Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition

I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Bill Amend’s FoxTrot for the 19th of May, 2019. Essays featuring FoxTrot, either the current (Sunday-only) strips or the 1990s-vintage reruns, should be at this link.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May, 2019. The strip originally ran the 17th of June, 1973. Comics Kingdom is printing both the current Funky Winkerbean strips and early-70s reprints. Essays that mention Funky Winkerbean, old or new, should appear at this link.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 10³. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10^-308 or as large as 10³⁰⁸, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10^-20, to a huge number, like 10²⁰ won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as . A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Mac King and Bill King’s Magic in a Minute for the 19th of May, 2019. So far as I know all these panels are new ones, although they do reuse gimmicks now and then. But the arithmetic and logic tricks featured in Magic In A Minute get discussed at this link, when they get mention from me at all.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Mark Anderson’s Andertoons for the 20th of May, 2019. Always glad to discuss Andertoons, as you can see from these essays.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.

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There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

Reading the Comics, September 6, 2016: Oh Thank Goodness We’re Back Edition

That’s a relief. After the previous week’s suspicious silence Comic Strip Master Command sent a healthy number of mathematically-themed comics my way. They cover a pretty normal spread of topics. So this makes for a nice normal sort of roundup.

Mac King and Bill King’s Magic In A Minute for the 4th is an arithmetic-magic-trick. Like most arithmetic-magic it depends on some true but, to me, dull bit of mathematics. In this case, that 81,234,567 minus 12,345,678 is equal to something. As a kid this sort of trick never impressed me because, well, anyone can do subtraction. I didn’t appreciate that the fun of stage magic in presenting well the mundane.

Jerry Scott and Jim Borgman’s Zits for the 5th is an ordinary mathematics-is-hard joke. But it’s elevated by the artwork, which shows off the expressive and slightly surreal style that makes the comic so reliable and popular. The formulas look fair enough, the sorts of things someone might’ve been cramming before class. If they’re a bit jumbled up, well, Pierce hasn’t been well.

Jerry Scott and Jim Borgman’s Zits for the 5th of September, 2016. It sure looks to me like there’s more things being explicitly multiplied by ‘1’ than are needed, but it might be the formulas got a little scrambled as Pierce vomited. We’ve all been there. Fun fact: apart from a bit in Calculus I where they drill you on differentiation formulas you never really need the secant. It makes a couple formulas a little more compact and that’s it, so if it’s been nagging at your mind go ahead and forget it.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 6th is an anthropomorphic-shapes joke and I feel like it’s been here before. Ah, yeah, there it is, from about this time last year. It’s a fair one to rerun.

Mustard and Boloney popped back in on the 8th with a strip I don’t have in my archive at least. It’s your standard Pi Pun, though. If they’re smart they’ll rerun it in March. I like the coloring; it’s at least a pleasant panel to look at.

Percy Crosby’s Skippy from the 9th of July, 1929 was rerun the 6th of September. It seems like a simple kid-saying-silly-stuff strip: what is the difference between the phone numbers Clinton 2651 and Clinton 2741 when they add to the same number? (And if Central knows what the number is why do they waste Skippy’s time correcting him? And why, 87 years later, does the phone yell at me for not guessing correctly whether I need the area code for a local number and whether I need to dial 1 before that?) But then who cares what the digits in a telephone number add to? What could that tell us about anything?

As phone numbers historically developed, the sum can’t tell us anything at all. But if we had designed telephone numbers correctly we could have made it … not impossible to dial a wrong number, but at least made it harder. This insight comes to us from information theory, which, to be fair, we have because telephone companies spent decades trying to work out solutions to problems like people dialing numbers wrong or signals getting garbled in the transmission. We can allow for error detection by schemes as simple as passing along, besides the numbers, the sum of the numbers. This can allow for the detection of a single error: had Skippy called for number 2641 instead of 2741 the problem would be known. But it’s helpless against two errors, calling for 2541 instead of 2741. But we could detect a second error by calculating some second term based on the number we wanted, and sending that along too.

By adding some more information, other modified sums of the digits we want, we can even start correcting errors. We understand the logic of this intuitively. When we repeat a message twice after sending it, we are trusting that even if one copy of the message is garbled the recipient will take the version received twice as more likely what’s meant. We can design subtler schemes, ones that don’t require we repeat the number three times over. But that should convince you that we can do it.

The tradeoff is obvious. We have to say more digits of the number we want. It isn’t hard to reach the point we’re ending more error-detecting and error-correcting numbers than we are numbers we want. And what if we make a mistake in the error-correcting numbers? (If we used a smart enough scheme, we can work out the error was in the error-correcting number, and relax.) If it’s important that we get the message through, we shrug and accept this. If there’s no real harm done in getting the message wrong — if we can shrug off the problem of accidentally getting the wrong phone number — then we don’t worry about making a mistake.

