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.
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.
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.
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 , 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.
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?
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.
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.
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Non-limit calculating e by hand. https://t.co/Kv80RotboJ Fun activity & easily reproducible. Anyone know the author?
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.
For college students that will be taking partial differential equations next semester, here is a very good online book. https://t.co/txtfbMaRKc
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.
Here's a cool way to paper-fold an ellipse:
1) Cut a circle and fold it so that the circumference falls on a fixed point inside 2) Repeat this procedure using random folds pic.twitter.com/TAU50pvgll
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.
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:
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.
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.
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.
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.
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.
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.
(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.
I’m thinking to do a second Mathematics A-To-Z Glossary. For those who missed it, last summer I had a fun string of several weeks in which I picked a mathematical term and explained it to within an inch of its life, or 950 words, whichever came first. I’m curious if there’s anything readers out there would like to see me attempt to explain. So, please, let me know of any requests. All requests must begin with a letter, although numbers might be considered.
Meanwhile since there’s been some golden ratio talk around these parts the last few days, I thought people might like to see this neat Algebra Fact of the Day:
In a comment on my “Gilded Ratios” essay fluffy wondered about a variation on the Golden and Golden-like ratios. What’s interesting about the Golden Ratio and similar numbers is that their reciprocal — one divided by them — is a whole number less than the original number. That is, 1 divided by 1.618(etc) is 0.618(etc), which is 1 less than the original number. 1 divided by 2.414(etc) is 0.414(etc), exactly 2 less than the original 2.414(etc). 1 divided by 3.302(etc) is 0.302(etc), exactly 3 less than the original 3.302(etc).
fluffy wondered about a variation. Is there some number x that’s exactly 2 less than 2 divided by x? Or a (presumably) differently number that’s exactly 3 less than 3 divided by it? Yes, there is.
Let me call the whole number difference — the 1 or 2 or 3 or so on, referred to above — by the name b. And let me call the other number — the one that’s b less than b divided by it — by the name x. Then a number x, for which b divided by x is exactly b less than itself, makes true the equation . This is slightly different from the equation used last time, but not very different. Multiply both sides by x, which we know not to be zero, and we get a polynomial.
Yes, quadratic formula, I see you waving your hand in the back there. And you’re right. There are two x’s that will make that equation true. The positive one is . The negative one you get by changing the + sign, just before the square root, to a – sign, but who cares about that root? Here’s the first several of the (positive) silver-leaf ratios:
Some More Numbers With Cute Reciprocals
Looking over those hypnotic rows of digits past the decimal inspires thoughts. The part beyond the decimal keeps rising, closer and closer to 1. Does it ever get past 1? That is, might (say) the silver-leaf number that’s 2,038 more than its reciprocal be 2,039.11111 (or something)?
No, it never does. There are a couple of ways to prove that, if you feel like. We can take the approach that’s easiest (to my eyes) to imagine. It takes a little algebraic grinding to complete. That is to look for the smallest number b for which the silver-leaf number, , is larger than . Follow that out and you realize that it’s any value of b for which 0 is greater than 4. Logically, therefore, we need to take b into a private room and have a serious talk about its job performance, what with it not existing.
A harder proof to imagine working out, but that takes no symbol manipulation, comes from thinking about these reciprocals. Let’s imagine we had some b for which its corresponding silver-leaf number x is more than b + 1. Then, x – b has to be greater than 1. But if x is greater than 1, then its reciprocal has to be less than 1. We again have to talk with b about how its nonexistence is keeping it from doing its job.
Are there other proofs? Most likely. I was satisfied by this point, and resolved not to work on it more until the shower. Updates after breakfast, I suppose.
I may have mentioned that I regard the Golden Ratio as a lot of bunk. If I haven’t, allow me to mention: the Golden Ratio is a lot of bunk. I concede it’s a cute number. I found it compelling when I first had a calculator that let me use the last answer for a new operation. You can pretty quickly find that 1.618033 (etc, and the next digit is a 9 by the way) has a reciprocal that’s 0.618033 (etc).
There’s no denying that. And there’s no denying that’s a neat pattern. But it is not some aesthetic ideal. When people evaluate rectangles that “look best” they go to stuff that’s a fair but not too much wider in one direction than the other. But people aren’t drawn to 1.618 (etc) any more reliably than they like 1.6, or 1.8, or 1.5, or other possible ratios. And it is not any kind of law of nature that the Golden Ratio will turn up. It’s often found within the error bars of a measurement, but so are a lot of numbers.
