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  • Joseph Nebus 6:00 pm on Sunday, 5 March, 2017 Permalink | Reply
    Tags: , , Joe Vanilla, Luann Againn, , , Poor Richard's Almanac, ,   

    Reading the Comics, March 4, 2017: Frazz, Christmas Trees, and Weddings Edition 


    It was another of those curious weeks when Comic Strip Master Command didn’t send quite enough comics my way. Among those they did send were a couple of strips in pairs. I can work with that.

    Samson’s Dark Side Of The Horse for the 26th is the Roman Numerals joke for this essay. I apologize to Horace for being so late in writing about Roman Numerals but I did have to wait for Cecil Adams to publish first.

    In Jef Mallett’s Frazz for the 26th Caulfield ponders what we know about Pythagoras. It’s hard to say much about the historical figure: he built a cult that sounds outright daft around himself. But it’s hard to say how much of their craziness was actually their craziness, how much was just that any ancient society had a lot of what seems nutty to us, and how much was jokes (or deliberate slander) directed against some weirdos. What does seem certain is that Pythagoras’s followers attributed many of their discoveries to him. And what’s certain is that the Pythagorean Theorem was known, at least a thing that could be used to measure things, long before Pythagoras was on the scene. I’m not sure if it was proved as a theorem or whether it was just known that making triangles with the right relative lengths meant you had a right triangle.

    Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.

    Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.

    Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.

    Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.

    'Has Carl given you any reason not to trust him?' 'No, not yet. But he might.' 'Fi ... you seek 100% certainty in people, but that doesn't exist. In the end,' and Dethany is drawn as her face on a pi symbol, 'we're *all* irrational numbers.'

    Bill Holbrook’s On The Fastrack for the 3rd of March, 2017. Fi’s dress isn’t one of those … kinds with the complicated pattern of holes in it. She got it torn while trying to escape the wedding and falling into the basement.

    Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.

    That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.

    'Fi ... humans are complicated. Like the irrational number pi, we can go on forever. You never get to the bottom of us! But right now, upstairs, there are two variables who *want* you in their lives. Assign values to them.' Carl, Fi's fiancee, is drawn as his face with a y; his kid as a face on an x.

    Bill Holbrook’s On The Fastrack for the 4th of March, 2017. I feel bad that I completely forgot Carl had a kid and that the face on the x doesn’t help me remember anything.

    Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.

     
  • Joseph Nebus 6:00 pm on Friday, 3 March, 2017 Permalink | Reply
    Tags: , , The Straight Dope   

    How To Use Roman Numerals (A Not Quite Useful Guide) 


    I haven’t got the chance to write a proper essay today, but did want to be sure people didn’t miss The Straight Dope this week. Cecil Adams gets the question “How did anyone do math in Roman numerals?” and does what he can to answer in a couple hundred words of newspaper space.

    It’ll disappoint you if you have visions of whipping through a quadratic equation written all in V’s and L’s and stuff. Roman numeral arithmetic is really easy for addition and subtraction. Multiplication and division turn into real challenges for which you need mechanical aid and the abacus. Adams describes this loosely, although not in enough detail that you’ll come away confident with your abacus. Fair enough. I’ve got a charming little abacus myself, someone’s gift to me, and I can’t use it even to the slight extent I can use a slide rule.

    The important thing, though, is that as a young know-it-all Cecil Adams’s first two books, The Straight Dope and The Return of the Straight Dope, were just magnificently important reading. Not as hefty as David Wallechinsky and Irving Wallace’s The People’s Almanac 2, but with a much higher fascinating-stuff-to-boring-stuff ratio. Stuff on Oak Island’s Treasure Pit and the (former) names of New York City boroughs and the like. I’m glad it’s still there.

     
    • Biff Sock Pow 10:16 pm on Friday, 3 March, 2017 Permalink | Reply

      Great post! I used to enjoy The Straight Dope when I first moved to Dallas in the 1980s. I want to say he was printed in the Dallas Observer, along with another literary titan of the time, Joe Bob Briggs (back when he just wrote spot-on movie reviews and wry observations about the local scene). But many of those brain cells from way back then are deceased, and so I could be misremembering the whole thing. Also, I still have my copy of The People’s Almanac (Roman numeral uno) that I purchased brand-spanking new back in 1975 or so. As a youth with lots of time on my hands, I read it cover to cover (all 1446 pages of it), slowing down in the more prurient parts, of course. It was an awesome book! Thanks for the trip down memory lane! Made me feel like I was XVI again.

      Liked by 1 person

      • Joseph Nebus 4:55 am on Saturday, 11 March, 2017 Permalink | Reply

        I’m glad hearing all this. I never saw The Straight Dope in newspapers, just in the book forms, tucked into the fascinating-miscellaneous books section of the library. And so read it a lot, over and over. The People’s Almanac 2 was one that my family had for some reason or other. We never had Almanac 1, and I never saw a copy. In college the newspaper office did briefly have a copy of The People’s Almanac 3 but I didn’t get the chance to absorb that nearly so well.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Tuesday, 28 February, 2017 Permalink | Reply
    Tags: , , , Julian Dates, , , US Naval Observatory   

    How To Work Out The Length Of Time Between Two Dates 


    September 1999 was a heck of a month you maybe remember. There that all that excitement of the Moon being blasted out of orbit thanks to the nuclear waste pile up there getting tipped over or something. And that was just as we were getting over the final new episode of Mystery Science Theater 3000‘s first airing. That episode was number 1003, Merlin’s Shop of Mystical Wonders, which aired a month after the season finale because of one of those broadcast rights tangles that the show always suffered through.

    Time moves on, and strange things happen, and show co-creator and first host Joel Hodgson got together a Kickstarter and a Netflix deal. The show’s Season Eleven is supposed to air starting the 14th of April, this year. The natural question: how long will we go, then, between new episodes of Mystery Science Theater 3000? Or more generally, how do you work out how long it is between two dates?

    The answer is dear Lord under no circumstances try to work this out yourself. I’m sorry to be so firm. But the Gregorian calendar grew out of a bunch of different weird influences. It’s just hard to keep track of all the different 31- and 30-day months between two events. And then February is all sorts of extra complications. It’s especially tricky if the interval spans a century year, like 2000, since the majority of those are not leap years, except that the one century year I’m likely to experience was. And then if your interval happens to cross the time the local region switched from the Julian to the Gregorian calendar —

    So my answer is don’t ever try to work this out yourself. Never. Just refuse the problem if you’re given it. If you’re a consultant charge an extra hundred dollars for even hearing the problem.

    All right, but what if you really absolutely must know for some reason? I only know one good answer. Convert the start and the end dates of your interval into Julian Dates and subtract one from the other. I mean subtract the smaller number from the larger. Julian Dates are one of those extremely minor points of calendar use. They track the number of days elapsed since noon, Universal Time, on the Julian-calendar date we call the 1st of January, 4713 BC. The scheme, for years, was set up in 1583 by Joseph Justus Scalinger, calendar reformer, who wanted for convenience an index year so far back that every historically known event would have a positive number. In the 19th century the astronomer John Herschel expanded it to date-counting.

    Scalinger picked the year from the convergence of a couple of convenient calendar cycles about how the sun and moon move as well as the 15-year indiction cycle that the Roman Empire used for tax matters (and that left an impression on European nations). His reasons don’t much matter to us. The specific choice means we’re not quite three-fifths of the way through the days in the 2,400,000’s. So it’s not rare to modify the Julian Date by subtracting 2,400,000 from it. The date starts from noon because astronomers used to start their new day at noon, which was more convenient for logging a whole night’s observations. Since astronomers started taking pictures of stuff and looking at them later they’ve switched to the new day starting at midnight like everybody else, but you know what it’s like changing an old system.

    This summons the problem: so how do I know many dates passed between whatever day I’m interested in and the Julian Calendar 1st of January, 4713 BC? Yes, there’s a formula. No, don’t try to use it. Let the fine people at the United States Naval Observatory do the work for you. They know what they’re doing and they’ve had this calculator up for a very long time without any appreciable scandal accruing to it. The system asks you for a time of day, because the Julian Date increases as the day goes on. You can just make something up if the time doesn’t matter. I normally leave it on midnight myself.

    So. The last episode of Mystery Science Theater 3000 to debut, on the 12th of September, 1999, did so on Julian Date 2,451,433. (Well, at 9 am Eastern that day, but nobody cares about that fine grain a detail.) The new season’s to debut on Netflix the 14th of April, 2017, which will be Julian Date 2,457,857. (I have no idea if there’s a set hour or if it’ll just become available at 12:01 am in whatever time zone Netflix Master Command’s servers are in.) That’s a difference of 6,424 days. You’re on your own in arguing about whether that means it was 6,424 or 6,423 days between new episodes.

    If you do take anything away from this, though, please let it be the warning: never try to work out the interval between dates yourself.

     
    • elkement (Elke Stangl) 9:31 am on Friday, 3 March, 2017 Permalink | Reply

      And I figured the routine date and time conversion mess you face as a software developer is a challenge ;-) …

      Like

      • Joseph Nebus 4:53 am on Saturday, 11 March, 2017 Permalink | Reply

        Oh you have no idea. In that one ancient database was designed with every column a string, and dates entered as literally, eg, ’03/10/2017′. That string of text. Which was all right when the date just had to be shown on-screen but then I had said it should be easy to include a date range, unaware of just what was in the database. Also, that there are so many mistakes too. Or people entering 00/00/0000 when the date wasn’t available.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Sunday, 26 February, 2017 Permalink | Reply
    Tags: , , Flo and Friends, , , , Promises Promises, , , , Tiger   

    Reading the Comics, February 23, 2017: The Week At Once Edition 


    For the first time in ages there aren’t enough mathematically-themed comic strips to justify my cutting the week’s roundup in two. No, I have no idea what I’m going to write about for Thursday. Let’s find out together.

    Jenny Campbell’s Flo and Friends for the 19th faintly irritates me. Flo wants to make sure her granddaughter understands that just because it takes people on average 14 minutes to fall asleep doesn’t mean that anyone actually does, by listing all sorts of reasons that a person might need more than fourteen minutes to sleep. It makes me think of a behavior John Allen Paulos notes in Innumeracy, wherein the statistically wise points out that someone has, say, a one-in-a-hundred-million chance of being killed by a terrorist (or whatever) and is answered, “ah, but what if you’re that one?” That is, it’s a response that has the form of wisdom without the substance. I notice Flo doesn’t mention the many reasons someone might fall asleep in less than fourteen minutes.

    But there is something wise in there nevertheless. For most stuff, the average is the most common value. By “the average” I mean the arithmetic mean, because that is what anyone means by “the average” unless they’re being difficult. (Mathematicians acknowledge the existence of an average called the mode, which is the most common value (or values), and that’s most common by definition.) But just because something is the most common result does not mean that it must be common. Toss a coin fairly a hundred times and it’s most likely to come up tails 50 times. But you shouldn’t be surprised if it actually turns up tails 51 or 49 or 45 times. This doesn’t make 50 a poor estimate for the average number of times something will happen. It just means that it’s not a guarantee.

    Gary Wise and Lance Aldrich’s Real Life Adventures for the 19th shows off an unusually dynamic camera angle. It’s in service for a class of problem you get in freshman calculus: find the longest pole that can fit around a corner. Oh, a box-spring mattress up a stairwell is a little different, what with box-spring mattresses being three-dimensional objects. It’s the same kind of problem. I want to say the most astounding furniture-moving event I’ve ever seen was when I moved a fold-out couch down one and a half flights of stairs single-handed. But that overlooks the caged mouse we had one winter, who moved a Chinese finger-trap full of crinkle paper up the tight curved plastic to his nest by sheer determination. The trap was far longer than could possibly be curved around the tube. We have no idea how he managed it.

