christoffel Symbols as one that new viewers could usefully try. Episode 16, ‘Geodesic Examples on Plane and Sphere, puts this work to good use.

And as we reach the end of the fairgrounds, please take a moment to try Find the Factors Puzzle number 121, a challenge from 2014 that still speaks to us today!

And do always stop and gaze in awe at the fantastic and amazing geometrical constructs of Robert Loves Pi. You shall never see stellations of its like elsewhere!


The Concessions Tent

With no thought of the risk to my life or limb I read the newspaper comics for mathematical topics they may illuminate! You may gape in awe at the results here. And furthermore this week and for the remainder of this calendar year of 2018 I dare to explain one and only one mathematical concept for each letter of our alphabet! I remind the sensitive patron that I have already done not one, not two, not three, but four previous entries all finding mathematical words for the letter “X” — will there be one come December? There is but one way you might ever know.

Denise Gaskins coordinates the Playful Mathematics Education Blog Carnival. Upcoming scheduled carnivals, including the chance to volunteer to host it yourself, or to recommend your site for mention, are listed here. And October’s 122nd Playful Mathematics Education Blog Carnival is scheduled to be hosted by Arithmophobia No More, and may this new host have the best of days!

The Mathematics Carnival is Coming


I apologize for falling even more silent than usual, and shall get to reviewing the past week’s comic strips soon. I had a big pile of life land on me, although, not so big a pile as landed on other people.

But people should know: the last week of the month is the time for the Mathematics Education Blog Carnival. It’s a collection of all sorts of fun mathematics. Educational? Sure. Playful? Sure. Tips? Games? Yes. The host for July was Joshua Greene, of Three J’s Learning. The host for this August is Iva Sallay, whose Find the Factors has been a reliably fun challenge, accompanied by neat little trivia about the whole numbers, for ages now. I hope you can look at her site, and at the carnival (when it posts), and that you enjoy.

And then now I’ve tipped my hand to who’s hosting the carnival for next month, and the last week of September.

Reading the Comics, July 7, 2018: Mutt and Jeff Relettering Scandal Edition


I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.

Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.

Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.

Lucy, going out in twilight with a pencil and sheet of paper: 'I'm going to count all the stars even if it kills me! People say I'm crazy, but I know I'm not , and that's what counts! I think I'll just sit here until it gets dark. This way I can take my time counting the stars. I'll mark 'em down as they come out. HA! There's the first one ... dum te ta te dum. There's another one. Two, three, four ... this is a cinch. Five, six, oh oh! SevenEightNineTen ... ElevenTwelveThirteen Oh, MY! They're coming out all over! SLOW DOWN! 21, 22, 23, 24, ... 35, 35, 40! Whew! (Gasp, gasp!) 41, 42 ... ' (Defeated Lucy sitting on the curb, exhausted, beneath the night sky.) 'Rats!'
Charles Schulz’s Peanuts Classics for the 6th of July, 2018. It originally ran the 4th of April, 1954. That is an adorable little adding machine and stool that Lucy has in the title panel there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.

It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.

Professor: 'Suppose you want to kill a baker. But, if you kill him, a bunch of starving people will get access to his bread. Should you do it anyway?' Caption: 'All moral dilemmas can be rephrased as evil-maximization problems.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of July, 2018. Confess I’m not sure the precise good-maximization reversal of this. I suppose it’s implying that the baker is refusing to give bread to starving people who can’t pay, and the hungry could alleviate the problem a while by eating the rich?

The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?

And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.

Jeff: 'You're such a fool, I'll bet you can't solve this simple problem!' Mutt: 'Which problem?' Jeff: 'If five men can eat a ham in five minutes, how long it will take ten men to eat that same ham?' Mutt: 'Well, some people eat slower.' Jeff: 'See? You just can't do it!' Mutt: 'Neither can you! It can't be solved!' Jeff: 'You say it can't be solved? Why?' Mutt: 'Because the first five men have ALREADY eaten the ham!'
Bud Fisher’s Mutt and Jeff for the 7th of July, 2018. So I found a previous iteration of this strip, from the 21st of February, 2015. They had relettered things, changing the wording slightly and making it overall somehow clunkier. The thing is, that 2015 strip looks to me like it might be a computer-lettered typeface too; look at the C’s, and the little loops on top of the letters. On the other hand, there’s some variation in the ? marks there. I understand relettering the more impenetrable old strips, especially if they don’t have the original material and have to go from archived newspaper prints. But the 2015 edition seems quite clear enough; why change that?

Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.

Teacher: 'Can you tell me how many triangles are in this diagram?' (It's an equilateral triangle, divided into thirds horizontally, and with the angle up top trisected, so that there are nine discrete figures inside.) Nancy, with a dozen scraps of used paper strewn around: 'Can you tell me how many pages we have to waste trying to solve this accursed puzzle?'
Olivia Jaimes’s Nancy for the 7th of July, 2018. There’s some real Old People Complaining in the comments, by the way, about how dare Nancy go sassing her elders like that. So, if you want to read those comments, judge wisely.

Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.


Care for more of this? You can catch all my Reading the Comics posts at this link. Essays with Saturday Morning Breakfast Cereal content are at this link. Essays with Peanuts are at this link. Those with Mutt and Jeff are at this link. And those with Nancy are here. Thank you.

Someone Else’s Homework: A Probability Question


My friend’s finished the last of the exams and been happy with the results. And I’m stuck thinking harder about a little thing that came across my Twitter feed last night. So let me share a different problem that we had discussed over the term.

It’s a probability question. Probability’s a great subject. So much of what people actually do involves estimating probabilities and making judgements based on them. In real life, yes, but also for fun. Like a lot of probability questions, this one is abstracted into a puzzle that’s nothing like anything anybody does for fun. But that makes it practical, anyway.

So. You have a bowl with fifteen balls inside. Five of the balls are labelled ‘1’. Five of the balls are labelled ‘2’. Five of the balls are labelled ‘3’. The balls are well-mixed, which is how mathematicians say that all of the balls are equally likely to be drawn out. Three balls are picked out, without being put back in. What’s the probability that the three balls have values which, together, add up to 6?

My friend’s instincts about this were right, knowing what things to calculate. There was part of actually doing one of these calculations that went wrong. And was complicated by my making a dumb mistake in my arithmetic. Fortunately my friend wasn’t shaken by my authority, and we got to what we’re pretty sure is the right answer.

Did The Greatest Generation Hosts Get As Drunk As I Expected?


I finally finished listening to Benjamin Ahr Harrison and Adam Pranica’s Greatest Generation podcast reviews of the first season of Star Trek: Deep Space Nine. (We’ve had fewer long car trips for this.) So I can return to my projection of how their drinking game would turn out.

Their plan was to make more exciting the discussion of some of Deep Space Nine‘s episodes by recording their reviews while drinking a lot. The plan was, for the fifteen episodes they had in the season, there would be a one-in-fifteen chance of doing any particular episode drunk. So how many drunk episodes would you expect to get, on this basis?

It’s a well-formed expectation value problem. There could be as few as zero or as many as fifteen, but some cases are more likely than others. Each episode could be recorded drunk or not-drunk. There’s an equal chance of each episode being recorded drunk. Whether one episode is drunk or not doesn’t depend on whether the one before was, and doesn’t affect whether the next one is. (I’ll come back to this.)

The most likely case was for there to be one drunk episode. The probability of exactly one drunk episode was a little over 38%. No drunk episodes was also a likely outcome. There was a better than 35% chance it would never have turned up. The chance of exactly two drunk episodes was about 19%. There drunk episodes had a slightly less than 6% chance of happening. Four drunk episodes a slightly more than 1% chance of happening. And after that you get into the deeply unlikely cases.

As the Deep Space Nine season turned out, this one-in-fifteen chance came up twice. It turned out they sort of did three drunk episodes, though. One of the drunk episodes turned out to be the first of two they planned to record that day. I’m not sure why they didn’t just swap what episode they recorded first, but I trust they had logistical reasons. As often happens with probability questions, the independence of events — whether a success for one affects the outcome of another — changes calculations.

There’s not going to be a second-season update to this. They’ve chosen to make a more elaborate recording game of things. They’ve set up a modified Snakes and Ladders type board with a handful of spots marked for stunts. Some sound like fun, such as recording without taking any notes about the episode. Some are, yes, drinking episodes. But this is all a very different and more complicated thing to project. If I were going to tackle that it’d probably be by running a bunch of simulations and taking averages from that.

Still from Deep Space Nine, season 6, episode 23, 'Profit and Lace', the sex-changed Quark feeling her breasts and looking horrified.
Real actual episode that was really actually made and really actually aired for real. I’m going to go ahead and guess that it hasn’t aged well.

Also I trust they’ve been warned about the episode where Quark has a sex change so he can meet a top Ferengi soda magnate after accidentally giving his mother a heart attack because gads but that was a thing that happened somehow.

A Bunch Of Tweets I’d Thought To Save


I’m slow about sharing them is all. It’s a simple dynamic: I want to write enough about each tweet that it’s interesting to share, and then once a little time has passed, I need to do something more impressive to be worth the wait. Eventually, nothing is ever shared. Let me try to fix that.

Just as it says: a link to Leonhard Euler’s Elements of Algebra, as rendered by Google Books. Euler you’ll remember from every field of mathematics ever. This 1770 textbook is one of the earliest that presents algebra that looks like, you know, algebra, the way we study it today. Much of that is because this book presented algebra so well that everyone wanted to imitate it.

An entry in the amusing and novel proofs. This one is John Conway’s candidate for most succinct published mathematics paper. It’s fun, at least as I understand fun to be.

This Theorem of the Day from back in November already is one about elliptic functions. Those came up several times in the Summer 2017 Mathematics A To Z. This day about the Goins-Maddox-Rusin Theorem on Heron Triangles, is dense reading even by the standards of the Theorem of the Day tweet (which fits each day’s theorem into a single slide). Still, it’s worth lounging about in the mathematics.

Elke Stangl, writing about one of those endlessly-to-me interesting subjects: phase space. This is a particular way of representing complicated physical systems. Set it up right and all sorts of physics problems become, if not easy, at least things there’s a standard set of tools for. Thermodynamics really encourages learning about such phase spaces, and about entropy, and here she writes about some of this.

So ‘e’ is an interesting number. At least, it’s a number that’s got a lot of interesting things built around it. Here, John Golden points out a neat, fun, and inefficient way to find the value of ‘e’. It’s kin to that scheme for calculating π inefficiently that I was being all curmudgeonly about a couple of Pi Days ago.

Jo Morgan comes to the rescue of everyone who tries to read old-time mathematics. There were a lot of great and surprisingly readable great minds publishing in the 19th century, but then you get partway through a paragraph and it might as well be Old High Martian with talk about diminishings and consequents and so on. So here’s some help.

As it says on the tin: a textbook on partial differential equations. If you find yourself adrift in the subject, maybe seeing how another author addresses the same subject will help, if nothing else for finding something familiar written in a different fashion.

And this is just fun: creating an ellipse as the locus of points that are never on the fold line when a circle’s folded by a particular rule.

Finally, something whose tweet origin I lost. It was from one of the surprisingly many economists I follow considering I don’t do financial mathematics. But it links to a bit of economic history: Origins of the Sicilian Mafia: The Market for Lemons. It’s 31 pages plus references. And more charts about wheat production in 19th century Sicily than I would have previously expected to see.

By the way, if you’re interested in me on Twitter, that would be @Nebusj. Thanks for stopping in, should you choose to.

Reading the Comics, January 3, 2018: Explaining Things Edition


There were a good number of mathematically-themed comic strips in the syndicated comics last week. Those from the first part of the week gave me topics I could really sink my rhetorical teeth into, too. So I’m going to lop those off into the first essay for last week and circle around to the other comics later on.

Jef Mallett’s Frazz started a week of calendar talk on the 31st of December. I’ve usually counted that as mathematical enough to mention here. The 1st of January as we know it derives, as best I can figure, from the 1st of January as Julius Caesar established for 45 BCE. This was the first Roman calendar to run basically automatically. Its length was quite close to the solar year’s length. It had leap days added according to a rule that should have been easy enough to understand (one day every fourth year). Before then the Roman calendar year was far enough off the solar year that they had to be kept in synch by interventions. Mostly, by that time, adding a short extra month to put things more nearly right. This had gotten all confusingly messed up and Caesar took the chance to set things right, running 46 BCE to 445 days long.