And at this point we’re only a few days into the week. I have enough hundreds of words on the close of the week I’ll put off posting that a couple of days. It’s quite good having the comics back to normal.

Theorem Thursday: Tutte’s Theorem, Magic, And Happy Endings

To wrap up my two-month Theorem Thursday project I have another request to fill. And it’s another from Gaurish, whose requests for the Leap Day A To Z inspired so many interesting posts.

Tutte’s Theorem

I admit I had not heard of Tutte’s Theorem before Gaurish’s request and I had to spend time working up to knowing what it was and why it was useful. I also admit I’m not completely sure I have it. But I’m happy to try to accept with grace corrections from the people in graph theory who know better.

It comes back to graphs. These are a bunch of points, “vertices”, which have curves, “edges” connecting pairs of them. This describes a lot of systems. Bunches of things that have to connect are naturally graphs. Connecting utilities to houses gave the first example I called up last week. The Internet’s made people vaguely familiar with the idea of bunches of computers linked to bunches of other computers. Social networks can be seen as graphs; each person is a vertex, her friendships edges connecting to other vertices.

In a graph not every vertex has to be connected to every other vertex. Not everyone needs to be friends with everyone else, which is a relief. I don’t want to point fingers but some of your friends are insufferable. I don’t know how you put up with them. You, you’re great. But the friends of a friend … yeeeeeeesh.

So we — mathematicians, anyway — get to wondering. Give me a graph. It’s got some vertices and some edges. Let’s look at only some of these edges. Each edge links to two vertices. Is there a subset of the edges that touch every one of the vertices exactly once? Sometimes there are; sometimes there aren’t. If there are, we have a “perfect matching”.

We look at this sort of thing because mathematicians learn to look at coverings. Coverings are what they sound like. What’s the smallest set of some standard item you need to … cover … all the stuff you’re interested in this problem? I think we’re bred to look for coverings in Real Analysis, because the idea of covering space with discs gives us measure. This gives us a rigorous idea of what length is, and what dimensions are, and lets us see there have to be more irrational than rational numbers and other interesting results like that. Mathematicians get used to looking for this sort of thing in other fields.

Tutte’s Theorem is about perfect matchings. It says what conditions a graph has to have to have a perfect matching. It’s based on striking subsets of the vertices from the original graph. The accompanying edges go too. What’s left might be connected, which means just what you think. You can get from any vertex in the decimated graph to any other vertex by following its edges. Or it might be disconnected, which means again just what you think. Mathematics is not always about complicated lingo.

Take the survivors. Count how many of the remaining connected components have an odd number of vertices. Is that less than or equal to the number of vertices you struck out? If it is, and if it is no matter how many vertices you struck out, and no matter how you arranged those vertices, then there’s a perfect matching.

This isn’t one for testing. I mean, consider what’s easier to do: start from your original graph and test out coloring in some edges to see if you can touch every edge the one time? Take every possible sub-graph of the original graph and count the number of connected sub-pieces with an odd number of vertices? … Well, maybe that isn’t so bad, if you set a computer to do the boring work. It’s going to take forever for your graph with 102 vertices, though, unless it’s a boring network. You have to test every possible removal all the way up to striking 102 vertices. (OK, it’s easy to show it’s true if 102 of 102 vertices are removed from the graph. Also if 101 of 102 vertices are removed. And 100 of 102 is also not hard. But that’s only a couple easy cases left.)

I don’t know the real work the theorem does. It has some neat and implications good for pop mathematics. Take a standard deck of 52 well-shuffled cards. Deal them out into thirteen piles of four cards each. Is it possible to select exactly one card from each pile so that, when done, there’s exactly one of each face value — Ace, Two, Three, Four, up through Queen and King — in the selected set? Indeed it is. I leave it to the Magic In A Minute cartoonists to turn this into a way to annoy a compliant monkey.

Another appealing use for it is in marriage problems. Well, marriage looked to mathematicians like a good source of bipartite graph problems. Society’s since noticed that nobody really hit on a compelling reason for “why does a woman have to marry a man, exactly”. I imagine the change to filter into mathematics textbooks sometime in the next seventy years. But please accept that this problem was formed and solved in the 1930s.

Suppose you have a group of women and a group of men, of equal size. Each woman would be happy married to some of the men in that group. Each man would be happy married to some of the women in that group. Is it possible to match women and men up in a way that everybody is married to at least someone they’d be happy with? We’re not guaranteeing that anyone gets their best possible pairing. We promise just that everyone marries someone they’ll be glad to.

It depends. Maybe this is best done by testing. Work through every possible subset of women. That is, look at every group of one woman, every group of two women, every group of three women, and so on. By group I mean what normal people mean by group, not what mathematicians mean. So look at your group of women. Count how many of the men at least one woman would be content marrying. Is that number equal to or larger than the number of women? Is that number equal to or larger than the number of women, however many women you picked and whichever women you did pick? If it did, great: everybody can marry someone they’re happy with.