The Golden Ratio is an irrational number, but basically all real numbers are irrational except for a few peculiar ones. Those peculiar ones happen to be the whole numbers and the rational numbers, which we find interesting, but which are the rare exception. It’s not a “transcendental number”, which is a kind of real number I don’t want to describe here. That’s a bit unusual, since basically all real numbers are transcendental numbers except for a few peculiar ones. Those peculiar ones include whole and rational numbers, and square roots and such, which we use so much we think they’re common. But not being transcendental isn’t that outstanding a feature. The Golden Ratio is one of those strange celebrities who’s famous for being a celebrity, and not for any actual accomplishment worth celebrating.
I started wondering: are there other Golden-Ratio-like numbers, though? The title of this essay gives what I suppose is the best name for this set. The Golden Ratio is interesting because its reciprocal — 1 divided by it — is equal to it minus 1. Is there another number whose reciprocal is equal to it minus 2? Another number yet whose reciprocal is equal to it minus 3?
So I looked. Is there a number between 2 and 3 whose reciprocal is it minus 2? Certainly there is. How do I know this?
Let me call this number, if it exists, x. The reciprocal of x is the number 1/x. The number x minus 2 is the number x – 2. We’ll pick up the pace in a little bit. Now imagine trying out every single number from 2 to 3, in order. The reciprocals 1/x start out at 1/2 and drop to 1/3. The subtracted numbers start out at 0 and grow to 1. There’s no gaps or sudden jumps or anything in either the reciprocals or the subtracted numbers. So there must be some x for which 1/x and x – 2 are the same number.
In the trade we call that an existence proof. It shows there’s got to be some answer. It doesn’t tell us much about what the answer is. Often it’s worth looking for an existence proof first. In this case, it’s probably overkill. But you can go from this to reasoning that there have to be Golden-Like-Ratio numbers between any two counting numbers. So, yes, there’s some number between 2,038 and 2,039 whose reciprocal is that number minus 2,038. That’s nice to know.
So what is the number that’s two more than its reciprocal? That’s whatever number or numbers make true the equation . That’s straightforward to solve. Multiply both sides by x, which won’t change whether the equation is true unless x is zero. (And x can’t be zero, or else we wouldn’t talk of 1/x except in hushed, embarrassed whispers.) This gets an equivalent equation . Subtract 1 from both sides, and we get and we’re set up to use the quadratic formula. The answer will be . The answer is about 2.414213562373095 (and on). (No, is not an answer; it’s not between 2 and 3.)
The number that’s three more than its reciprocal? We’ll call that x again, trusting that we remember this is a different number with the same name. For that we need to solve and that turns into the equation . And so and so it’s about 3.30277563773200. Yes, there’s another possible answer we rule out because it isn’t between 3 and 4.
We can do the same thing to find another number, named x, that’s four more than its reciprocal. That starts with and gets eventually to or about 4.23606797749979. We could go on like this. The number x that’s 2,038 more than its reciprocal is about 2038.00049082160.
If your eyes haven’t just slid gently past the equations you noticed the pattern. Suppose instead of saying 2 or 3 or 4 or 2038 we say the number b. b is some whole number, any that we like. The number whose reciprocal is exactly b less than it is the number x that makes true the equation . And that leads to the finding the number that makes the equation true.
And, what the heck. Here’s the first twenty or so gilded numbers. You can read this either as a list of the numbers I’ve been calling x — 1.618034, 2.414214, 3.302776 — or as an ordered list of the reciprocals of x — 0.618034, 0.414214, 0.302276 — as you like. I’ll call that the gilt; you add it to the whole number to its left to get that a number that, cutely, has a reciprocal that’s the same after the decimal.
I did think about including a graph of these numbers, but the appeal of them is that you can take the reciprocal and see digits not changing. A graph doesn’t give you that.
Some Numbers With Cute Reciprocals
None of these are important numbers. But they are pretty, and that can be enough on a quiet day.
I’m sure you’re intrigued too. It’s not the case that any old splitting of a strand of spaghetti will give you three pieces you can make into a triangle. You need the lengths of the three pieces to satisfy what’s imaginatively called the Triangle Inequality. That inequality demands the lengths of any two sides have to be greater than the length of the third side. So, suppose we start with spaghetti that’s 12 inches long, and we have it cut into pieces 5, 4, and 3 inches long: that’s fine. If we have it cut into pieces 9, 2, and 1 inches long, we’re stuck.