But why 445 and not, say, 443 or 457? And I find on research that my recollection might not be right. That is, I recall that the plan was to set the 1st of January, Reformed, to the first new moon after the winter solstice. A choice that makes sense only for that one year, but, where to set the 1st is literally arbitrary. While that apparently passes astronomical muster (the new moon as seen from Rome then would be just after midnight the 2nd of January, but hitting the night of 1/2 January is good enough), there’s apparently dispute about whether that was the objective. It might have been to set the winter solstice to the 25th of December. Or it might have been that the extra days matched neatly the length of two intercalated months that by rights should have gone into earlier years. It’s a good reminder of the difficulty of reading motivation.

Brian Fies’s The Last Mechanical Monster for the 1st of January, 2018, continues his story about the mad scientist from the Fleischer studios’ first Superman cartoon, back in 1941. In this panel he’s describing how he realized, over the course of his long prison sentence, that his intelligence was fading with age. He uses the ability to do arithmetic in his head as proof of that. These types never try naming, like, rulers of the Byzantine Empire. Anyway, to calculate the cube root of 50,653 in his head? As he used to be able to do? … guh. It’s not the sort of mental arithmetic that I find fun.

But I could think of a couple ways to do it. The one I’d use is based on a technique called Newton-Raphson iteration that can often be used to find where a function’s value is zero. Raphson here is Joseph Raphson, a late 17th century English mathematician known for the Newton-Raphson method. Newton is that falling-apples fellow. It’s an iterative scheme because you start with a guess about what the answer would be, and do calculations to make the answer better. I don’t say this is the best method, but it’s the one that demands me remember the least stuff to re-generate the algorithm. And it’ll work for any positive number ‘A’ and any root, to the ‘n’-th power.

So you want the n-th root of ‘A’. Start with your current guess about what this root is. (If you have no idea, try ‘1’ or ‘A’.) Call that guess ‘x’. Then work out this number:

\frac{1}{n}\left( (n - 1) \cdot x + \frac{A}{x^{n - 1}} \right)

Ta-da! You have, probably, now a better guess of the n-th root of ‘A’. If you want a better guess yet, take the result you just got and call that ‘x’, and go back calculating that again. Stop when you feel like your answer is good enough. This is going to be tedious but, hey, if you’re serving a prison term of the length of US copyright you’ve got time. (It’s possible with this sort of iterator to get a worse approximation, although I don’t think that happens with n-th root process. Most of the time, a couple more iterations will get you back on track.)

But that’s work. Can we think instead? Now, most n-th roots of whole numbers aren’t going to be whole numbers. Most integers aren’t perfect powers of some other integer. If you think 50,653 is a perfect cube of something, though, you can say some things about it. For one, it’s going to have to be a two-digit number. 103 is 1,000; 1003 is 1,000,000. The second digit has to be a 7. 73 is 343. The cube of any number ending in 7 has to end in 3. There’s not another number from 1 to 9 with a cube that ends in 3. That’s one of those things you learn from playing with arithmetic. (A number ending in 1 cubes to something ending in 1. A number ending in 2 cubes to something ending in 8. And so on.)

So the cube root has to be one of 17, 27, 37, 47, 57, 67, 77, 87, or 97. Again, if 50,653 is a perfect cube. And we can do better than saying it’s merely one of those nine possibilities. 40 times 40 times 40 is 64,000. This means, first, that 47 and up are definitely too large. But it also means that 40 is just a little more than the cube root of 50,653. So, if 50,653 is a perfect cube, then it’s most likely going to be the cube of 37.

Bill Watterson’s Calvin and Hobbes rerun for the 2nd is a great sequence of Hobbes explaining arithmetic to Calvin. There is nothing which could be added to Hobbes’s explanation of 3 + 8 which would make it better. I will modify Hobbes’s explanation of what the numerator. It’s ridiculous to think it’s Latin for “number eighter”. The reality is possibly more ridiculous, as it means “a numberer”. Apparently it derives from “numeratus”, meaning, “to number”. The “denominator” comes from “de nomen”, as in “name”. So, you know, “the thing that’s named”. Which does show the terms mean something. A poet could turn “numerator over denominator” into “the number of parts of the thing we name”, or something near enough that.

Hobbes continues the next day, introducing Calvin to imaginary numbers. The term “imaginary numbers” tells us their history: they looked, when first noticed in formulas for finding roots of third- and fourth-degree polynomials, like obvious nonsense. But if you carry on, following the rules as best you can, that nonsense would often shake out and you’d get back to normal numbers again. And as generations of mathematicians grew up realizing these acted like numbers we started to ask: well, how is an imaginary number any less real than, oh, the square root of six?

Hobbes’s particular examples of imaginary numbers — “eleventenn” and “thirty-twelve” — are great-sounding compositions. They put me in mind, as many of Watterson’s best words do, of a 1960s Peanuts in which Charlie Brown is trying to help Sally practice arithmetic. (I can’t find it online, as that meme with edited text about Sally Brown and the sixty grapefruits confounds my web searches.) She offers suggestions like “eleventy-Q” and asks if she’s close, which Charlie Brown admits is hard to say.

Cherry Trail: 'Good morning, honey! Where's Dad?' Mark Trail: 'He's out on the porch reading the paper!' Cherry: 'Rusty sure is excited about our upcoming trip to Mexico!' Mark: 'Did you get everything worked out with the school?' Cherry: 'Rusty will need to do some math assignments, but he'll get credit for his other subjects since it's an educational trip!'
James Allen’s Mark Trail for the 3rd of January, 2018. James Allen has changed many things about the comic strip since Jack Elrod’s retirement, as I have observed over on the other blog. There are less ruthlessly linear stories. There’s no more odd word balloon placement implying that giant squirrels are talking about the poachers. Mark Trail sometimes has internal thoughts. I’m glad that he does still choose to over-emphasize declarations like “[Your Dad]’s out on the porch reading the paper!” There are some traditions.

And finally, James Allen’s Mark Trail for the 3rd just mentions mathematics as the subject that Rusty Trail is going to have to do some work on instead of allowing the experience of a family trip to Mexico to count. This is of extremely marginal relevance, but it lets me include a picture of a comic strip, and I always like getting to do that.

Reading the Comics, December 9, 2017: Zach Weinersmith Wants My Attention Edition


If anything dominated the week in mathematically-themed comic strips it was Zach Weinersmith’s Saturday Morning Breakfast Cereal. I don’t know how GoComics selects the strips to (re?)print on their site. But there were at least four that seemed on-point enough for me to mention. So, okay. He’s got my attention. What’s he do with it?

On the 3rd of December is a strip I can say is about conditional probability. The mathematician might be right that the chance someone will be murdered by a serial killer are less than one in ten million. But that is the chance of someone drawn from the whole universe of human experiences. There are people who will never be near a serial killer, for example, or who never come to his attention or who evade his interest. But if we know someone is near a serial killer, or does attract his interest? The information changes the probability. And this is where you get all those counter-intuitive and somewhat annoying logic puzzles about, like, the chance someone’s other child is a girl if the one who just walked in was, and how that changes if you’re told whether the girl who just entered was the elder.

On the 5th is a strip about sequences. And built on the famous example of exponential growth from doubling a reward enough times. Well, you know these things never work out for the wise guy. The “Fibonacci Spiral” spoken of in the next-to-last panel is a spiral, like you figure. The dimensions of the spiral are based on those of golden-ratio rectangles. It looks a great deal like a logarithmic spiral to the untrained eye. Also to the trained eye, but you knew that. I think it’s supposed to be humiliating that someone would call such a spiral “random”. But I admit I don’t get that part.

The strip for the 6th has a more implicit mathematical content. It hypothesizes that mathematicians, given the chance, will be more interested in doing recreational puzzles than even in eating and drinking. It’s amusing, but I’ll admit I’ve found very few puzzles all that compelling. This isn’t to say there aren’t problems I keep coming back to because I’m curious about them, just that they don’t overwhelm my common sense. Don’t ask me when I last received actual pay for doing something mathematical.

And then on the 9th is one more strip, about logicians. And logic puzzles, such as you might get in a Martin Gardner collection. The problem is written out on the chalkboard with some shorthand logical symbols. And they’re symbols both philosophers and mathematicians use. The letter that looks like a V with a crossbar means “for all”. (The mnemonic I got was “it’s an A-for-all, upside-down”. This paired with the other common symbol, which looks like a backwards E and means there exists: “E-for-exists, backwards”. Later I noticed upside-down A and backwards E could both be just 180-degree-rotated A and E. But try saying “180-degree-rotated” in a quick way.) The curvy E between the letters ‘x’ and ‘S’ means “belongs to the set”. So that first line says “for all x that belong to the set S this follows”. Writing out “isLiar(x)” instead of, say, “L(x)”, is more a philosopher’s thing than a mathematician’s. But it wouldn’t throw anyway. And the T just means emphasizing that this is true.

And that is as much about Saturday Morning Breakfast Cereal as I have to say this week.

Sam Hurt’s Eyebeam for the 4th tells a cute story about twins trying to explain infinity to one another. I’m not sure I can agree with the older twin’s assertion that infinity means there’s no biggest number. But that’s just because I worry there’s something imprecise going on there. I’m looking forward to the kids learning about negative numbers, though, and getting to wonder what’s the biggest negative real number.

Percy Crosby’s Skippy for the 4th starts with Skippy explaining a story problem. One about buying potatoes, in this case. I’m tickled by how cranky Skippy is about boring old story problems. Motivation is always a challenge. The strip originally ran the 7th of October, 1930.

Dave Whamond’s Reality Check for the 6th uses a panel of (gibberish) mathematics as an example of an algorithm. Algorithms are mathematical, in origin at least. The word comes to us from the 9th century Persian mathematician Al-Khwarizmi’s text about how to calculate. The modern sense of the word comes from trying to describe the methods by which a problem can be solved. So, legitimate use of mathematics to show off the idea. The symbols still don’t mean anything.

Joe: 'Grandpa, what's 5x7?' Grandpa: 'Why do you wanna know?' Joe: 'I'm testing your memory.' Grandpa: 'Oh! The answer's 35.' Joe: 'Thanks! Now what is 8x8?' Grandpa: 'Joe, is that last night's homework?' Joe: 'We're almost done! Only 19 more!'
Rick Detorie’s One Big Happy for the 7th of December, 2017. And some attention, please, for Ruthie there. She’s completely irrelevant to the action, but it makes sense for her to be there if Grandpa is walking them to school, and she adds action — and acting — to the scenes.

Rick Detorie’s One Big Happy for the 7th has Joe trying to get his mathematics homework done at the last minute. … And it’s caused me to reflect on how twenty multiplication problems seems like a reasonable number to do. But there’s only fifty multiplications to even do, at least if you’re doing the times tables up to the 10s. No wonder students get so bored seeing the same problems over and over. It’s a little less dire if you’re learning times tables up to the 12s, but not that much better. Yow.

Olivia Walch’s Imogen Quest for the 8th looks pretty legitimate to me. It’s going to read as gibberish to people who haven’t done parametric functions, though. Start with the plane and the familiar old idea of ‘x’ and ‘y’ representing how far one is along a horizontal and a vertical direction. Here, we’re given a dummy variable ‘t’, and functions to describe a value for ‘x’ and ‘y’ matching each value of ‘t’. The plot then shows all the points that ever match a pair of ‘x’ and ‘y’ coordinates for some ‘t’. The top drawing is a shape known as the cardioid, because it kind of looks like a Valentine-heart. The lower figure is a much more complicated parametric equation. It looks more anatomically accurate,

Still no sign of Mark Anderson’s Andertoons and the drought is worrying me, yes.

But they’re still going on the cartoonist’s web site, so there’s that.

Reading the Comics, November 8, 2017: Uses Of Mathematics Edition


Was there an uptick in mathematics-themed comic strips in the syndicated comics this past week? It depends how tight a definition of “theme” you use. I have enough to write about that I’m splitting the week’s load. And I’ve got a follow-up to that Wronski post the other day, so I’m feeling nice and full of content right now. So here goes.

Zach Weinersmith’s Saturday Morning Breakfast Cereal posted the 5th gets my week off to an annoying start. Science and mathematics and engineering people have a tendency to be smug about their subjects. And to see aptitude or interest in their subjects as virtue, or at least intelligence. (If they see a distinction between virtue and intelligence.) To presume that an interest in the field I like is a demonstration of intelligence is a pretty nasty and arrogant move.

And yes, I also dislike the attitude that school should be about training people. Teaching should be about letting people be literate with the great thoughts people have had. Mathematics has a privileged spot here. The field, as we’ve developed it, seems to build on human aptitudes for number and space. It’s easy to find useful sides to it. Doesn’t mean it’s vocational training.

Lincoln Peirce’s Big Nate on the 6th discovered mathematics puzzles. And this gave him the desire to create a new mathematical puzzle that he would use to get rich. Good luck with that. Coming up with interesting enough recreational mathematics puzzles is hard. Presenting it in a way that people will buy is another, possibly greater, challenge. It takes luck and timing and presentation, just as getting a hit song does. Sudoku, for example, spent five years in the Dell Magazine puzzle books before getting a foothold in Japanese newspapers. And then twenty years there before being noticed in the English-speaking puzzle world. Big Nate’s teacher tries to encourage him, although that doesn’t go as Mr Staples might have hoped. (The storyline continues to the 11th. Spoiler: Nate does not invent the next great recreational mathematics puzzle.)

Jef Mallett’s Frazz for the 7th start out in a mathematics class, at least. I suppose the mathematical content doesn’t matter, though. Mallett’s making a point about questions that, I confess, I’m not sure I get. I’ll leave it for wiser heads to understand.

Mike Thompson’s Grand Avenue for the 8th is a subverted word-problem joke. And I suppose a reminder about the need for word problems to parse as things people would do, or might be interested in. I can’t go along with characterizing buying twelve candy bars “gluttonous” though. Not if they’re in a pack of twelve or something like that. I may be unfair to Grand Avenue. Mind, until a few years ago I was large enough my main method of getting around was “being rolled by Oompa-Loompas”, so I could be a poor judge.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 8th does a rounding joke. It’s not much, but I’ve included appearances of this joke before and it seems unfair to skip it this time.

Calculus For Breakfast


Robert Austin, of the RobertLovesPi blog, got to thinking about one of those interesting mathematics problems. It starts with the equations that describe the volume and the surface area of a sphere.

If the sphere has radius r, then the surface area of the sphere is 4πr2. And the volume is (4/3)πr3. What’s interesting about this is that there’s a relationship between these two expressions. The first is the derivative of the second. The derivative is one of the earliest things one learns in calculus. It describes how much a quantity changes with a tiny change in something it depends on.

And this got him to thinking about the surface area of a cube. Call the length of a cube’s side s. Its surface is six squares, each of them with a side of length s. So the surface area of each of the six squares is s2, which is obvious when you remember we call raising something to the second power “squaring”. Its total surface area then is 6s2. But its volume is is s3. This is why we even call raising something to the third power “cubing”. And the derivative of s3 is 3s2. (If you don’t know calculus, but you suspect you see a pattern here, you’re learning calculus. If you’re not sure about the pattern, let me tell you that the derivative of s4 would be 4s3, and the derivative of (1/3)s2 would be (2/3)s.)

There’s an obvious flaw there, and Austin’s aware of it. But it got him pondering different ways to characterize how big a cube is. He can find one that makes the relationship between volume and surface area work out like he expects. But the question remains, why that? And what about other shapes?

I think that’s an interesting discussion to have, and mean to think about it some more myself. And I wanted to point people who’d be interested over there to join in.

Proportional Dice


So, here’s a nice probability problem that recently made it to my Twitter friends page:

(By the way, I’m @Nebusj on Twitter. I’m happy to pick up new conversational partners even if I never quite feel right starting to chat with someone.)

Schmidt does assume normal, ordinary, six-sided dice for this. You can work out the problem for four- or eight- or twenty- or whatever-sided dice, with most likely a different answer.

But given that, the problem hasn’t quite got an answer right away. Reasonable people could disagree about what it means to say “if you roll a die four times, what is the probability you create a correct proportion?” For example, do you have to put the die result in a particular order? Or can you take the four numbers you get and arrange them any way at all? This is important. If you have the numbers 1, 4, 2, and 2, then obviously 1/4 = 2/2 is false. But rearrange them to 1/2 = 2/4 and you have something true.

We can reason this out. We can work out how many ways there are to throw a die four times, and so how many different outcomes there are. Then we count the number of outcomes that give us a valid proportion. That count divided by the number of possible outcomes is the probability of a successful outcome. It’s getting a correct count of the desired outcomes that’s tricky.