    J R Faulkner’s Promises, Promises for the 20th jokes that one could use Roman numerals to obscure calculations. So you could. Roman numerals are terrible things for doing arithmetic, at least past addition and subtraction. This is why accountants and mathematicians abandoned them pretty soon after learning there were alternatives.

    Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. Probably anything would do for the blackboard problem, but something geometry reads very well.

    Jef Mallett’s Frazz for the 21st makes some comedy out of the sort of arithmetic error we all make. It’s so easy to pair up, like, 7 and 3 make 10 and 8 and 2 make 10. It takes a moment, or experience, to realize 78 and 32 will not make 100. Forgive casual mistakes.

    Bud Fisher’s Mutt and Jeff rerun for the 22nd is a similar-in-tone joke built on arithmetic errors. It’s got the form of vaudeville-style sketch compressed way down, which is probably why the third panel could be made into a satisfying final panel too.

    'How did you do on the math test?' 'Terrible.' 'Will your mom be mad?' 'Maybe. But at least she'll know I didn't cheat!'

    Bud Blake’s Tiger for the 23rd of February, 2017. I want to blame the colorists for making Hugo’s baby tooth look so weird in the second and third panels, but the coloring is such a faint thing at that point I can’t. I’m sorry to bring it to your attention if you didn’t notice and weren’t bothered by it before.

    Bud Blake’s Tiger rerun for the 23rd just name-drops mathematics; it could be any subject. But I need some kind of picture around here, don’t I?

    Mike Baldwin’s Cornered for the 23rd is the anthropomorphic numerals joke for the week.

     
  • Joseph Nebus 6:00 pm on Thursday, 23 February, 2017 Permalink | Reply
    Tags: , , compulsions, , Over The Hedge, , , Wide Open   

    Reading the Comics, February 15, 2017: SMBC Does Not Cut In Line Edition 


    On reflection, that Saturday Morning Breakfast Cereal I was thinking about was not mathematically-inclined enough to be worth including here. Helping make my mind up on that was that I had enough other comic strips to discuss here that I didn’t need to pad my essay. Yes, on a slow week I let even more marginal stuff in. Here’s the comic I don’t figure to talk about. Enjoy!

    Jack Pullan’s Boomerangs rerun for the 16th is another strip built around the “algebra is useless in real life” notion. I’m too busy noticing Mom in the first panel saying “what are you doing play [sic] video games?” to respond.

    Ruben Bolling’s Super-Fun-Pak Comix excerpt for the 16th is marginal, yeah, but fun. Numeric coincidence and numerology can sneak into compulsions with terrible ease. I can believe easily the need to make the number of steps divisible by some favored number.

    Rich Powell’s Wide Open for the 16th is a caveman science joke, and it does rely on a chalkboard full of algebra for flavor. The symbols come tantalizingly close to meaningful. The amount of kinetic energy, K or KE, of a particle of mass m moving at speed v is indeed K = \frac{1}{2} m v^2 . Both 16 and 32 turn up often in the physics of falling bodies, at least if we’re using feet to measure. a = -\frac{k}{m} x turns up in physics too. It comes from the acceleration of a mass on a spring. But an equation of the same shape turns up whenever you describe things that go through tiny wobbles around the normal value. So the blackboard is gibberish, but it’s a higher grade of gibberish than usual.

    Rick Detorie’s One Big Happy rerun for the 17th is a resisting-the-word-problem joke, made fresher by setting it in little Ruthie’s playing at school.

    T Lewis and Michael Fry’s Over The Hedge for the 18th mentions the three-body problem. As Verne the turtle says, it’s a problem from physics. The way two objects — sun and planet, planet and moon, pair of planets, whatever — orbit each other if they’re the only things in the universe is easy. You can describe it all perfectly and without using more than freshman physics majors know. Introduce a third body, though, and we don’t know anymore. Chaos can happen.

    Emphasis on can. There’s no good way to solve the “general” three-body problem, the one where the star and planets can have any sizes and any starting positions and any starting speeds. We can do well for special cases, though. If you have a sun, a planet, and a satellite — each body negligible compared to the other — we can predict orbits perfectly well. If the bodies have to stay in one plane of motion, instead of moving in three-dimensional space, we can do pretty well. If we know two of the bodies orbit each other tightly and the third is way off in the middle of nowhere we can do pretty well.

    But there’s still so many interesting cases for which we just can’t be sure chaos will not break out. Three interacting bodies just offer so much more chance for things to happen. (To mention something surely coincidental, it does seem to be a lot easier to write good comedy, or drama, with three important characters rather than two. Any pair of characters can gang up on the third, after all. I notice how much more energetic Over The Hedge became when Hammy the Squirrel joined RJ and Verne as the core cast.)

    Dave Whamond’s Reality Check for the 18th is your basic mathematics-illiteracy joke, done well enough.

     
  • Joseph Nebus 6:00 pm on Tuesday, 21 February, 2017 Permalink | Reply
    Tags: automated proofs, , , ,   

    One Way To Get Your Own Theorem 


    While doing some research to better grouse about Ken Keeler’s Futurama theorem I ran across an amusing site I hadn’t known about. It is Theory Mine, a site that allows you to hire — and name — a genuine, mathematically sound theorem. The spirit of the thing is akin to that scam in which you “name” a star. But this is more legitimate in that, you know, it’s got any legitimacy. For this, you’re buying naming rights from someone who has any rights to sell. By convention the discoverer of a theorem can name it whatever she wishes, and there’s one chance in ten that anyone else will use the name.

    I haven’t used it. I’ve made my own theorems, thanks, and could put them on a coffee mug or t-shirt if I wished to make a particularly boring t-shirt. But I’m delighted by the scheme. They don’t have a team of freelance mathematicians whipping up stuff and hoping it isn’t already known. Not for the kinds of prices they charge. This should inspire the question: well, where do the theorems come from?

    The scheme uses an automated reasoning system. I don’t know the details of how it works, but I can think of a system by which this might work. It goes back to the Crisis of Foundations, the time in the late 19th/early 20th century when logicians got very worried that we were still letting physical intuitions and unstated assumptions stay in our mathematics. One solution: turn everything into symbols, icons with no connotations. The axioms of mathematics become a couple basic symbols. The laws of logical deduction become things we can do with the symbols, converting one line of symbols into a related other line. Every line we get is a theorem. And we know it’s correct. To write out the theorem in this scheme is to write out its proof, and to feel like you’re touching some deep magic. And there’s no human frailties in the system, besides the thrill of reeling off True Names like that.

    You may not be sure what this works like. It may help to compare it to a slightly-fun number coding scheme. I mean the one where you start with a number, like, ‘1’. Then you write down how many times and which digit appears. There’s a single ‘1’ in that string, so you would write down ’11’. And repeat: In ’11’ there’s a sequence of two ‘1’s, so you would write down ’21’. And repeat: there’s a single ‘2’ and a single ‘1’, so you then write down ‘1211’. And again: there’s a single ‘1’, a single ‘2’, and then a double ‘1’, so you next write ‘111221’. And so on until you get bored or die.

    When we do this for mathematics we start with a couple different basic units. And we also start with several things we may do at most symbols. So there’s rarely a single line that follows from the previous. There’s an ever-expanding tree of known truths. This may stave off boredom but I make no promises about death.

    The result of this is pages and pages that look like Ancient High Martian. I don’t feel the thrill of doing this. Some people do, though. And as recreational mathematics goes I suppose it’s at least as good as sudoku. Anyway, this kind of project, rewarding indefatigability and thoroughness, is perfect for automation anyway. Let the computer work out all the things we can prove are true.

    If I’m reading Theory Mine’s description correctly they seem to be doing something roughly like this. If they’re not, well, you go ahead and make your own rival service using my paragraphs as your system. All I ask is one penny for every use of L’Hôpital’s Rule, a theorem named for Guillaume de l’Hôpital and discovered by Johann Bernoulli. (I have heard that Bernoulli was paid for his work, but I do not know that’s true. I have now explained why, if we suppose that to be true, my prior sentence is a very funny joke and you should at minimum chuckle.)

    This should inspire the question: what do we need mathematicians for, then? It’s for the same reason we need writers, when it would be possible to automate the composing of sentences that satisfy the rules of English grammar. I mean if there were rules to English grammar. That we can identify a theorem that’s true does not mean it has even the slightest interest to anyone, ever. There’s much more that could be known than that we could ever care about.

    You can see this in Theory Mine’s example of Quentin’s Theorem. Quentin’s Theorem is about an operation you can do on a set whose elements consist of the non-negative whole numbers with a separate value, which they call color, attached. You can add these colored-numbers together according to some particular rules about how the values and the colors add. The order of this addition normally matters: blue two plus green three isn’t the same as green three plus blue two. Quentin’s Theorem finds cases where, if you add enough colored-numbers together, the order doesn’t matter. I know. I am also staggered by how useful this fact promises to be.

    Yeah, maybe there is some use. I don’t know what it is. If anyone’s going to find the use it’ll be a mathematician. Or a physicist who’s found some bizarre quark properties she wants to codify. Anyway, if what you’re interested in is “what can you do to make a vertical column stable?” then the automatic proof generator isn’t helping you at all. Not without a lot of work put in to guiding it. So we can skip the hard work of finding and proving theorems, if we can do the hard work of figuring out where to look for these theorems instead. Always the way.

    You also may wonder how we know the computer is doing its work right. It’s possible to write software that is logically proven to be correct. That is, the software can’t produce anything but the designed behavior. We don’t usually write software this way. It’s harder to write, because you have to actually design your software’s behavior. And we can get away without doing it. Usually there’s some human overseeing the results who can say what to do if the software seems to be going wrong. Advocates of logically-proven software point out that we’re getting more software, often passing results on to other programs. This can turn a bug in one program into a bug in the whole world faster than a responsible human can say, “I dunno. Did you try turning it off and on again?” I’d like to think we could get more logically-proven software. But I also fear I couldn’t write software that sound and, you know, mathematics blogging isn’t earning me enough to eat on.

    Also, yes, even proven software will malfunction if the hardware the computer’s on malfunctions. That’s rare, but does happen. Fortunately, it’s possible to automate the checking of a proof, and that’s easier to do than creating a proof in the first place. We just have to prove we have the proof-checker working. Certainty would be a nice thing if we ever got it, I suppose.

     
    • mathtuition88 5:01 am on Wednesday, 22 February, 2017 Permalink | Reply

      Computers are getting more amazing!

      Like

      • Joseph Nebus 4:19 pm on Saturday, 25 February, 2017 Permalink | Reply

        They are astounding, which makes it only the more baffling that we can’t get iTunes to reliably download new episodes of a podcast we’re subscribed to and listen to every week.

        Liked by 1 person

    • Henry Game 9:40 am on Wednesday, 22 February, 2017 Permalink | Reply

      One day I’d like you to explain the magic of numbers, vortex maths etc to me. I am interested in numerology, ancient geography, Metatron’s cube and all that, but, for some reason, I have never studied maths.
      Maybe you could inspire me and advise me where to start?
      I thoroughly enjoy your posts, when I come across them, but half the time I am blown away. 😂

      Liked by 1 person

      • Joseph Nebus 4:27 pm on Saturday, 25 February, 2017 Permalink | Reply

        Well, hm. I’m not sure about literal magic of numbers, as in numerology and the like. For what’s wonderful about mathematics … I’m still not perfectly sure. I think I’d give a try of Courant and Robbins’s What Is Mathematics?, originally published in 1940 but still in print and updated, and your library (or university library) will have copies. It’s a little survey of a lot of the fields of mathematics. And it’s mostly episodic, so if one section isn’t doing anything for you it’s fine to skip to the next, or just to pick a section arbitrarily and see what’s going on there.

        And I’m glad you enjoy stuff around here, but if you do get stuck on something please say so! It’s very hard for me to guess what people don’t know, and there’s usually a good post to be made in explaining why something confused someone.

        Liked by 1 person

      • mathtuition88 4:34 pm on Saturday, 25 February, 2017 Permalink | Reply

        I just checked out metatron’s cube, looks really cool.

        Like

  • Joseph Nebus 6:00 pm on Sunday, 19 February, 2017 Permalink | Reply
    Tags: , fandom, , , , , , , Wandering Melon   

    Reading the Comics, February 15, 2017: SMBC Cuts In Line Edition 


    It’s another busy enough week for mathematically-themed comic strips that I’m dividing the harvest in two. There’s a natural cutting point since there weren’t any comics I could call relevant for the 15th. But I’m moving a Saturday Morning Breakfast Cereal of course from the 16th into this pile. That’s because there’s another Saturday Morning Breakfast Cereal of course from after the 16th that I might include. I’m still deciding if it’s close enough to on topic. We’ll see.

    John Graziano’s Ripley’s Believe It Or Not for the 12th mentions the “Futurama Theorem”. The trivia is true, in that writer Ken Keeler did create a theorem for a body-swap plot he had going. The premise was that any two bodies could swap minds at most one time. So, after a couple people had swapped bodies, was there any way to get everyone back to their correct original body? There is, if you bring two more people in to the body-swapping party. It’s clever.

    From reading comment threads about the episode I conclude people are really awestruck by the idea of creating a theorem for a TV show episode. The thing is that “a theorem” isn’t necessarily a mind-boggling piece of work. It’s just the name mathematicians give when we have a clearly-defined logical problem and its solution. A theorem and its proof can be a mind-wrenching bit of work, like Fermat’s Last Theorem or the Four-Color Map Theorem are. Or it can be on the verge of obvious. Keeler’s proof isn’t on the obvious side of things. But it is the reasoning one would have to do to solve the body-swap problem the episode posited without cheating. Logic and good story-telling are, as often, good partners.

    Teresa Burritt’s Frog Applause is a Dadaist nonsense strip. But for the 13th it hit across some legitimate words, about a 14 percent false-positive rate. This is something run across in hypothesis testing. The hypothesis is something like “is whatever we’re measuring so much above (or so far below) the average that it’s not plausibly just luck?” A false positive is what it sounds like: our analysis said yes, this can’t just be luck, and it turns out that it was. This turns up most notoriously in medical screenings, when we want to know if there’s reason to suspect a health risk, and in forensic analysis, when we want to know if a particular person can be shown to have been a particular place at a particular time. A 14 percent false positive rate doesn’t sound very good — except.

    Suppose we are looking for a rare condition. Say, something one person out of 500 will have. A test that’s 99 percent accurate will turn up positives for the one person who has got it and for five of the people who haven’t. It’s not that the test is bad; it’s just there are so many negatives to work through. If you can screen out a good number of the negatives, though, the people who haven’t got the condition, then the good test will turn up fewer false positives. So suppose you have a cheap or easy or quick test that doesn’t miss any true positives but does have a 14 percent false positive rate. That would screen out 430 of the people who haven’t got whatever we’re testing for, leaving only 71 people who need the 99-percent-accurate test. This can make for a more effective use of resources.

    Gary Wise and Lance Aldrich’s Real Life Adventures for the 13th is an algebra-in-real-life joke and I can’t make something deeper out of that.

    Mike Shiell’s The Wandering Melon for the 13th is a spot of wordplay built around statisticians. Good for taping to the mathematics teacher’s walls.

    Eric the Circle for the 14th, this one by “zapaway”, is another bit of wordplay. Tans and tangents.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th identifies, aptly, a difference between scientists and science fans. Weinersmith is right that loving trivia is a hallmark of a fan. Expertise — in any field, not just science — is more about recognizing patterns of problems and concepts, ways to bring approaches from one field into another, this sort of thing. And the digits of π are great examples of trivia. There’s no need for anyone to know the 1,681st digit of π. There’s few calculations you could ever do when you needed more than three dozen digits. But if memorizing digits seems like fun then π is a great set to learn. e is the only other number at all compelling.

    The thing is, it’s very hard to become an expert in something without first being a fan of it. It’s possible, but if a field doesn’t delight you why would you put that much work into it? So even though the scientist might have long since gotten past caring how many digits of π, it’s awfully hard to get something memorized in the flush of fandom out of your head.

    I know you’re curious. I can only remember π out to 3.14158926535787962. I might have gotten farther if I’d tried, but I actually got a digit wrong, inserting a ‘3’ before that last ’62’, and the effort to get that mistake out of my head obliterated any desire to waste more time memorizing digits. For e I can only give you 2.718281828. But there’s almost no hope I’d know that far if it weren’t for how e happens to repeat that 1828 stanza right away.

     
  • Joseph Nebus 6:00 pm on Thursday, 16 February, 2017 Permalink | Reply
    Tags: , , Crock, , , Mr Lowe, , , ,   

    Reading the Comics, February 11, 2017: Trivia Edition 


    And now to wrap up last week’s mathematically-themed comic strips. It’s not a set that let me get into any really deep topics however hard I tried overthinking it. Maybe something will turn up for Sunday.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 7th tries setting arithmetic versus celebrity trivia. It’s for the old joke about what everyone should know versus what everyone does know. One might question whether Kardashian pet eating habits are actually things everyone knows. But the joke needs some hyperbole in it to have any vitality and that’s the only available spot for it. It’s easy also to rate stuff like arithmetic as trivia since, you know, calculators. But it is worth knowing that seven squared is pretty close to 50. It comes up when you do a lot of estimates of calculations in your head. The square root of 10 is pretty near 3. The square root of 50 is near 7. The cube root of 10 is a little more than 2. The cube root of 50 a little more than three and a half. The cube root of 100 is a little more than four and a half. When you see ways to rewrite a calculation in estimates like this, suddenly, a lot of amazing tricks become possible.

    Leigh Rubin’s Rubes for the 7th is a “mathematics in the real world” joke. It could be done with any mythological animals, although I suppose unicorns have the advantage of being relatively easy to draw recognizably. Mermaids would do well too. Dragons would also read well, but they’re more complicated to draw.

    Mark Pett’s Mr Lowe rerun for the 8th has the kid resisting the mathematics book. Quentin’s grounds are that how can he know a dated book is still relevant. There’s truth to Quentin’s excuse. A mathematical truth may be universal. Whether we find it interesting is a matter of culture and even fashion. There are many ways to present any fact, and the question of why we want to know this fact has as many potential answers as it has people pondering the question.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th is a paean to one of the joys of numbers. There is something wonderful in counting, in measuring, in tracking. I suspect it’s nearly universal. We see it reflected in people passing around, say, the number of rivets used in the Chrysler Building or how long a person’s nervous system would reach if stretched out into a line or ever-more-fanciful measures of stuff. Is it properly mathematics? It’s delightful, isn’t that enough?

    Scott Hilburn’s The Argyle Sweater for the 10th is a Fibonacci Sequence joke. That’s a good one for taping to the walls of a mathematics teacher’s office.

    'Did you ever take a date to a drive-in movie in high school?' 'Once, but she went to the concession stand and never came back.' 'Did you wonder why?' 'Yeah, but I kept on doing my math homework.'

    Bill Rechin’s Crock rerun for the 11th of February, 2017. They actually opened a brand-new drive-in theater something like forty minutes away from us a couple years back. We haven’t had the chance to get there. But we did get to one a fair bit farther away where yes, we saw Turbo, that movie about the snail that races in the Indianapolis 500. The movie was everything we hoped for and it’s just a shame Roger Ebert died too young to review it for us.

    Bill Rechin’s Crock rerun for the 11th is a name-drop of mathematics. Really anybody’s homework would be sufficiently boring for the joke. But I suppose mathematics adds the connotation that whatever you’re working on hasn’t got a human story behind it, the way English or History might, and that it hasn’t got the potential to eat, explode, or knock a steel ball into you the way Biology, Chemistry, or Physics have. Fair enough.

     
  • Joseph Nebus 6:00 pm on Wednesday, 15 February, 2017 Permalink | Reply
    Tags: , , , nerfing, , , Superleague   

    How Much I Did Lose In Pinball 


    A follow-up for people curious how much I lost at the state pinball championships Saturday: I lost at the state pinball championships Saturday. As I expected I lost in the first round. I did beat my expectations, though. I’d figured I would win one, maybe two games in our best-of-seven contest. As it happened I won three games and I had a fighting chance in game seven.

    I’d mentioned in the previous essay about how much contingency there is especially in a short series like this one. My opponent picked the game I expected she would to start out. And she got an awful bounce on the first ball, while I got a very lucky bounce that started multiball on the last. So I won, but not because I was playing better. The seventh game was one that I had figured she might pick if she needed to crush me, and if I had gotten a better bounce on the first ball I’d still have had an uphill struggle. Just less of one.

    After the first round I got into a set of three “tie-breaking” rounds, used to sort out which of the sixteen players ranked as number 11 versus number 10. Each of those were a best-of-three series. I did win one series and lost two others, dropping me into 12th place. Over the three series I had four wins and four losses, so I can’t say that I mismatched there.

    Where I might have been mismatched is the side tournament. This was a two-hour marathon of playing a lot of games one after the other. I finished with three wins and 13 losses, enough to make me wonder whether I somehow went from competent to incompetent in the hour or so between the main and the side tournament. Of course not, based on a record like that, but — can I prove it?

    Meanwhile a friend pointed out The New York Times covering the New York State pinball championship:

    The article is (at least for now) at https://www.nytimes.com/2017/02/12/nyregion/pinball-state-championship.html. What my friend couldn’t have known, and what shows how networked people are, is that I know one of the people featured in the article, Sean “The Storm” Grant. Well, I knew him, back in college. He was an awesome pinball player even then. And he’s only got more awesome since.

    How awesome? Let me give you some background. The International Flipper Pinball Association (IFPA) gives players ranking points. These points are gathered by playing in leagues and tournaments. Each league or tournament has a certain point value. That point value is divided up among the players, in descending order from how they finish. How many points do the events have? That depends on how many people play and what their ranking is. So, yes, how much someone’s IFPA score increases depends on the events they go to, and the events they go to depend on their score. This might sound to you like there’s a differential equation describing all this. You’re close: it’s a difference equation, because these rankings change with the discrete number of events players go to. But there’s an interesting and iterative system at work there.

    (Points only expire with time. The system is designed to encourage people to play a lot of things and keep playing them. You can’t lose ranking points by playing, although it might hurt your player-versus-player rating. That’s calculated by a formula I don’t understand at all.)

    Anyway, Sean Grant plays in the New York Superleague, a crime-fighting band of pinball players who figured out how to game the IFPA rankings system. They figured out how to turn the large number of people who might visit a Manhattan bar and casually play one or two games into a source of ranking points for the serious players. The IFPA, combatting this scheme, just this week recalculated the Superleague values and the rankings of everyone involved in it. It’s fascinating stuff, in that way a heated debate over an issue you aren’t emotionally invested in can be.

    Anyway. Grant is such a skilled player that he lost more points in this nerfing than I have gathered in my whole competitive-pinball-playing career.

    So while I knew I’d be knocked out in the first round of the Michigan State Championships I’ll admit I had fantasies of having an impossibly lucky run. In that case, I’d have gone to the nationals and been turned into a pale, silverball-covered paste by people like Grant.

    Thanks again for all your good wishes, kind readers. Now we start the long road to the 2017 State Championships, to be held in February of next year. I’m already in 63rd place in the state for the year! (There haven’t been many events for the year yet, and the championship and side tournament haven’t posted their ranking scores yet.)

     
  • Joseph Nebus 6:00 pm on Sunday, 12 February, 2017 Permalink | Reply
    Tags: Agnes, , , , , Lay Lines, , Pooch Cafe, Rabbits Against Magic,   

    Reading the Comics, February 6, 2017: Another Pictureless Half-Week Edition 


    Got another little flood of mathematically-themed comic strips last week and so once again I’ll split them along something that looks kind of middle-ish. Also this is another bunch of GoComics.com-only posts. Since those seem to be accessible to anyone whether or not they’re subscribers indefinitely far into the future I don’t feel like I can put the comics directly up and will trust you all to click on the links that you find interesting. Which is fine; the new GoComics.com design makes it annoyingly hard to download a comic strip. I don’t think that was their intention. But that’s one of the two nagging problems I have with their new design. So you know.

    Tony Cochran’s Agnes for the 5th sees a brand-new mathematics. Always dangerous stuff. But mathematicians do invent, or discover, new things in mathematics all the time. Part of the task is naming the things in it. That’s something which takes talent. Some people, such as Leonhard Euler, had the knack a great novelist has for putting names to things. The rest of us muddle along. Often if there’s any real-world inspiration, or resemblance to anything, we’ll rely on that. And we look for terminology that evokes similar ideas in other fields. … And, Agnes would like to know, there is mathematics that’s about approximate answers, being “right around” the desired answer. Unfortunately, that’s hard. (It’s all hard, if you’re going to take it seriously, much like everything else people do.)

    Scott Hilburn’s The Argyle Sweater for the 5th is the anthropomorphic numerals joke for this essay.

    Carol Lay’s Lay Lines for the 6th depicts the hazards of thinking deeply and hard about the infinitely large and the infinitesimally small. They’re hard. Our intuition seems well-suited to handing a modest bunch of household-sized things. Logic guides us when thinking about the infinitely large or small, but it takes a long time to get truly conversant and comfortable with it all.

    Paul Gilligan’s Pooch Cafe for the 6th sees Poncho try to argue there’s thermodynamical reasons for not being kind. Reasoning about why one should be kind (or not) is the business of philosophers and I won’t overstep my expertise. Poncho’s mathematics, that’s something I can write about. He argues “if you give something of yourself, you inherently have less”. That seems to be arguing for a global conservation of self-ness, that the thing can’t be created or lost, merely transferred around. That’s fair enough as a description of what the first law of thermodynamics tells us about energy. The equation he reads off reads, “the change in the internal energy (Δ U) equals the heat added to the system (U) minus the work done by the system (W)”. Conservation laws aren’t unique to thermodynamics. But Poncho may be aware of just how universal and powerful thermodynamics is. I’m open to an argument that it’s the most important field of physics.

    Jonathan Lemon’s Rabbits Against Magic for the 6th is another strip Intro to Calculus instructors can use for their presentation on instantaneous versus average velocities. There’s been a bunch of them recently. I wonder if someone at Comic Strip Master Command got a speeding ticket.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th is about numeric bases. They’re fun to learn about. There’s an arbitrariness in the way we represent concepts. I think we can understand better what kinds of problems seem easy and what kinds seem harder if we write them out different ways. But base eleven is only good for jokes.

     
    • davekingsbury 10:01 pm on Monday, 13 February, 2017 Permalink | Reply

      He argues “if you give something of yourself, you inherently have less”. That seems to be arguing for a global conservation of self-ness, that the thing can’t be created or lost, merely transferred around.

      How, I wonder, to marry that with Juliet’s declaration of love for Juliet?

      “My bounty is as boundless as the sea,
      My love as deep; the more I give to thee,
      The more I have, for both are infinite.”

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      • Joseph Nebus 11:08 pm on Thursday, 16 February, 2017 Permalink | Reply

        Oh, well, infinities are just trouble no matter what. Anything can happen with them.

        I suppose there’s also the question of how the Banach-Tarski Paradox affects love.

        Liked by 1 person

    • Downpuppy (@Downpuppy) 12:30 am on Tuesday, 14 February, 2017 Permalink | Reply

      Agnes is the first Fuzzy Math reference I’ve seen in about 10 years.

      Squirrel Girl counted to 31 on one hand to defeat Count Nefario, but SMBC is more an ASL snub

      Like

      • Joseph Nebus 11:12 pm on Thursday, 16 February, 2017 Permalink | Reply

        I’m a little surprised fuzzy mathematics doesn’t get used for more comic strips, but I don’t suppose it lends itself to too many different jokes. On the other hand, neither does Pi Day and we’ll see a bunch of those over the coming month.

        I had expected, really, Saturday Morning Breakfast Cereal to go with 1,024 as a natural base if you use your hands in a particularly digit-efficient way.

        Like

  • Joseph Nebus 6:00 pm on Friday, 10 February, 2017 Permalink | Reply
    Tags: , , , , , The Simpsons   

    How Much Can I Expect To Lose In Pinball? 


    This weekend, all going well, I’ll be going to the Michigan state pinball championship contest. There, I will lose in the first round.

    I’m not trying to run myself down. But I know who I’m scheduled to play in the first round, and she’s quite a good player. She’s the state’s highest-ranked woman playing competitive pinball. So she starts off being better than me. And then the venue is one she gets to play in more than I do. Pinball, a physical thing, is idiosyncratic. The reflexes you build practicing on one table can betray you on a strange machine. She’s had more chance to practice on the games we have and that pretty well settles the question. I’m still showing up, of course, and doing my best. Stranger things have happened than my winning a game. But I’m going in with I hope realistic expectations.

    That bit about having realistic expectations, though, makes me ask what are realistic expectations. The first round is a best-of-seven match. How many games should I expect to win? And that becomes a probability question. It’s a great question to learn on, too. Our match is straightforward to model: we play up to seven times. Each time we play one or the other wins.

    So we can start calculating. There’s some probability I have of winning any particular game. Call that number ‘p’. It’s at least zero (I’m not sure to lose) but it’s less than one (I’m not sure to win). Let’s suppose the probability of my winning never changes over the course of seven games. I will come back to the card I palmed there. If we’re playing 7 games, and I have a chance ‘p’ of winning any one of them, then the number of games I can expect to win is 7 times ‘p’. This is the number of wins you might expect if you were called on in class and had no idea and bluffed the first thing that came to mind. Sometimes that works.

    7 times p isn’t very enlightening. What number is ‘p’, after all? And I don’t know exactly. The International Flipper Pinball Association tracks how many times I’ve finished a tournament or league above her and vice-versa. We’ve played in 54 recorded events together, and I’ve won 23 and lost 29 of them. (We’ve tied twice.) But that isn’t all head-to-head play. It counts matches where I’m beaten by someone she goes on to beat as her beating me, and vice-versa. And it includes a lot of playing not at the venue. I lack statistics and must go with my feelings. I’d estimate my chance of beating her at about one in three. Let’s say ‘p’ is 1/3 until we get evidence to the contrary. It is “Flipper Pinball” because the earliest pinball machines had no flippers. You plunged the ball into play and nudged the machine a little to keep it going somewhere you wanted. (The game Simpsons Pinball Party has a moment where Grampa Simpson says, “back in my day we didn’t have flippers”. It’s the best kind of joke, the one that is factually correct.)

    Seven times one-third is not a difficult problem. It comes out to two and a third, raising the question of how one wins one-third of a pinball game. Most games involve playing three rounds, called balls, is the obvious observation. But this one-third of a game is an average. Imagine the two of us playing three thousand seven-game matches, without either of us getting the least bit better or worse or collapsing of exhaustion. I would expect to win seven thousand of the games, or two and a third games per seven-game match.

    Ah, but … that’s too high. I would expect to win two and a third games out of seven. But we probably won’t play seven. We’ll stop when she or I gets to four wins. This makes the problem hard. Hard is the wrong word. It makes the problem tedious. At least it threatens to. Things will get easy enough, but we have to go through some difficult parts first.

    There are eight different ways that our best-of-seven match can end. She can win in four games. I can win in four games. She can win in five games. I can win in five games. She can win in six games. I can win in six games. She can win in seven games. I can win in seven games. There is some chance of each of those eight outcomes happening. And exactly one of those will happen; it’s not possible that she’ll win in four games and in five games, unless we lose track of how many games we’d played. They give us index cards to write results down. We won’t lose track.

    It’s easy to calculate the probability that I win in four games, if the chance of my winning a game is the number ‘p’. The probability is p4. Similarly it’s easy to calculate the probability that she wins in four games. If I have the chance ‘p’ of winning, then she has the chance ‘1 – p’ of winning. So her probability of winning in four games is (1 – p)4.

    The probability of my winning in five games is more tedious to work out. It’s going to be p4 times (1 – p) times 4. The 4 here is the number of different ways that she can win one of the first four games. Turns out there’s four ways to do that. She could win the first game, or the second, or the third, or the fourth. And in the same way the probability she wins in five games is p times (1 – p)4 times 4.

    The probability of my winning in six games is going to be p4 times (1 – p)2 times 10. There are ten ways to scatter four wins by her among the first five games. The probability of her winning in six games is the strikingly parallel p2 times (1 – p)4 times 10.

    The probability of my winning in seven games is going to be p4 times (1 – p)3 times 20, because there are 20 ways to scatter three wins among the first six games. And the probability of her winning in seven games is p3 times (1 – p)4 times 20.

    Add all those probabilities up, no matter what ‘p’ is, and you should get 1. Exactly one of those four outcomes has to happen. And we can work out the probability that the series will end after four games: it’s the chance she wins in four games plus the chance I win in four games. The probability that the series goes to five games is the probability that she wins in five games plus the probability that I win in five games. And so on for six and for seven games.

    So that’s neat. We can figure out the probability of the match ending after four games, after five, after six, or after seven. And from that we can figure out the expected length of the match. This is the expectation value. Take the product of ‘4’ and the chance the match ends at four games. Take the product of ‘5’ and the chance the match ends at five games. Take the product of ‘6’ and the chance the match ends at six games. Take the product of ‘7’ and the chance the match ends at seven games. Add all those up. That’ll be, wonder of wonders, the number of games a match like this can be expected to run.

    Now it’s a matter of adding together all these combinations of all these different outcomes and you know what? I’m not doing that. I don’t know what the chance is I’d do all this arithmetic correctly is, but I know there’s no chance I’d do all this arithmetic correctly. This is the stuff we pirate Mathematica to do. (Mathematica is supernaturally good at working out mathematical expressions. A personal license costs all the money you will ever have in your life plus ten percent, which it will calculate for you.)

    Happily I won’t have to work it out. A person appearing to be a high school teacher named B Kiggins has worked it out already. Kiggins put it and a bunch of other interesting worksheets on the web. (Look for the Voronoi Diagramas!)

    There’s a lot of arithmetic involved. But it all simplifies out, somehow. Per Kiggins’ work, the expected number of games in a best-of-seven match, if one of the competitors has the chance ‘p’ of winning any given game, is:

    E(p) = 4 + 4\cdot p + 4\cdot p^2 + 4\cdot p^3 - 52\cdot p^4 + 60\cdot p^5 - 20\cdot p^6

    Whatever you want to say about that, it’s a polynomial. And it’s easy enough to evaluate it, especially if you let the computer evaluate it. Oh, I would say it seems like a shame all those coefficients of ‘4’ drop off and we get weird numbers like ’52’ after that. But there’s something beautiful in there being four 4’s, isn’t there? Good enough.

    So. If the chance of my winning a game, ‘p’, is one-third, then we’d expect the series to go 5.5 games. This accords well with my intuition. I thought I would be likely to win one game. Winning two would be a moral victory akin to championship.

    Let me go back to my palmed card. This whole analysis is based on the idea that I have some fixed probability of winning and that it isn’t going to change from one game to the next. If the probability of winning is entirely based on my and my opponents’ abilities this is fair enough. Neither of us is likely to get significantly more or less skilled over the course of even seven matches. We won’t even play long enough to get fatigued. But ability isn’t everything.

    But our abilities aren’t everything. We’re going to be playing up to seven different tables. How each table reacts to our play is going to vary. Some tables may treat me better, some tables my opponent. Luck of the draw. And there’s an important psychological component. It’s easy to get thrown and to let a bad ball wreck the rest of one’s game. It’s hard to resist feeling nervous if you go into the last ball from way behind your opponent. And it seems as if a pinball knows you’re nervous and races out of play to help you calm down. (The best pinball players tend to have outstanding last balls, though. They don’t get rattled. And they spend the first several balls building up to high-value shots they can collect later on.) And there will be freak events. Last weekend I was saved from elimination in a tournament by the pinball machine spontaneously resetting. We had to replay the game. I did well in the tournament, but it was the freak event that kept me from being knocked out in the first round.

    That’s some complicated stuff to fit together. I suppose with enough data we could possibly model how much the differences between pinball machines affects the outcome. That’s what sabermetrics is all about. Representing how severely I’ll build a little bad luck into a lot of bad luck? Oh, that’s hard.

    Too hard to deal with, at least not without much more sports psychology and modelling of pinball players than we have data to do. The supposition that my chance of winning is fixed for the duration of the match may not be true. But we won’t be playing enough games to be able to tell the difference. The assumption that my chance of winning doesn’t change over the course of the match may be false. But it’s near enough, and it gets us some useful information. We have to know not to demand too much precision from our model.

    And seven games isn’t statistically significant. Not when players are as closely matched as we are. I could be worse and still get a couple wins in when they count; I could play better than my average and still get creamed four games straight. I’ll be trying my best, of course. But I expect my best is one or two wins, then getting to the snack room and waiting for the side tournament to start. Shall let you know if something interesting happens.

     
    • ksbeth 6:03 pm on Friday, 10 February, 2017 Permalink | Reply

      Woo hoo! Good luck )

      Liked by 1 person

    • vagabondurges 7:33 pm on Friday, 10 February, 2017 Permalink | Reply

      Best of luck! I am loving these pinball posts! And there’s a pinball place in Alameda, CA that you’ve just inspired me to visit again.

      Liked by 1 person

      • Joseph Nebus 4:45 am on Saturday, 11 February, 2017 Permalink | Reply

        Thank you! I’m sorry I don’t find more excuses to write about pinball, since there’s so much about it I do like. And I’m glad you’re feeling inspired; I hope it’s a good visit.

        The secrets are: plunge the ball softly, let the ball bounce back and forth on the flippers until it’s moving slowly, and hold the flipper up until the ball comes to a rest so you can aim. So much of pinball is about letting things calm down so you can understand what’s going on and what you want to do next.

        Liked by 1 person

    • mathtuition88 12:32 am on Saturday, 11 February, 2017 Permalink | Reply

      Good luck and all the best!

      Like

    • davekingsbury 3:52 pm on Saturday, 11 February, 2017 Permalink | Reply

      All this work and you’ll tell me you’re not a betting man …

      Like

      • Joseph Nebus 11:05 pm on Thursday, 16 February, 2017 Permalink | Reply

        I honestly am not. The occasional lottery ticket is my limit. But probability questions are so hard to resist. They usually involve very little calculation but demand thoughtful analysis. It’s great.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Thursday, 9 February, 2017 Permalink | Reply
    Tags: , , , Dogs of C Kennel, , , , Nest Heads,   

    Reading the Comics, February 3, 2017: Counting Edition 


    And now I can close out last week’s mathematically-themed comic strips. Two of them are even about counting, which is enough for me to make that the name of this set.

    John Allen’s Nest Heads for the 2nd mentions a probability and statistics class and something it’s supposed to be good for. I would agree that probability and statistics are probably (I can’t find a better way to write this) the most practically useful mathematics one can learn. At least once you’re past arithmetic. They’re practical by birth; humans began studying them because they offer guidance in uncertain situations. And one can use many of their tools without needing more than arithmetic.

    I’m not so staunchly anti-lottery as many mathematics people are. I’ll admit I play it myself, when the jackpot is large enough. When the expectation value of the prize gets to be positive, it’s harder to rationalize not playing. This happens only once or twice a year, but it’s fun to watch and see when it happens. I grant it’s a foolish way to use two dollars (two tickets are my limit), but you know? My budget is not so tight I can’t spend four dollars foolishly a year. Besides, I don’t insist on winning one of those half-billion-dollar prizes. I imagine I’d be satisfied if I brought in a mere $10,000.

    'Hey, Ruthie's Granny, how old are you?' 'You can't count that high, James.' 'I can too!' 'Fine! Start at one and I'll tell you when you get to my age.' '1, 2, 3, 4, 11, 22, 88, 99, 200, a gazillion!' 'Very good! It's somewhere between 22 and a gazillion!' 'Gazowie!'

    Rick Detorie’s One Big Happy for the 3rd of February, 2017. A ‘gazillion’ is actually a surprisingly low number, hovering as it does somewhere around 212. Fun fact!

    Rick Detorie’s One Big Happy for the 3rd continues my previous essay’s bit of incompetence at basic mathematics, here, counting. But working out that her age is between 22 an a gazillion may be worth doing. It’s a common mathematical challenge to find a correct number starting from little information about it. Usually we find it by locating bounds: the number must be larger than this and smaller than that. And then get the bounds closer together. Stop when they’re close enough for our needs, if we’re numerical mathematicians. Stop when the bounds are equal to each other, if we’re analytic mathematicians. That can take a lot of work. Many problems in number theory amount to “improve our estimate of the lowest (or highest) number for which this is true”. We have to start somewhere.

    Samson’s Dark Side of the Horse for the 3rd is a counting-sheep joke and I was amused that the counting went so awry here. On looking over the strip again for this essay, though, I realize I read it wrong. It’s the fences that are getting counted, not the sheep. Well, it’s a cute little sheep having the same problems counting that Horace has. We don’t tend to do well counting more than around seven things at a glance. We can get a bit farther if we can group things together and spot that, say, we have four groups of four fences each. That works and it’s legitimate; we’re counting and we get the right count out of it. But it does feel like we’re doing something different from how we count, say, three things at a glance.

    Mick Mastroianni and Mason MastroianniDogs of C Kennel for the 3rd is about the world’s favorite piece of statistical mechanics, entropy. There’s room for quibbling about what exactly we mean by thermodynamics saying all matter is slowly breaking down. But the gist is fair enough. It’s still mysterious, though. To say that the disorder of things is always increasing forces us to think about what we mean by disorder. It’s easy to think we have an idea what we mean by it. It’s hard to make that a completely satisfying definition. In this way it’s much like randomness, which is another idea often treated as the same as disorder.

    Bill Amend’s FoxTrot Classics for the 3rd reprinted the comic from the 10th of February, 2006. Mathematics teachers always want to see how you get your answers. Why? … Well, there are different categories of mistakes someone can make. One can set out trying to solve the wrong problem. One can set out trying to solve the right problem in a wrong way. One can set out solving the right problem in the right way and get lost somewhere in the process. Or one can be doing just fine and somewhere along the line change an addition to a subtraction and get what looks like the wrong answer. Each of these is a different kind of mistake. Knowing what kinds of mistakes people make is key to helping them not make these mistakes. They can get on to making more exciting mistakes.

     
  • Joseph Nebus 6:00 pm on Sunday, 5 February, 2017 Permalink | Reply
    Tags: , , , , , , Pajama Diaries, ,   

    Reading the Comics, February 2, 2017: I Haven’t Got A Jumble Replacement Source Yet 


    If there was one major theme for this week it was my confidence that there must be another source of Jumble strips out there. I haven’t found it, but I admit not making it a priority either. The official Jumble site says I can play if I activate Flash, but I don’t have enough days in the year to keep up with Flash updates. And that doesn’t help me posting mathematics-relevant puzzles here anyway.

    Mark Anderson’s Andertoons for January 29th satisfies my Andertoons need for this week. And it name-drops the one bit of geometry everyone remembers. To be dour and humorless about it, though, I don’t think one could likely apply the Pythagorean Theorem. Typically the horizontal axis and the vertical axis in a graph like this measure different things. Squaring the different kinds of quantities and adding them together wouldn’t mean anything intelligible. What would even be the square root of (say) a squared-dollars-plus-squared-weeks? This is something one learns from dimensional analysis, a corner of mathematics I’ve thought about writing about some. I admit this particular insight isn’t deep, but everything starts somewhere.

    Norm Feuti’s Gil rerun for the 30th is a geometry name-drop, listing it as the sort of category Jeopardy! features. Gil shouldn’t quit so soon. The responses for the category are “What is the Pythagorean Theorem?”, “What is acute?”, “What is parallel?”, “What is 180 degrees?” (or, possibly, 360 or 90 degrees), and “What is a pentagon?”.

    Parents' Glossary Of Terms: 'Mortifraction': That utter shame when you realize you can no longer do math in your head. Parent having trouble making change at a volunteer event.

    Terri Libenson’s Pajama Diaries for the 1st of February, 2017. You know even for a fundraising event $17.50 seems a bit much for a hot dog and bottled water. Maybe the friend’s 8-year-old child is way off too.

    Terri Libenson’s Pajama Diaries for the 1st of February shows off the other major theme of this past week, which was busy enough that I have to again split the comics post into two pieces. That theme is people getting basic mathematics wrong. Mostly counting. (You’ll see.) I know there’s no controlling what people feel embarrassed about. But I think it’s unfair to conclude you “can no longer” do mathematics in your head because you’re not able to make change right away. It’s normal to be slow or unreliable about something you don’t do often. Inexperience and inability are not the same thing, and it’s unfair to people to conflate them.

    Gordon Bess’s Redeye for the 21st of September, 1970, got rerun the 1st of February. And it’s another in the theme of people getting basic mathematics wrong. And even more basic mathematics this time. There’s more problems-with-counting comics coming when I finish the comics from the past week.

    'That was his sixth shot!' 'Good! OK, Paleface! You've had it now!' (BLAM) 'I could never get that straight, does six come after four or after five?'

    Gordon Bess’s Redeye for the 21st of September, 1970. Rerun the 1st of February, 2017. I don’t see why they’re so worried about counting bullets if being shot just leaves you a little discombobulated.

    Dave Whamond’s Reality Check for the 1st hopes that you won’t notice the label on the door is painted backwards. Just saying. It’s an easy joke to make about algebra, also, that it should put letters in to perfectly good mathematics. Letters are used for good reasons, though. We’ve always wanted to work out the value of numbers we only know descriptions of. But it’s way too wordy to use the whole description of the number every time we might speak of it. Before we started using letters we could use placeholder names like “re”, meaning “thing” (as in “thing we want to calculate”). That works fine, although it crashes horribly when we want to track two or three things at once. It’s hard to find words that are decently noncommittal about their values but that we aren’t going to confuse with each other.

    So the alphabet works great for this. An individual letter doesn’t suggest any particular number, as long as we pretend ‘O’ and ‘I’ and ‘l’ don’t look like they do. But we also haven’t got any problem telling ‘x’ from ‘y’ unless our handwriting is bad. They’re quick to write and to say aloud, and they don’t require learning to write any new symbols.

    Later, yes, letters do start picking up connotations. And sometimes we need more letters than the Roman alphabet allows. So we import from the Greek alphabet the letters that look different from their Roman analogues. That’s a bit exotic. But at least in a Western-European-based culture they aren’t completely novel. Mathematicians aren’t really trying to make this hard because, after all, they’re the ones who have to deal with the hard parts.

    Bu Fisher’s Mutt and Jeff rerun for the 2nd is another of the basic-mathematics-wrong jokes. But it does get there by throwing out a baffling set of story-problem-starter points. Particularly interesting to me is Jeff’s protest in the first panel that they couldn’t have been doing 60 miles an hour as they hadn’t been out an hour. It’s the sort of protest easy to use as introduction to the ideas of average speed and instantaneous speed and, from that, derivatives.

     
  • Joseph Nebus 6:00 pm on Friday, 3 February, 2017 Permalink | Reply
    Tags: , January, , ,   

    How January 2017 Treated My Mathematics Blog 


    My mathematics blog finally broke the psychologically important barrier of 1,000 page views in January! It’s an important barrier to me. WordPress’s statistics say I drew 1,031 page views in January, the largest number since July. In December 2016 I’d puttered around 956, and November 923. This came from only 586 distinct readers, about the same as in December (589) and November (575), but that just implies more archive-binging going on.

    What’s baffling and a tiny bit disheartening about this is it came on one of my least-written months. I posted only 11 articles in January, compared to 21 in December and November. This was my laziest month since September. I have no idea what’s the most economical balance between time spent writing and instances of being read. But given two of the top-five articles this past month I suspect I got identified as an authority on a couple of questions.

    Part of why I suspect that’s so: there were only 97 pages liked around here in January. That’s the lowest count I’ve seen in the past twelve months, and it’s down a fair bit from December’s 136 and November’s 157. Maybe I need a couple more posts per month to encourage reader engagement. Or maybe not. There were 33 comments in January, not that different to December’s 29 and November’s 35. I think that a lot of January’s comments were examinations of December’s readership review. That counts, of course, although it suggests people have more fun talking about blogging than they do about mathematics. I can’t fault them; there’s a natural limit to how much there is to say about a comic strip filling a blackboard full of mathematics symbols.

    According to Insights my most popular day for page views here was Thursday, which throws me. It’s usually Sundays, when there’s always a Reading the Comics post. But for January it was Thursdays, with 16 percent of page views. That’s not very much above 1/7th of the days, though, so I suspect there’s not much link between what day it is and whether anybody reads me. The most popular hour, with 10 percent of page views, was yet again 6:00 pm, which I’m assuming is 6 pm Universal Time. I set most posts to appear at 6:00 pm Universal Time.

    So here’s what was popular around here in January:

    Here’s always-liked list of countries and number of page views from each. And for another month running India’s a top-five country; I don’t know why that should satisfy me so. Singapore comes in pretty high too, but I can explain why that satisfies me. I used to work there.

    Country Views
    United States 598
    United Kingdom 94
    Hong Kong SAR China 33
    India 33
    Philippines 30
    Germany 24
    Singapore 22
    Canada 19
    Austria 16
    Slovenia 13
    Spain 12
    France 11
    Taiwan 10
    Australia 7
    Puerto Rico 7
    Japan 6
    Israel 5
    Russia 5
    Pakistan 4
    South Africa 4
    Bosnia & Herzegovina 3
    Egypt 3
    Greece 3
    New Zealand 3
    Norway 3
    Poland 3
    Portugal 3
    Sweden 3
    Ukraine 3
    Brazil 2
    Croatia 2
    Denmark 2
    Indonesia 2
    Ireland 2
    Nepal 2
    Nigeria 2
    Northern Mariana Islands 2
    Saudi Arabia 2
    Switzerland 2
    Thailand 2
    Bangladesh 1 (*)
    Belgium 1 (*)
    Estonia 1
    Finland 1
    Italy 1
    Kuwait 1 (*)
    Lithuania 1
    Malaysia 1
    Mexico 1
    Netherlands 1
    Paraguay 1
    South Korea 1
    Trinidad and Tobago 1

    Bangladesh, Belgium, and Kuwait were single-reader countries last month. No country’s on a three-month single-reader streak. There were 53 countries altogether sending me readers, up from December’s 50 and November’s 46. I make that out as 13 single-view countries, technically down from December and November’s 15. The mysterious “European Union” reader is gone again.

    Search terms were the usual meager set of things, including:

    • comics strip of production function
    • little iodine comics
    • 5 color map theory (way easier than the four-color map theorem, plus it let me rag on New England so I’m glad someone was looking)
    • how to do pinball league and how does pinball league work (get some players and some pinball machines, and have them play each other. It’s easy and fun! Try to get it in a bar somewhere, as that’s good for giving the place a pleasant casual air; but there’s interesting probability stuff going on in the topic)
    • origin is the gateway to your entire gaming universe (with and without a period on the end)
    • what engineering taught in school dy/dx what society expect him to do mason image (um … I don’t know?)
    • urban legend venn diagram (I know of no urban legends about Venn diagrams and would be delighted if someone shared one. I also don’t know any Venn diagrams showing elements in common among various urban legends, but that would probably be a neat way of organizing at least some tales and I’d be glad at least for seeing those).

    February starts with my blog having 45,135 page views from 19,475 admitted distinct viewers. (My first year or so WordPress didn’t record unique visitors in any way they’ve told us about.) I seem to start February with 646 WordPress.com followers and I don’t know how that compares to the start of January. I failed to keep track of that. I do wonder how many of those are active yet.

    If you’d like to follow my blog here please click the buttons on the upper-right corner of the page. You can have new posts e-mailed to you, or you can follow in the WordPress reader, which gives me the chance to fix my stupid typoes. And I’m on Twitter as @Nebus, with usually just a couple posts a day. I don’t understand those folks who have 86 things to tweet about every hour day and night either. Thank you, won’t you please?

     
  • Joseph Nebus 6:00 pm on Wednesday, 1 February, 2017 Permalink | Reply
    Tags: cables, , , podcasts   

    Mathematics Stuff To Read Or Listen To 


    I concede January was a month around here that could be characterized as “lazy”. Not that I particularly skimped on the Reading the Comics posts. But they’re relatively easy to do: the comics tell me what to write about, and I could do a couple paragraphs on most anything, apparently.

    While I get a couple things planned out for the coming month, though, here’s some reading for other people.

    The above links to a paper in the Proceedings of the National Academy of Sciences. It’s about something I’ve mentioned when talking about knot before. And it’s about something everyone with computer cables or, like the tweet suggests, holiday lights finds. The things coil up. Spontaneous knotting of an agitated string by Dorian M Raymer and Douglas E Smith examines when these knots are likely to form, and how likely they are. It’s not a paper for the lay audience, but there are a bunch of fine pictures. The paper doesn’t talk about Christmas lights, no matter what the tweet does, but the mathematics carries over to this.

    MathsByAGirl, meanwhile, had a post midmonth listing a couple of mathematics podcasts. I’m familiar with one of them, BBC Radio 4’s A Brief History of Mathematics, which was a set of ten-to-twenty-minute sketches of historically important mathematics figures. I’ll trust MathsByAGirl’s taste on other podcasts. I’d spent most of this month finishing off a couple of audio books (David Hackett Fischer’s Washington’s Crossing which I started listening to while I was in Trenton for a week, because that’s the sort of thing I think is funny, and Robert Louis Stevenson’s Doctor Jekyll and Mister Hyde And Other Stories) and so fell behind on podcasts. But now there’s some more stuff to listen forward to.

    And then I’ll wrap up with this from KeplerLounge. It looks to be the start of some essays about something outside the scope of my Why Stuff Can Orbit series. (Which I figure to resume soon.) We start off talking about orbits as if planets were “point masses”. Which is what the name suggests: a mass that fills up a single point, with no volume, no shape, no features. This makes the mathematics easier. The mathematics is just as easy if the planets are perfect spheres, whether hollow or solid. But real planets are not perfect spheres. They’re a tiny bit blobby. And they’re a little lumpy as well. We can ignore that if we’re doing rough estimates of how orbits work. But if we want to get them right we can’t ignore that anymore. And this essay describes some of how we go about dealing with that.

     
  • Joseph Nebus 6:00 pm on Sunday, 29 January, 2017 Permalink | Reply
    Tags: , , , , , , , Randy Glasbergen, ,   

    Reading the Comics, January 28, 2017: Chuckle Brothers Edition 


    The week started out quite busy and I was expecting I’d have to split my essay again. It didn’t turn out that way; Comic Strip Master Command called a big break on mathematically-themed comics from Tuesday on. And then nobody from Comics Kingdom or from Creators.com needed inclusion either. I just have a bunch of GoComics links and a heap of text here. I bet that changes by next week. Still no new Jumble strips.

    Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 22nd was their first anthropomorphic numerals joke of the week.

    Kevin Fagan’s Drabble for the 22nd uses arithmetic as the sort of problem it’s easy to get clearly right or clearly wrong. It’s a more economical use of space than (say) knowing how many moons Saturn’s known to have. (More than we thought there were as long ago as Thursday.) I do like that there’s a decent moral to this on the way to the punch line.

    Bill Amend’s FoxTrot for the 22nd has Jason stand up for “torus” as a better name for doughnuts. You know how nerdy people will like putting a complicated word onto an ordinary thing. But there are always complications. A torus ordinarily describes the shape made by rotating a circle around an axis that’s in the plane of the circle. The result is a surface, though, the shell of a doughnut and none of the interior. If we’re being fussy. I don’t know of a particular name for the torus with its interior and suspect that, if pressed, a mathematician would just say “torus” or maybe “doughnut”.

    We can talk about toruses in two dimensions; those look just like circles. The doughnut-shell shape is a torus in three dimensions. There’s torus shapes made by rotating spheres, or hyperspheres, in four or more dimensions. I’m not going to draw them. And we can also talk about toruses by the number of holes that go through them. If a normal torus is the shape of a ring-shaped pool toy, a double torus is the shape of a two-seater pool toy, a triple torus something I don’t imagine exists in the real world. A quadruple torus could look, I imagine, like some pool toys Roller Coaster Tycoon allows in its water parks. I’m saying nothing about whether they’re edible.

    Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 23rd was their second anthropomorphic numerals joke of the week. I suppose sometimes you just get an idea going.

    Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 23rd jokes about mathematics skills versus life. The growth is fine enough; after all, most of us are at, or get to, our best at something while we’re training in it or making regular use of it. So the joke peters out into the usual “I never use mathematics in real life” crack, which, eh. I agree it’s what I feel like my mathematics skills have done ever since I got my degree, at any rate.

    Teresa Burritt’s Frog Applause for the 24th describes an extreme condition which hasn’t been a problem for me. I’m not an overindulgey type.

    Randy Glasbergen’s Glasbergen Cartoons rerun for the 26th is the pie chart joke for this week.

    Michael Fry’s Committed rerun for the 28th just riffs on the escalation of hyperbole, and what sure looks like an exponential growth of hyperbolic numbers. There’s a bit of scientific notation in the last panel. The “1 x” part isn’t necessary. It doesn’t change the value of the expression “1 x 1026”. But it might be convenient to use the “1 x” anyway. Scientific notation is about separating the size of the number from the interesting digits that the number has. Often when you compare numbers you’re interested in the size or else you’re interested in the important digits. Get into that habit and it’s not worth making an exception just because the interesting digits turn out to be boring in this case.

     
  • Joseph Nebus 6:00 pm on Thursday, 26 January, 2017 Permalink | Reply
    Tags: , Clear Blue Water, Hi and Lois, , , One Big Family, ,   

    Reading the Comics, January 21, 2017: Homework Edition 


    Now to close out what Comic Strip Master Command sent my way through last Saturday. And I’m glad I’ve shifted to a regular schedule for these. They ordered a mass of comics with mathematical themes for Sunday and Monday this current week.

    Karen Montague-Reyes’s Clear Blue Water rerun for the 17th describes trick-or-treating as “logarithmic”. The intention is to say that the difficulty in wrangling kids from house to house grows incredibly fast as the number of kids increases. Fair enough, but should it be “logarithmic” or “exponential”? Because the logarithm grows slowly as the number you take the logarithm of grows. It grows all the slower the bigger the number gets. The exponential of a number, though, that grows faster and faster still as the number underlying it grows. So is this mistaken?

    I say no. It depends what the logarithm is, and is of. If the number of kids is the logarithm of the difficulty of hauling them around, then the intent and the mathematics are in perfect alignment. Five kids are (let’s say) ten times harder to deal with than four kids. Sensible and, from what I can tell of packs of kids, correct.

    'Anne has six nickels. Sue has 41 pennies. Who has more money?' 'That's not going to be easy to figure out. It all depends on how they're dressed!'

    Rick Detorie’s One Big Happy for the 17th of January, 2017. The section was about how the appearance and trappings of wealth matter for more than the actual substance of wealth so everyone’s really up to speed in the course.

    Rick Detorie’s One Big Happy for the 17th is a resisting-the-word-problem joke. There’s probably some warning that could be drawn about this in how to write story problems. It’s hard to foresee all the reasonable confounding factors that might get a student to the wrong answer, or to see a problem that isn’t meant to be there.

    Bill Holbrook’s On The Fastrack for the 19th continues Fi’s story of considering leaving Fastrack Inc, and finding a non-competition clause that’s of appropriate comical absurdity. As an auditor there’s not even a chance Fi could do without numbers. Were she a pure mathematician … yeah, no. There’s fields of mathematics in which numbers aren’t all that important. But we never do without them entirely. Even if we exclude cases where a number is just used as an index, for which Roman numerals would be almost as good as regular numerals. If nothing else numbers would keep sneaking in by way of polynomials.

    'Uh, Fi? Have you looked at the non-compete clause in your contract?' 'I wouldn't go to one of Fastrack's competitors.' 'No, but, um ... you'd better read this.' 'I COULDN'T USE NUMBERS FOR TWO YEARS???' 'Roman numerals would be okay.'

    Bill Holbrook’s On The Fastrack for the 19th of January, 2017. I feel like someone could write a convoluted story that lets someone do mathematics while avoiding any actual use of any numbers, and that it would probably be Greg Egan who did it.

    Dave Whamond’s Reality Check for the 19th breaks our long dry spell without pie chart jokes.

    Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959 uses calculus as stand-in for what college is all about. Lois’s particular example is about a second derivative. Suppose we have a function named ‘y’ and that depends on a variable named ‘x’. Probably it’s a function with domain and range both real numbers. If complex numbers were involved then the variable would more likely be called ‘z’. The first derivative of a function is about how fast its values change with small changes in the variable. The second derivative is about how fast the values of the first derivative change with small changes in the variable.

    'I hope our kids are smart enough to win scholarships for college.' 'We can't count on that. We'll just have to save the money!' 'Do you know it costs about $10,000 to send one child through college?!' 'That's $40,000 we'd have to save!' Lois reads to the kids: (d^2/dx^2)y = 6x - 2.

    Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959. Fortunately Lois discovered the other way to avoid college costs: simply freeze the ages of your children where they are now, so they never face student loans. It’s an appealing plan until you imagine being Trixie.

    The ‘d’ in this equation is more of an instruction than it is a number, which is why it’s a mistake to just divide those out. Instead of writing it as \frac{d^2 y}{dx^2} it’s permitted, and common, to write it as \frac{d^2}{dx^2} y . This means the same thing. I like that because, to me at least, it more clearly suggests “do this thing (take the second derivative) to the function we call ‘y’.” That’s a matter of style and what the author thinks needs emphasis.

    There are infinitely many possible functions y that would make the equation \frac{d^2 y}{dx^2} = 6x - 2 true. They all belong to one family, though. They all look like y(x) = \frac{1}{6} 6 x^3 - \frac{1}{2} 2 x^2 + C x + D , where ‘C’ and ‘D’ are some fixed numbers. There’s no way to know, from what Lois has given, what those numbers should be. It might be that the context of the problem gives information to use to say what those numbers should be. It might be that the problem doesn’t care what those numbers should be. Impossible to say without the context.

     
    • Joshua K. 6:26 am on Monday, 30 January, 2017 Permalink | Reply

      Why is the function in the Hi & Lois discussion stated as y(x) = (1/6)6x^3 – (1/2)2x^2 + Cx +D? Why not just y(x) = x^3 – x^2 + Cx + D?

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      • Joseph Nebus 5:43 pm on Friday, 3 February, 2017 Permalink | Reply

        Good question! I actually put a fair bit of thought into this. If I were doing the problem myself I’d have cut right to x^3 – x^2 + Cx + D. But I thought there’s a number of people reading this for whom calculus is a perfect mystery and I thought that if I put an intermediate step it might help spot the pattern at work, that the coefficients in front of the x^3 and x^2 terms don’t vanish without cause.

        That said, I probably screwed up by writing them as 1/6 and 1/2. That looks too much like I’m just dividing by what the coefficients are. If I had taken more time to think out the post I should have written 1/(23) and 1/(12). This might’ve given a slightly better chance at connecting the powers of x and the fractions in the denominator. I’m not sure how much help that would give, since I didn’t describe how to take antiderivatives here. But I think it’d be a better presentation and I should remember that in future situations like that.

        Like

  • Joseph Nebus 6:00 pm on Sunday, 22 January, 2017 Permalink | Reply
    Tags: , BC, dummy variables, Edge City, , , , ,   

    Reading the Comics, January 16, 2017: Numerals Edition 


    Comic Strip Master Command decreed that last week should be busy again. So I’m splitting its strips into two essays. It’s a week that feels like it had more anthropomorphic numerals jokes than usual, but see if I actually count these things.

    2 asks 4: 'Six, six, six, can't you think of anything but six?'

    Mike Peters’s Mother Goose and Grimm for the 15th of January, 2017. I understand that sometimes you just have to use the idea you have instead of waiting for something that can best use the space available, but really, a whole Sunday strip for a single panel? And a panel that’s almost a barren stage?

    Mike Peters’s Mother Goose and Grimm for the 15th I figured would be the anthropomorphic numerals joke for the week. Shows what I know. It is an easy joke, but I do appreciate the touch of craft involved in picking the numerals. The joke is just faintly dirty if the numbers don’t add to six. If they were a pair of 3’s, there’d be the unwanted connotations of a pair of twins talking about all this. A 6 and a 0 would make at least one character weirdly obsessed. So it has to be a 4 and a 2, or a 5 and a 1. I imagine Peters knew this instinctively, at this point in his career. It’s one of the things you learn in becoming an expert.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 15th is mostly physical comedy, with a touch of — I’m not sure what to call this kind of joke. The one where a little arithmetic error results in bodily harm. In this sort of joke it’s almost always something not being carried that’s the error. I suppose that’s a matter of word economy. “Forgot to carry the (number)” is short, and everybody’s done it. And even if they don’t remember making this error, the phrasing clarifies to people that it’s a little arithmetic mistake. I think in practice mistaking a plus for a minus (or vice-versa) is the more common arithmetic error. But it’s harder to describe that clearly and concisely.

    Jef Mallett’s Frazz for the 15th puzzled me. I hadn’t heard this thing the kid says about how if you can “spew ten random lines from a classic movie” to convince people you’ve seen it. (I don’t know the kid’s name; it happens.) I suppose that it would be convincing, though. I certainly know a couple lines from movies I haven’t seen, what with living in pop culture and all that. But ten would be taxing for all but the most over-saturated movies, like any of the Indiana Jones films. (There I’m helped by having played the 90s pinball machine a lot.) Anyway, knowing ten random mathematics things isn’t convincing, especially since you can generate new mathematical things at will just by changing a number. But I would probably be convinced that someone who could describe what’s interesting about ten fields of mathematics had a decent understanding of the subject. That requires remembering more stuff, but then, mathematics is a bigger subject than even a long movie is.

    In Bill Holbrook’s On The Fastrack for the 16th Fi speaks of tallying the pluses and minuses of her life. Trying to make life into something that can be counted is an old decision-making technique. I think Benjamin Franklin explained how he found it so useful. It’s not a bad approach if a choice is hard. The challenging part is how to weight each consideration. Getting into fractions seems rather fussy to me, but some things are just like that. There is the connotation here that a fraction is a positive number smaller than 1. But the mathematically-trained (such as Fi) would be comfortable with fractions larger than 1. Or also smaller than zero. “Fraction” is no more bounded than “real number”. So, there’s the room for more sweetness here than might appear to the casual reader.

    'In a couple of weeks I'm getting married, so I'm taking stock of my life, adding up the pluses and minuses that factor into my goals.' 'Am I a positive or a negative integer?' 'You're a fraction.' 'How presumptuous of me.'

    Bill Holbrook’s On The Fastrack for the 16th of January, 2017. Were I in Dethany’s position I would have asked about being a positive or negative number, but then that would leave Holbrook without a third panel. Dethany knows what her author needs most.

    Scott Hilburn’s The Argyle Sweater for the 16th is the next anthropomorphic numerals joke for this week. I’m glad Hilburn want to be in my pages more. 5’s concern about figuring out x might be misplaced. We use variables for several purposes. One of them is as a name to give a number whose value we don’t know but wish to work out, and that’s how we first see them in high school algebra. But a variable might also be a number whose value we don’t particularly care about and will never try to work out. This could be because the variable is a parameter, with a value that’s fixed for a problem but not what we’re interested in. We don’t typically use ‘x’ for that, though; usually parameter are something earlier in the alphabet. That’s merely convention, but it is convention that dates back to René Descartes. Alternatively, we might use ‘x’ as a dummy variable. A dummy variable serves the same role that falsework on a building or a reference for an artistic sketch does. We use dummy variables to organize and carry out work, but we don’t care what its values are and we don’t even see the dummy variable in the final result. A dummy variable can be any name, but ‘x’ and ‘t’ are popular choices.

    Terry LaBan and Patty LaBan’s Edge City rerun for the 16th plays on the idea that mathematics people talk in algebra. Funny enough, although, “the opposing defense is a variable of 6”? That’s an idiosyncratic use of “variable”. I’m going to suppose that Charles is just messing with Len’s head because, really, it’s fun doing a bit of that.

     
  • Joseph Nebus 6:00 pm on Thursday, 19 January, 2017 Permalink | Reply
    Tags: , , , , , , , The Daily Drawing, The Elderberries, Ziggy   

    Reading the Comics, January 14, 2017: Maybe The Last Jumble? Edition 


    So now let me get to the other half of last week’s comics. Also, not to spoil things, but this coming week is looking pretty busy so I may have anothe split-week Reading the Comics coming up. The shocking thing this time is that the Houston Chronicle has announced it’s discontinuing its comics page. I don’t know why; I suppose because they’re fed up with people coming loyally to a daily feature. I will try finding alternate sources for the things I had still been reading there, but don’t know if I’ll make it.

    I’m saddened by this. Back in the 90s comics were just coming onto the Internet. The Houston Chronicle was one of a couple newspapers that knew what to do with them. It, and the Philadelphia Inquirer and the San Jose Mercury-News, had exactly what we wanted in comics: you could make a page up of all the strips you wanted to read, and read them on a single page. You could even go backwards day by day in case you missed some. The Philadelphia Inquirer was the only page that let you put the comics in the order you wanted, as opposed to alphabetical order by title. But if you were unafraid of opening up URLs you could reorder the Houston Chronicle page you built too.

    And those have all faded away. In the interests of whatever interest is served by web site redesigns all these papers did away with their user-buildable comics pages. The Chronicle was the last holdout, but even they abolished their pages a few years ago, with a promise for a while that they’d have a replacement comics-page scheme up soon. It never came and now, I suppose, never will.

    Most of the newspapers’ sites had become redundant anyway. Comics Kingdom and GoComics.com offer user-customizable comics pages, with a subscription model that makes it clear that money ought to be going to the cartoonists. I still had the Chronicle for a few holdouts, like Joe Martin’s strips or the Jumble feature. And from that inertia that attaches to long-running Internet associations.

    So among the other things January 2017 takes away from us, it is taking the last, faded echo of the days in the 1990s when newspapers saw comics as awesome things that could be made part of their sites.