The Triangle Inequality is often known as the Cauchy Inequality, or the Cauchy-Schwarz Inequality, or the Cauchy-Bunyakovsky-Schwarz Inequality, or if that’s getting too long the CBS Inequality. And some pranksters reorder it to the Cauchy-Schwarz-Bunyakovsky Inequality. The Cauchy (etc) Inequality isn’t quite the same thing as the Triangle Inequality. But it’s an important and useful idea, and the Cauchy (etc) Inequality has the Triangle Inequality as one of its consequences. The name of it so overflows with names because mathematics history is complicated. Augustin-Louis Cauchy published the first proof of it, but for the special case of sums of series. Viktor Bunyakovsky proved a similar version of it for integrals, and has a name that’s so very much fun to say. Hermann Amandus Schwarz first put the proof into its modern form. So who deserves credit for it? Good question. If it influences your decision, know that Cauchy was incredibly prolific and has plenty of things named for him already. He’s got, without exaggeration, about eight hundred papers to his credit. Collecting all his work into a definitive volume took from 1882 to 1974.
Back to the spaghetti. The problem’s a fun one and if you follow the Twitter link above you’ll see the gritty work of mathematicians reasoning out the problem. As with every probability problem ever, the challenge is defining exactly what you’re looking for the probability of. This we call finding the “sample space”, the set of all the possible outcomes and how likely each of them are. Subtle changes in how you think of the problem will change whether you are right.
Smith cleans things up a bit, but preserves the essence of how the answer worked out. The answer that looks most likely correct was developed partly by reasoning and partly by numerical simulation. Numerical simulation is a great blessing for probability problems. Often the easiest way to figure out how likely something is will be trying it. But this does require working out the sample space, and what parts of the sample space represent what you’re interested in. With the information the numerical simulation provided, Smith was able to go back and find an analytic, purely reason-based, answer that looks plausible.
I have a combination lock at work. There are three digits, all in the range 1 – 40; they’re all prime numbers. They’re X+Y, X+2Y, X+3Y — where X and Y are positive integers.
If I told you what X was but not Y, you wouldn’t be able to tell me the combination. If I told you what Y was but not X, you wouldn’t be able to tell me the combination. Now, what’s the combination?
I did work out the puzzle. It did make me notice a couple of strings of uniformly-spaced prime numbers I hadn’t done before, too, such as 3-13-23. (However, 3-13-23 isn’t one of the possible answers, because of the constraints of the problem. There aren’t positive X and Y for which X + Y = 3, X + 2Y = 13, and X + 3Y = 23.)
As with the Singapore Birthday Problem, this is a puzzle based on reasoning about the information we have. Mercifully there aren’t actually many prime numbers below 40, so if you want you can take the brute force approach and find all the strings of uniformly-spaced prime numbers. Then you can find what one matches the rules in ChefMongoose’s second paragraph.
I confess I wasn’t that systematic. I had a strong suspicion what the starting number of the sequence had to be, and then did some tests to be sure. I credit that to just having stared at lot at the smaller prime numbers in my life, so I’d had some intuitive feel for it. That’s a dangerous way to work. My intuitive feel, for example, hadn’t warned me about 3-13-23. But then there aren’t other trios of prime numbers spaced by ten, so that set would be ruled out by the “If I told you what Y was but not X” constraint. But now I know how to get stuff out of ChefMongoose’s work locker, you know, just in case.
My friend ChefMongoose sent a neat little puzzle that came in a dream. I wanted to share it.
So! It’s not often that my dreams give me math puzzles. Here’s one: You are on floor 20 of a hotel. The stairs are blocked.
There are four elevators in front of you with display panels saying ‘3’, ‘4’, ‘7’, and ’35’. They will take you up that many floors, then the number will double. Going down an elevator will take you down that many floors, then the number will halve.
(The dream didn’t tell me what will happen if you can’t halve the number. For good puzzle logic, let’s assume the elevator goes down that much, then breaks.)
There is no basement, the hotel has an infinite amount of floors. Your challenge: get to floor 101. Can it be done?
(And I have no idea if it can be done. But apparently I, Riker, and Worf were trying to do it.)
The puzzle caught my imagination. It so happens the dream set things up so that this is possible: I worked out one path, and ChefMongoose found another.
ChefMongoose was right, of course, that something has to be done about halving the floor steps. I’d thought to make it half of either one less than or one more than the count. That is, going down 7 would turn the elevator into one that goes down either 3 or 4 floors. (My solution turned out not to need either.)