Parlor tricks, I admit, but pleasant ones. What are its real uses? At this point I am really synthesizing my readings on the subject rather than speaking from things I’m confident about. If I understand right, though, Tutte’s Theorem is a foundation for the study of graph toughness. That is what you’d guess from the name. It’s about how easy it is to break up a graph into disconnected pieces. It’s easy to imagine real networks with strength or weakness. Image the person that holds a complicated group of friends together, for example, and whose removal devastates it. Consider the electrical network with a few vulnerable points that allow small problems to become nationwide blackouts. I believe it’s relevant to the study of proteins. Those long strands of blocks of molecules that then fold back on themselves. (I haven’t seen a source that says this, but it can’t imagine why it shouldn’t. I am open to correction from sneering protein researchers.) I would be surprised if the theorem has nothing to say about when a strand of Christmas tree lights will get too tangled to fix.

Let me close with a puzzle, a delightful little one. It regards one of the graphs we met last week. K₅ is a complete graph with five vertices. Each vertex is connected to all four of its siblings. It can’t have a perfect matching. Only a graph with an even number of vertices can. Each edge connects to two vertices, after all. So — what is the subset that breaks Tutte’s theorem? It must be possible to strike some set of vertices from K₅ so that the number of stricken vertices is smaller than the number of remaining connected components with an odd number of vertices. What’s that set?

Go ahead and ponder it a while. If you give up I don’t blame you. The answer is at the other end of this link. If you’d like a hint, let me offer this, which you might be able to mouse over to reveal.

It is obvious once you’ve heard what the subset of vertices is, and it is not K₅. The rest of this paragraph is padding so that the size of the spoiler doesn’t give matters away. And by the way I’d like to thank the world at large for doing such a great job not spoiling Star Wars: The Force Awakens. So why could you not give some similar consideration for Star Trek Beyond? I stopped reading TrekBBS for a month partly to avoid spoilers and then I started getting them everywhere I looked. Not cool, world.

Good luck!

Reading the Comics, July 6, 2016: Another Busy Week Edition

It’s supposed to be the summer vacation. I don’t know why Comic Strip Master Command is so eager to send me stuff. Maybe my standards are too loose. This doesn’t even cover all of last week’s mathematically-themed comics. I’ll need another that I’ve got set for Tuesday. I don’t mind.

Corey Pandolph and Phil Frank and Joe Troise’s The Elderberries rerun for the 3rd features one of my favorite examples of applied probability. The game show Deal or No Deal offered contestants the prize within a suitcase they picked, or a dealer’s offer. The offer would vary up or down as non-selected suitcases were picked, giving the chance for people to second-guess themselves. It also makes a good redemption game. The banker’s offer would typically be less than the expectation value, what you’d get on average from all the available suitcases. But now and then the dealer offered more than the expectation value and I got all ready to yell at the contestants.

This particular strip focuses on a smaller question: can you pick which of the many suitcases held the grand prize? And with the right setup, yes, you can pick it reliably.

Mac King and Bill King’s Magic in a Minute for the 3rd uses a bit of arithmetic to support a mind-reading magic trick. The instructions say to start with a number from 1 to 10 and do various bits of arithmetic which lead inevitably to 4. You can prove that for an arbitrary number, or you can just try it for all ten numbers. That’s tedious but not hard and it’ll prove the inevitability of 4 here. There aren’t many countries with names that start with ‘D’; Denmark’s surely the one any American (or European) reader is likeliest to name. But Dominica, the Dominican Republic, and Djibouti would also be answers. (List Of Countries Of The World.com also lists Dhekelia, which I never heard of either.) Anyway, with Denmark forced, ‘E’ almost begs for ‘elephant’. I suppose ’emu’ would do too, or ‘echidna’. And ‘elephant’ almost forces ‘grey’ for a color, although ‘white’ would be plausible too. A magician has to know how things like this work.

Werner Wejp-Olsen’s feature Inspector Danger’s Crime Quiz for the 4th features a mathematician as victim of the day’s puzzle murder. I admit I’m skeptical of deathbed identifications of murderers like this, but it would spoil a lot of puzzle mysteries if we disallowed them. (Does anyone know how often a deathbed identification actually happens?) I can’t make the alleged answer make any sense to me. Danger of the trade in murder puzzles.

Kris Straub’s Starship for the 4th uses mathematics as a stand-in for anything that’s hard to study and solve. I’m amused.

John Hambrock’s The Brilliant Mind of Edison lee for the 6th of July, 2016. I’m a little surprised the last panel wasn’t set on a duplicate Earth where things turned out a little differently.

John Hambrock’s The Brilliant Mind of Edison lee for the 6th is about the existentialist dread mathematics can inspire. Suppose there is a chance, within any given volume of space, of Earth being made. Well, it happened at least once, didn’t it? If the universe is vast enough, it seems hard to argue that there wouldn’t be two or three or, really, infinitely many versions of Earth. It’s a chilling thought. But it requires some big suppositions, most importantly that the universe actually is infinite. The observable universe, the one we can ever get a signal from, certainly isn’t. The entire universe including the stuff we can never get to? I don’t know that that’s infinite. I wouldn’t be surprised if it’s impossible to say, for good reason. Anyway, I’m not worried about it.

Jim Meddick’s Monty for the 6th is part of a storyline in which Monty is worshipped by tiny aliens who resemble him. They’re a bit nerdy, and calculate before they understand the relevant units. It’s a common mistake. Understand the problem before you start calculating.

Reading the Comics, April 10, 2016: Four-Digit Prime Number Edition

In today’s installment of Reading The Comics, mathematics gets name-dropped a bunch in strips that aren’t really about my favorite subject other than my love. Also, I reveal the big lie we’ve been fed about who drew the Henry comic strip attributed to Carl Anderson. Finally, I get a question from Queen Victoria. I feel like this should be the start of a podcast.

Patrick Roberts’ Todd the Dinosaur for the 6th of April, 2016.

Patrick Roberts’ Todd the Dinosaur for the 6th of April just name-drops mathematics. The flash cards suggest it. They’re almost iconic for learning arithmetic. I’ve seen flash cards for other subjects. But apart from learning the words of other languages I’ve never been able to make myself believe they’d work. On the other hand, I haven’t used flash cards to learn (or teach) things myself.

Joe Martin’s Boffo for the 7th of April, 2016. I bet the link expires in early May.

Joe Martin’s Boffo for the 7th of April is a solid giggle. (I have a pretty watery giggle myself.) There are unknowable, or at least unprovable, things in mathematics. Any logic system with enough rules to be interesting has ideas which would make sense, and which might be true, but which can’t be proven. Arithmetic is such a system. But just fractions and long division by itself? No, I think we need something more abstract for that.

Carl Anderson’s Henry for the 7th of April, 2016.

Carl Anderson’s Henry for the 7th of April is, of course, a rerun. It’s also a rerun that gives away that the “Carl Anderson” credit is a lie. Anderson turned over drawing the comic strip in 1942 to John Liney, for weekday strips, and Don Trachte for Sundays. There is no possible way the phrase “New Math” appeared on the cover of a textbook Carl Anderson drew. Liney retired in 1979, and Jack Tippit took over until 1983. Then Dick Hodgins, Jr, drew the strip until 1990. So depending on how quickly word of the New Math penetrated Comic Strip Master Command, this was drawn by either Liney, Tippit, or possibly Hodgins. (Peanuts made New Math jokes in the 60s, but it does seem the older the comic strip the longer it takes to mention new stuff.) I don’t know when these reruns date from. I also don’t know why Comics Kingdom is fibbing about the artist. But then they went and cancelled The Katzenjammer Kids without telling anyone either.

Eric the Circle for the 8th, this one by “lolz”, declares that Eric doesn’t like being graphed. This is your traditional sort of graph, one in which points with coordinates x and y are on the plot if their values make some equation true. For a circle, that equation’s something like (x – a)² + (y – b)² = r². Here (a, b) are the coordinates for the point that’s the center of the circle, and r is the radius of the circle. This looks a lot like Eric is centered on the origin, the point with coordinates (0, 0). It’s a popular choice. Any center is as good. Another would just have equations that take longer to work with.

Richard Thompson’s Cul de Sac rerun for the 10th is so much fun to look at that I’m including it even though it just name-drops mathematics. The joke would be the same if it were something besides fractions. Although see Boffo.

Norm Feuti’s Gil rerun for the 10th takes on mathematics’ favorite group theory application, the Rubik’s Cube. It’s the way I solved them best. This approach falls outside the bounds of normal group theory, though.

Mac King and Bill King’s Magic in a Minute for the 10th shows off a magic trick. It’s also a non-Rubik’s-cube problem in group theory. One of the groups that a mathematics major learns, after integers-mod-four and the like, is the permutation group. In this, the act of swapping two (or more) things is a thing. This puzzle restricts the allowed permutations down to swapping one item with the thing next to it. And thanks to that, an astounding result emerges. It’s worth figuring out why the trick would work. If you can figure out the reason the first set of switches have to leave a penny on the far right then you’ve got the gimmick solved.

Pab Sungenis’s New Adventures of Queen Victoria for the 10th made me wonder just how many four-digit prime numbers there are. If I haven’t worked this out wrong, there’s 1,061 of them.

Some Cards Stuff

Paper that proved seven shuffles will sufficiently randomize a deck of cards https://t.co/9zVcWJbE7K via @madcaptenor

— Probability Fact (@ProbFact) February 18, 2016

I’m not good at shuffling cards. I can eventually scramble up a deck tolerably well, given time, but I can’t do a good riffle shuffle. Nor can I do any of the moves that show competence at card-shuffling. That thing where people make a little semicircular arch of cards in their hands? I can’t even understand how that works, never mind do it myself.

That said, I do know how many shuffles it takes to randomize a deck of cards. I mean a standard deck of 52. It’s seven. I learned that ages ago, but never saw it proved. Best I could work out was that each shuffling would, if done perfectly, mix the upper and lower halves of the old deck. So I want what had been (say) the first and second card to be mixed up arbitrarily far from one another. One shuffle might get one or two cards between the old first and second. Two shuffles might double that; three shuffles that again, and so on. By six shuffles there could be anywhere up to 64 cards between the first and second, and that’s … surely enough, right? Then one more for good luck? It’s not rigorous but you can see where that satisfies.

The Probability Fact of the Day tweet above gives a real explanation. It links to the paper Trailing the Dovetail Shuffle to its Lair, by Dr Dave Bayer and Dr Persi Diaconis. It first appeared in the Annals of Applied Probability, 1992, Vol 2, Number 2, 294 – 313. It gives an actual proof of why seven shuffles are what’s needed.

I’m sad to admit the paper isn’t one you can jump into without mathematical training. Even the symbols may seem bizarre: it uses π not for that thing to do with circles. Instead it’s used as a variable name, the way we might use ‘x’ in ordinary algebra. In this context it stands for ‘permutation’. That’s a standard thing to do in this field of mathematics. It just looks funny if you’re coming in cold.

A permutation, here, means how you change the order of things. For example, suppose we start out with five things, which I’ll label with the letters of the alphabet. Suppose they start out in the order A B C D E. (We could use other labels, but letters are convenient.) I can apply a permutation π to this ordered list of letters. Suppose that afterwards they end up in the order C A B E D. Then the permutation π did this: it moved the first thing to the second spot. It moved the second thing to the third spot. It moved the third thing to the first spot. It moved the fourth thing to the fifth spot. It moved the fifth thing to the fourth spot. There are several ways to describe this efficiently. I could say, for example, that the permutation π is (2 3 1 5 4). (If you don’t see why that works, think about it a while.) There’s other ways to write this. We don’t need them just now.

You can chain permutations together. If we did the same swapping of order on C A B E D, we would get the list B C A D E. That’s the same list we would have gotten if we had started with A B C D E and done a different permutation once. It’s what we would get if we had done (3 1 2 4 5). We can think of this as what you get if we “multiply” π by π. Permutations, these directions of how to shuffle a list of things, can work a lot like numbers.

There’s more interesting things in here, even if you don’t follow the argument. I admit I get lost somewhere in section 3. I’m hoping someone asks me about the baker’s transformation. But it does describe some impressive-sounding magic tricks to be done with a slightly shuffled deck. And it gives this great puzzle, as well as answering it.

Suppose someone has a well-shuffled deck of cards. She deals them one at a time. You try to guess what card is coming up next. And you never make the foolish mistake of predicting a card that’s already come up. Most of the time you’ll be wrong. But at least you’ll end on a success. After 51 cards have been dealt you will call the final one right.

How many cards would you expect, on average, to call correctly, out of these 52?

Reading the Comics, September 10, 2015: Back To School Edition

I assume that Comic Strip Master Command ordered many mathematically-themed comic strips to coincide with the United States school system getting back up to full. That or they knew I’d have a busy week. This is only the first part of comic strips that have appeared since Tuesday.

Mel Henze’s Gentle Creatures for the 7th and the 8th of September use mathematical talk to fill out the technobabble. It’s a cute enough notion. These particular strips ran last year, and I talked about them then. The talk of a “Lagrangian model” interests me. It name-checks a real and important and interesting scientist who’s not Einstein or Stephen Hawking. But I’m still not aware of any “Lagrangian model” that would be relevant to starship operations.

Jon Rosenberg’s Scenes from a Multiverse for the 7th of September speaks of a society of “powerful thaumaturgic diagrammers” who used Venn diagrams not wisely but too well. The diagrammers got into trouble when one made “a Venn diagram that showed the intersection of all the Venns and all the diagrams”. I imagine this not to be a rigorous description of what happened. But Venn diagrams match up well with many logic problems. And self-referential logic, logic statements that describe their own truth or falsity, is often problematic. So I would accept a story in which Venn diagrams about Venn diagrams leads to trouble. The motif of tying logic and mathematics into magic is an old one. I understand it. A clever mathematical argument often feels like magic, especially the surprising ones. To me, the magical theorems are those that prove a set of seemingly irrelevant lemmas. Then, with that stock in hand, the theorem goes on to the main point in a few wondrous lines. If you can do that, why not transmute lead, or accidentally retcon a society out of existence?

Mark Anderson’s Andertoons for the 8th of September just delights me. Occasionally I feel a bit like Mark Anderson’s volunteer publicity department. A panel like this, though, makes me feel that he deserves it.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 8th of September is the first anthropomorphic-geometric-figures joke we’ve had here in a while.

Mike Baldwin’s Cornered for the 9th of September is a drug testing joke, and a gambling joke. Both are subjects driven by probabilities. Any truly interesting system is always changing. If we want to know whether something affects the system we have to know whether we can make a change that’s bigger than the system does on its own. And this gives us drug-testing and other statistical inference tests. If we apply a drug, or some treatment, or whatever, how does the system change? Does it change enough, consistently, that it’s not plausible that the change just happened by chance? Or by some other influence?

You might have noticed a controversy going around psychology journals. A fair number of experiments were re-run, by new experimenters following the original protocols as closely as possible. Quite a few of the reported results didn’t happen again, or happened in a weaker way. That’s produced some handwringing. No one thinks deliberate experimental fraud is that widespread in the field. There may be accidental fraud, people choosing data or analyses that heighten the effect they want to prove, or that pick out any effect. However, it may also simply be chance again. Psychology experiments tend to have a lower threshold of “this is sufficiently improbable that it indicates something is happening” than, say, physics has. Psychology has a harder time getting the raw data. A supercollider has enormous startup costs, but you can run the thing for as long as you like. And every electron is the same thing. A test of how sleep deprivation affects driving skills? That’s hard. No two sleepers or drivers are quite alike, even at different times of the day. There’s not an obvious cure. Independent replication of previously done experiments helps. That’s work that isn’t exciting — necessary as it is, it’s also repeating what others did — and it’s harder to get people to do it, or pay for it. But in the meantime it’s harder to be sure what interesting results to trust.

Ruben Bolling’s Super-Fun-Pak Comix for the 9th of September is another Chaos Butterfly installment. I don’t want to get folks too excited for posts I technically haven’t written yet, but there is more Chaos Butterfly soon.

Rick Stromoski’s Soup To Nutz for the 10th of September has Royboy guess the odds of winning a lottery are 50-50. Silly, yes, but only because we know that anyone is much more likely to lose a lottery than to win it. But then how do we know that?

Since the rules of a lottery are laid out clearly we can reason about the probability of winning. We can calculate the number of possible outcomes of the game, and how many of them count as winning. Suppose each of those possible outcomes are equally likely. Then the probability of winning is the number of winning outcomes divided by the number of probable outcomes. Quite easy.

— Of course, that’s exactly what Royboy did. There’s two possible outcomes, winning or losing. Lacking reason to think they aren’t equally likely he concluded a win and a loss were just as probable.

We have to be careful what we mean by “an outcome”. What we probably mean for a drawn-numbers lottery is the number of ways the lottery numbers can be drawn. For a scratch-off card we mean the number of tickets that can be printed. But we’re still stuck with this idea of “equally likely” outcomes. I suspect we know what we mean by this, but trying to say what that is clearly, and without question-begging, is hard. And even this works only because we know the rules by which the lottery operates. Or we can look them up. If we didn’t know the details of the lottery’s workings, past the assumption that it has consistently followed rules, what could we do?

Well, that’s what we have probability classes for, and particularly the field of Bayesian probability. This field tries to estimate the probabilities of things based on what actually happens. Suppose Royboy played the lottery fifty times and lost every time. That would smash the idea that his chances were 50-50, although that would not yet tell him what the chances really are.

Reading the Comics, June 30, 2015: Fumigating The Theater Edition

One of my favorite ever episodes of The Muppet Show when I was a kid had the premise the Muppet Theater was being fumigated and so they had to put on a show from the train station instead. (It was the Loretta Lynn episode, third season, number eight.) I loved seeing them try to carry on as normal when not a single thing was as it should be. Since then — probably before, too, but I don’t remember that — I’ve loved seeing stuff trying to carry on in adverse circumstances.

Why this is mentioned here is that Sunday night my computer had a nasty freeze and some video card mishaps. I discovered that my early-2011 MacBook Pro might be among those recalled earlier this year for a service glitch. My computer is in for what I hope is a simple, free, and quick repair. But obviously I’m not at my best right now. I might be even longer than usual answering people and goodness knows how the statistics survey of June will go.

Anyway. Rick Kirkman and Jerry Scott’s Baby Blues (June 26) is a joke about motivating kids to do mathematics. And about how you can’t do mathematics over summer vacation.

Rick Kirkman and Jerry Scott’s Baby Blues for the 26th of June, 2015.

Ruben Bolling’s Tom The Dancing Bug (June 26) features a return appearance of Chaos Butterfly. Chaos Butterfly does what Chaos Butterfly does best.

Charles Schulz’s Peanuts Begins (June 26; actually just the Peanuts of March 23, 1951) uses arithmetic as a test of smartness. And as an example of something impractical.

Alex Hallatt’s Arctic Circle (June 28) is a riff on the Good Will Hunting premise. That movie’s particular premise — the janitor solves an impossible problem left on the board — is, so far as I know, something that hasn’t happened. But it’s not impossible. Training will help one develop reasoning ability. Training will provide context and definitions and models to work from. But that’s not essential. All that’s essential is the ability to reason. Everyone has that ability; everyone can do mathematics. Someone coming from outside the academy could do first-rate work. However, I’d bet on the person with the advanced degree in mathematics. There is value in training.

Alex Hallatt’s Arctic Circle for the 28th of June, 2015.

But as many note, the Good Will Hunting premise has got a kernel of truth in it. In 1939, George Dantzig, a grad student in mathematics at University of California/Berkeley, came in late to class. He didn’t know that two problems on the board were examples of unproven theorems, and assumed them to be homework. So he did them, though he apologized for taking so long to do them. Before you draw too much inspiration from this, though, remember that Dantzig was a graduate student almost ready to start work on a PhD thesis. And the problems were not thought unsolvable, just conjectures not yet proven. Snopes, as ever, provides some explanation of the legend and some of the variant ways the story is told.

Mac King and Bill King’s Magic In A Minute (June 28) shows off a magic trick that you could recast as a permutations problem. If you’ve been studying group theory, and many of my Mathematics A To Z terms have readied you for group theory, you can prove why this trick works.

Guy Gilchrist’s Nancy (June 28) carries on Baby Blues‘s theme of mathematics during summer vacation being simply undoable.

Piers Baker’s Ollie and Quentin for December 28, 2014, and repeated on June 28, 2015.

Piers Baker’s Ollie and Quentin (June 28) is a gambler’s fallacy-themed joke. It was run — on ComicsKingdom, back then — back in December, and I talked some more about it then.

Mike Twohy’s That’s Life (June 28) is about the perils of putting too much attention into mental arithmetic. It’s also about how perilously hypnotic decimals are: if the pitcher had realized “fourteen million over three years” must be “four and two-thirds million per year” he’d surely have been less distracted.

Reading the Comics, March 4, 2015: Driving Me Crazy Edition

I like it when there are themes to these collections of mathematical comics, but since I don’t decide what subjects cartoonists write about — Comic Strip Master Command does — it depends on luck and my ability to dig out loose connections to find any. Sometimes, a theme just drops into my lap, though, as with today’s collection: several cartoonists tossed off bits that had me double-checking their work and trying to figure out what it was I wasn’t understanding. Ultimately I came to the conclusion that they just made mistakes, and that’s unnerving since how could a mathematical error slip through the rigorous editing and checking of modern comic strips?

Mac and Bill King’s Magic in a Minute (March 1) tries to show off how to do a magic trick based on parity, using the spots on a die to tell whether it was turned in one direction or another. It’s a good gimmick, and parity — whether something is odd or even — can be a great way to encode information or to do simple checks against slight errors. That said, I believe the Kings made a mistake in describing the system: I can’t figure out how the parity of the three sides of a die facing you could not change, from odd to even or from even to odd, as the die is rotated one turn. I believe they mean that you should just count the dots on the vertical sides, so that for example in the “Howdy Do It?” panel in the lower right corner, add two and one to make three. But with that corrected it should be a good trick.

Reading The Comics, November 9, 2014: Finally, A Picture Edition

I knew if I kept going long enough some cartoonist not on Gocomics.com would have to mention mathematics. That finally happened with one from Comics Kingdom, and then one from the slightly freak case of Rick Detorie’s One Big Happy. Detorie’s strip is on Gocomics.com, but a rerun from several years ago. He has a different one that runs on the normal daily pages. This is for sound economic reasons: actual newspapers pay much better than the online groupings of them (considering how cheap Comics Kingdom and Gocomics are for subscribers I’m not surprised) so he doesn’t want his current strips run on Gocomics.com. As for why his current strips do appear on, for example, the fairly good online comics page of AZcentral.com, that’s a good question, and one that deserves a full answer.

Vic Lee’s Pardon My Planet for the 6th of November, 2014.

Vic Lee’s Pardon My Planet (November 9), which broke the streak of Comics Kingdom not making it into these pages, builds around a quote from Einstein I never heard of before but which sounds like the sort of vaguely inspirational message that naturally attaches to famous names. The patient talks about the difficulty of finding something in “the middle of four-dimensional curved space-time”, although properly speaking it could be tricky finding anything within a bounded space, whether it’s curved or not. The generic mathematics problem you’d build from this would be to have some function whose maximum in a region you want to find (if you want the minimum, just multiply your function by minus one and then find the maximum of that), and there’s multiple ways to do that. One obvious way is the mathematical equivalent of getting to the top of a hill by starting from wherever you are and walking the steepest way uphill. Another way is to just amble around, picking your next direction at random, always taking directions that get you higher and usually but not always refusing directions that bring you lower. You can probably see some of the obvious problems with either approach, and this is why finding the spot you want can be harder than it sounds, even if it’s easy to get started looking.

Reuben Bolling’s Super Fun-Pak Comix (November 6), which is technically a rerun since the Super Fun-Pak Comix have been a longrunning feature in his Tom The Dancing Bug pages, is primarily a joke about the Heisenberg Uncertainty Principle, that there is a limit to what information one can know about the universe. This limit can be understood mathematically, though. The wave formulation of quantum mechanics describes everything there is to know about a system in terms of a function, called the state function and normally designated Ψ, the value of which can vary with location and time. Determining the location or the momentum or anything about the system is done by a process called “applying an operator to the state function”. An operator is a function that turns one function into another, which sounds like pretty sophisticated stuff until you learn that, like, “multiply this function by minus one” counts.

In quantum mechanics anything that can be observed has its own operator, normally a bit tricker than just “multiply this function by minus one” (although some are not very much harder!), and applying that operator to the state function is the mathematical representation of making that observation. If you want to observe two distinct things, such as location and momentum, that’s a matter of applying the operator for the first thing to your state function, and then taking the result of that and applying the operator for the second thing to it. And here’s where it gets really interesting: it doesn’t have to, but it can depend what order you do this in, so that you get different results applying the first operator and then the second from what you get applying the second operator and then the first. The operators for location and momentum are such a pair, and the result is that we can’t know to arbitrary precision both at once. But there are pairs of operators for which it doesn’t make a difference. You could, for example, know both the momentum and the electrical charge of Scott Baio simultaneously to as great a precision as your Scott-Baio-momentum-and-electrical-charge-determination needs are, and the mathematics will back you up on that.

Ruben Bolling’s Tom The Dancing Bug (November 6), meanwhile, was a rerun from a few years back when it looked like the Large Hadron Collider might never get to working and the glitches started seeming absurd, as if an enormous project involving thousands of people and millions of parts could ever suffer annoying setbacks because not everything was perfectly right the first time around. There was an amusing notion going around, illustrated by Bolling nicely enough, that perhaps the results of the Large Hadron Collider would be so disastrous somehow that the universe would in a fit of teleological outrage prevent its successful completion. It’s still a funny idea, and a good one for science fiction stories: Isaac Asimov used the idea in a short story dubbed “Thiotimoline and the Space Age”, published 1959, which resulted in the attempts to manipulate a compound which dissolves before it adds water might have accidentally sent hurricanes Carol, Edna, and Diane into New England in 1954 and 1955.

Chip Sansom’s The Born Loser (November 7) gives me a bit of a writing break by just being a pun strip that you can save for next March 14.

Dan Thompson’s Brevity (November 7), out of reruns, is another pun strip, though with giant monsters.

Francesco Marciuliano’s Medium Large (November 7) is about two of the fads of the early 80s, those of turning everything into a breakfast cereal somehow and that of playing with Rubik’s Cubes. Rubik’s Cubes have long been loved by a certain streak of mathematicians because they are a nice tangible representation of group theory — the study of things that can do things that look like addition without necessarily being numbers — that’s more interesting than just picking up a square and rotating it one, two, three, or four quarter-turns. I still think it’s easier to just peel the stickers off (and yet, the die-hard Rubik’s Cube Popularizer can point out there’s good questions about polarity you can represent by working out the rules of how to peel off only some stickers and put them back on without being detected).

Rick Detorie’s One Big Happy for the 9th of November, 2014.

Rick Detorie’s One Big Happy (November 9), and I’m sorry, readers about a month in the future from now, because that link’s almost certainly expired, is another entry in the subject of word problems resisted because the thing used to make the problem seem less abstract has connotations that the student doesn’t like.

Fred Wagner’s Animal Crackers (November 9) is your rare comic that could be used to teach positional notation, although when you actually pay attention you realize it doesn’t actually require that.

Mac and Bill King’s Magic In A Minute (November 9) shows off a mathematically-based slight-of-hand trick, describing a way to make it look like you’re reading your partner-monkey’s mind. This is probably a nice prealgebra problem to work out just why it works. You could also consider this a toe-step into the problem of encoding messages, finding a way to send information about something in a way that the original information can be recovered, although obviously this particular method isn’t terribly secure for more than a quick bit of stage magic.

Reading the Comics, October 25, 2012

As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with gocomics.com strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.