    Lorie Ransom’s The Daily Drawing for the 11th is almost but not quite the anthropomorphized-numerals joke for this installment. It’s certainly the most numerical duck content I’ve got on record.

    Tom II Wilson’s Ziggy for the 11th is an Early Pi Day joke. Cosmically there isn’t any reason we couldn’t use π in take-a-number dispensers, after all. Their purpose is to give us some certain order in which to do things. We could use any set of numbers which can be put in order. So the counting numbers work. So do the integers. And the real numbers. But practicality comes into it. Most people have probably heard that π is a bit bigger than 3 and a fair bit smaller than 4. But pity the two people who drew e^{\pi} and \pi^{e} figuring out who gets to go first. Still, I won’t be surprised if some mathematics-oriented place uses a gimmick like this, albeit with numbers that couldn’t be confused. At least not confused by people who go to mathematics-oriented places. That would be for fun rather than cake.

    CTEFH -OOO-; ITODI OOO--; RAWDON O--O-O; FITNAN OO--O-. He wanted to expand his collection and the Mesopotamian abacus would make a OOOO OOOOOOOO.

    the Jumble for the 11th of January, 2017. This link’s all but sure to die the 1st of February, so, sorry about that. Mesopotamia did have the abacus, although I don’t know that the depiction is anything close to what the actual ones looked like. I’d imagine they do, at least within the limits of what will be an understandable drawing.

    I can’t promise that the Jumble for the 11th is the last one I’ll ever feature here. I might find where David L Hoyt and Jeff Knurek keep a linkable reference to their strips and point to them. But just in case of the worst here’s an abacus gag for you to work on.

    Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries for the 12th is, I have to point out, a rerun. So if you’re trying to do the puzzle the reference to “the number of the last president” isn’t what you’re thinking of. It is an example of the conflation of intelligence with skill at arithmetic. It’s also an example the conflation of intelligence with a mastery of trivia. But I think it leans on arithmetic more. I am not sure when this strip first appeared. “The last president” might have been Bill Clinton (42) or George W Bush (43). But this means we’re taking the square root of either 33 or 34. And there’s no doing that in your head. The square root of a whole number is either a whole number — the way the square root of 36 is — or else it’s an irrational number. You can work out the square root of a non-perfect-square by hand. But it’s boring and it’s worse than just writing “\sqrt{33} ” or “\sqrt{34} ”. Except in figuring out if that number is larger than or smaller than five or six. It’s good for that.

    Dave Blazek’s Loose Parts for the 13th is the actuary joke for this installment. Actuarial studies are built on one of the great wonders of statistics: that it is possible to predict how often things will happen. They can happen to a population, as in forecasts of how many people will be in traffic accidents or fires or will lose their jobs or will move to a new city. We may have no idea to whom any of these will happen, and they may have no way of guessing, but the enormous number of people and great number of things that can combine to make a predictable state of affairs. I suppose it’s imaginable that a group could study its dynamics well enough to identify who screws up the most and most seriously. So they might be able to say what the odds are it is his fault. But I imagine in practice it’s too difficult to define screw-ups or to assign fault consistently enough to get the data needed.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is another multiverse strip, echoing the Dinosaur Comics I featured here Sunday. I’ll echo my comments then. If there is a multiverse — again, there is not evidence for this — then there may be infinitely many versions of every book of the Bible. This suggests, but it does not mandate, that there should be every possible incarnation of the Bible. And a multiverse might be a spendthrift option anyway. Just allow for enough editions, and the chance that any of them will have a misprint at any word or phrase, and we can eventually get infinitely many versions of every book of the Bible. If we wait long enough.

     
  • Joseph Nebus 6:00 pm on Tuesday, 17 January, 2017 Permalink | Reply
    Tags: capitals, , , ,   

    48 Altered States 


    I saw this intriguing map produced by Brian Brettschneider.

    He made it on and for Twitter, as best I can determine. I found it from a stray post in Usenet newsgroup soc.history.what-if, dedicated to ways history could have gone otherwise. It also covers ways that it could not possibly have gone otherwise but would be interesting to see happen. Very different United States state boundaries are part of the latter set of things.

    The location of these boundaries is described in English and so comes out a little confusing. It’s hard to make concise. Every point in, say, this alternate Missouri is closer to Missouri’s capital of … uhm … Missouri City than it is to any other state’s capital. And the same for all the other states. All you kind readers who made it through my recent A To Z know a technical term for this. This is a Voronoi Diagram. It uses as its basis points the capitals of the (contiguous) United States.

    It’s an amusing map. I mean amusing to people who can attach concepts like amusement to maps. It’d probably be a good one to use if someone needed to make a Risk-style grand strategy game map and didn’t want to be to beholden to the actual map.

    No state comes out unchanged, although a few don’t come out too bad. Maine is nearly unchanged. Michigan isn’t changed beyond recognition. Florida gets a little weirder but if you showed someone this alternate shape they’d recognize the original. No such luck with alternate Tennessee or alternate Wyoming.

    The connectivity between states changes a little. California and Arizona lose their border. Washington and Montana gain one; similarly, Vermont and Maine suddenly become neighbors. The “Four Corners” spot where Utah, Colorado, New Mexico, and Arizona converge is gone. Two new ones look like they appear, between New Hampshire, Massachusetts, Rhode Island, and Connecticut; and between Pennsylvania, Maryland, Virginia, and West Virginia. I would be stunned if that weren’t just because we can’t zoom far enough in on the map to see they’re actually a pair of nearby three-way junctions.

    I’m impressed by the number of borders that are nearly intact, like those of Missouri or Washington. After all, many actual state boundaries are geographic features like rivers that a Voronoi Diagram doesn’t notice. How could Ohio come out looking anything like Ohio?

    The reason comes to historical subtleties. At least once you get past the original 13 states, basically the east coast of the United States. The boundaries of those states were set by colonial charters, with boundaries set based on little or ambiguous information about what the local terrain was actually like, and drawn to reward or punish court factions and favorites. Never mind the original thirteen (plus Maine and Vermont, which we might as well consider part of the original thirteen).

    After that, though, the United States started drawing state boundaries and had some method to it all. Generally a chunk of territory would be split into territories and later states that would be roughly rectangular, so far as practical, and roughly similar in size to the other states carved of the same area. So for example Missouri and Alabama are roughly similar to Georgia in size and even shape. Louisiana, Arkansas, and Missouri are about equal in north-south span and loosely similar east-to-west. Kansas, Nebraska, South Dakota, and North Dakota aren’t too different in their north-to-south or east-to-west spans.

    There’s exceptions, for reasons tied to the complexities of history. California and Texas get peculiar shapes because they could. Michigan has an upper peninsula for quirky reasons that some friend of mine on Twitter discovers every three weeks or so. But the rough guide is that states look a lot more similar to one another than you’d think from a quick look. Mark Stein’s How The States Got Their Shapes is an endlessly fascinating text explaining this all.

    If there is a loose logic to state boundaries, though, what about state capitals? Those are more quirky. One starts to see the patterns when considering questions like “why put California’s capital in Sacramento instead of, like, San Francisco?” or “Why Saint Joseph instead Saint Louis or Kansas City?” There is no universal guide, but there are some trends. Generally states end up putting their capitals in a city that’s relatively central, at least to the major population centers around the time of statehood. And, generally, not in one of the state’s big commercial or industrial centers. The desire to be geographically central is easy to understand. No fair making citizens trudge that far if they have business in the capital. Avoiding the (pardon) first tier of cities has subtler politics to it; it’s an attempt to get the government somewhere at least a little inconvenient to the money powers.

    There’s exceptions, of course. Boston is the obviously important city in Massachusetts, Salt Lake City the place of interest for Utah, Denver the equivalent for Colorado. Capitals relocated; Atlanta is Georgia’s eighth(?) I think since statehood. Sometimes they were weirder. Until 1854 Rhode Island rotated between five cities, to the surprise of people trying to name a third city in Rhode Island. New Jersey settled on Trenton as compromise between the East and West Jersey capitals of Perth Amboy and Burlington. But if you look for a city that’s fairly central but not the biggest in the state you get to the capital pretty often.

    So these are historical and cultural factors which combine to make a Voronoi Diagram map of the United States strange, but not impossibly strange, compared to what has really happened. Things are rarely so arbitrary as they seem at first.

     
    • Matthew Wright 6:49 pm on Tuesday, 17 January, 2017 Permalink | Reply

      New Zealand’s provincial borders were devised at much the same time as the midwestern and western US and in much the same way. Some guy with a map that only vaguely showed rivers, and a ruler. Well, when I say ‘some guy’ I mean George Grey, Edward Eyre and their factotum, Alfred Domett among only a handful of others. Early colonial New Zealand was like that. The civil service consisted of about three people (all of them Domett) and because the franchise system meant some voting districts might have as few as 25 electors, anybody had at least a 50/50 chance of becoming Prime Minister.

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      • Joseph Nebus 3:45 pm on Saturday, 21 January, 2017 Permalink | Reply

        I am intrigued and delighted to learn this! For all that I do love maps and seeing how borders evolve over time I’m stronger on United States and Canadian province borders; they’re just what was easily available when I grew up. (Well, and European boundaries, but I don’t think there’s a single one of them that’s based on anything more than “this is where the armies stood on V-E Day”.)

        Would you have a recommendation on a pop history of New Zealand for someone who knows only, mostly, that I guess confederation with Australia was mooted in 1900 but refused since the islands are actually closer to the Scilly Isles than they are Canberra for crying out loud?

        Liked by 1 person

    • Matthew Wright 8:43 pm on Saturday, 21 January, 2017 Permalink | Reply

      Europe has had so many boundary changes since Roman times that I wouldn’t be surprised if there’s a tradition for governments to issue people with an eraser and pot of paint to update their maps – and, no question, their history IS the history of those boundary changes. Certainly it explains their wars…

      On matters NZ, I wrote just such a book – it was first published in 2004 and has been through a couple of editions (I updated it in 2012). My publishers, Bateman, put it up on Kindle:

      It’s ‘publisher priced’ but I’d thoroughly recommend it! :-) The parallels between NZ’s settler period and the US ‘midwestern’ expansion through to California at the same time are direct.

      The reasons why NZ never joined Australia in 1900 have been endlessly debated and never answered but probably had something to do with the way NZ was socially re-identifying itself with Britain at the time. The British ignored the whole thing for defence/strategic purposes, deploying just one RN squadron to Sydney as the ‘mid point’ of Australasia. Sydney-siders liked it, but everybody from Perth to Wellington was annoyed. I wrote my thesis on the political outcome, way back when.

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      • Joseph Nebus 6:19 am on Saturday, 28 January, 2017 Permalink | Reply

        Aw, thank you kindly! I’d thought you might have something suitable.

        The organizing of territory that white folks told themselves was unsettled is a process I find interesting, I suppose because I’ve always wondered about how one goes about establishing systems. I think it’s similar to my interest in how nations devastated by wars get stuff like trash collection and fire departments and regional power systems running again. The legal system for at least how the United States organized territory is made clear enough in public schools (at least to students who pay attention, like me), but it isn’t easy to find the parallel processes in other countries. Now and then I try reading about Canada and how two of every seven sections of land in (now) Quebec and Ontario was reserved to the church and then I pass out and by the time I wake up again they’re making infrastructure promises to Prince Edward Island.

        I’m not surprised that from the British side of things the organization of New Zealand and Australia amounted to a bit of afterthought and trusting things would work out all right. I have read a fair bit (for an American) about the British Empire and it does feel like all that was ever thought about was India and the route to India and an ever-widening corridor of imagined weak spots on the route to India. The rest of the world was, pick some spot they had already, declare it “the Gibraltar of [ Geographic Region ]” and suppose there’d be a ship they could send there if they really had to.

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