It looks lucky that ChefMongoose, Riker, and Worf picked a set of elevator moves, and rules, and starting, and ending floors that had a solution. Is it, though? Suppose we wanted to get to, say, floor 35? … Well, that’s possible. (Up 7, up 14, down 4, down 2.) 34 obviously follows. (Down 1 more.) 36? (Up 7, up 3, up 6.) 38? (Up 35, down 7, down 3, down 4, down 2, down 1.) The universe of reachable floors is bigger than it might seem at first.
The elevator problem had nagged at me with the thought it was related to some famous mathematical problem. At least that it was a type of one. ChefMongoose worked out what I was thinking of, the Collatz Conjecture. But on further reflection that’s the wrong parallel. This elevator problem is more akin to the McNuggets Problem. (When McDonald’s first sold Chicken McNuggets they were in packs of six, nine, and twenty. So what is the largest number of McNuggets that could not be bought by some combination of packages?) The doubling and halving of floor range makes the problem different, though. I am curious if there are finitely many unreachable floors. I’m also curious whether allowing negative numbers — basement floors — would change what floors are accessible.
The Collatz Conjecture is a fun one. It’s almost a game. Start with a positive whole number. If it’s even, divide it in half. If it’s odd, multiply it by three and add one. Then repeat.
If we start with 1, that’s odd, so we triple it and add one, giving us 4. Even, so cut in half: 2. Even again, so cut in half: 1. That’s going to bring us back to 4.
If we start with 2, we know where this is going. Cut in half: 1. Triple and add one: 4. Cut in half: 2. And repeat.
Starting with 3 suggests something new might happen. Triple 3 and add one: 10. Halve that: 5. Triple and add one: 16. Halve: 8. Halve: 4. Halve: 2. Halve: 1.
4 we’re already a bit sick of at this point. 5 — well, we just worked 5 out. That’ll go 5, 16, 8, 4, 2, 1, etc. Start from 6: we halve it to 3 and then we just worked out 3.
7 jumps right up to 22, then 11, then 34 — what a interesting number there — and then 52, 26, 13, 40, 20, 10 and we’ve seen that routine already. 10, 5, 16, 8, 4, 2, 1.
The Collatz Conjecture is that whatever positive whole number you start from will lead, eventually, to the 4, 2, 1 cycle. It may take a while to get there. I was working the numbers in my head while falling asleep the other night and got to wondering what exactly was 27’s problem anyway. (It takes over a hundred steps to settle down, and gets to numbers as high as 9,232 before finishing.)
Nobody knows whether it’s true. It seems plausible that it might be false. We can imagine a number that doesn’t. At least I can imagine there’s some number, let me call it N, and suppose it’s odd. Then triple that and add one, so we get an even number; halve that maybe a couple times until we get an odd number, and triple that and add one and get back the original N. You might have fun trying out numbers and seeing if you can find a loop like that.
Just do that for fun, though. Mathematicians have tested out every number less than 1,152,921,504,606,846,976. (It’s a round number in binary.) They all end in that 4, 2, 1 cycle. So it seems hard to believe that 1,152,921,504,606,846,977 and onward wouldn’t. We just don’t know that’s so.
If you allow zero, then that’s a valid but very short cycle: 0 halves to 0 and never budges. If you allow negative numbers, then there are at least three more cycles. They start from -1, from -5, and from -17. It’s not known whether there are any more in the negative integers.
The conjecture’s named for Lothar Collatz, 1910 – 1990, a German mathematician who specialized in numerical analysis. That’s the study of how to do calculations that meaningfully reflect the mathematics we would like to know. The Collatz Conjecture is, to the best of my knowledge, a novelty act. I don’t know of any interesting or useful results that depend on it being true (or false). It’s just a question easy to ask and understand, and that we don’t know how to solve. But those are fun to have around too.
Iva Sallay’s Find The Factors page I’ve mentioned before, since it provides a daily factorization puzzle. That’s a fun recreational mathematics puzzle even if the level 6’s and sometimes level 5’s will sometimes feel impossible. I wanted to point out, though, there’s also talk about the factoring of numbers, and ways to represent that factoring, that’s also interesting and attractive to look at. It can also include neat bits of trivia about numbers and their representation. In this example 585 presents some interesting facets, including several ways that it’s a palindromic number. If you don’t care for, or aren’t interested in, the factoring puzzles you might find it worth visiting for the trivia alone.
This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.
As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.
Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?
Since I’ve run out of letters there’s little dignified to do except end the Summer 2015 Mathematics A to Z. I’m still organizing my thoughts about the experience. I’m quite glad to have done it, though.
For the sake of good organization, here’s the set of pages that this project’s seen created: