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  • Joseph Nebus 6:00 pm on Sunday, 19 March, 2017 Permalink | Reply
    Tags: 2 Cows And A Chicken, Archie, , Arlo and Janis, Lard's World Peace Tips, , Off The Mark, , , , Working Daze   

    Reading the Comics, March 18, 2017: Pi Day Edition 


    No surprise what the recurring theme for this set of mathematics-mentioning comic strips is. Look at the date range. But here goes.

    Henry Scarpelli and Craig Boldman’s Archie rerun for the 13th uses algebra as the thing that will stun a class into silence. I know the silence. As a grad student you get whole minutes of instructions on how to teach a course before being sent out as recitation section leader for some professor. And what you do get told is the importance of asking students their thoughts and their ideas. This maybe works in courses that are obviously friendly to opinions or partially formed ideas. But in Freshman Calculus? It’s just deadly. Even if you can draw someone into offering an idea how we might start calculating a limit (say), they’re either going to be exactly right or they’re going to need a lot of help coaxing the idea into something usable. I’d like to have more chatty classes, but some subjects are just hard to chat about.

    Mr Weatherby walks past a silent class. 'What a well-behaved class! ... Flutesnoot, how do you get them to be so quiet and still?' 'I just asked for a volunteer to solve an algebra problem!'

    Henry Scarpelli and Craig Boldman’s Archie rerun for the 13th of March, 2017. I didn’t know the mathematics teacher’s name and suppose that “Flutesnoot” is as plausible as anything. Anyway, I admire his ability to stand in front of a dead-silent class. The stage fright the scenario produces is powerful. At least when I was taught how to teach we got nothing about stage presence or how to remain confident during awkward pauses. What I know I learned from a half-year Drama course in high school.

    Steve Skelton’s 2 Cows And A Chicken for the 13th includes some casual talk about probability. As normally happens, they figure the chances are about 50-50. I think that’s a default estimate of the probability of something. If you have no evidence to suppose one outcome is more likely than the other, then that is a reason to suppose the chance of something is 50 percent. This is the Bayesian approach to probability, in which we rate things as more or less likely based on what information we have about how often they turn out. It’s a practical way of saying what we mean by the probability of something. It’s terrible if we don’t have much reliable information, though. We need to fall back on reasoning about what is likely and what is not to save us in that case.

    Scott Hilburn’s The Argyle Sweater lead off the Pi Day jokes with an anthropomorphic numerals panel. This is because I read most of the daily comics in alphabetical order by title. It is also because The Argyle Sweater is The Argyle Sweater. Among π’s famous traits is that it goes on forever, in decimal representations, yes. That’s not by itself extraordinary; dull numbers like one-third do that too. (Arguably, even a number like ‘2’ does, if you write all the zeroes in past the decimal point.) π gets to be interesting because it goes on forever without repeating, and without having a pattern easily describable. Also because it’s probably a normal number but we don’t actually know that for sure yet.

    Mark Parisi’s Off The Mark panel for the 14th is another anthropomorphic numerals joke and nearly the same joke as above. The answer, dear numeral, is “chained tweets”. I do not know that there’s a Twitter bot posting the digits of π in an enormous chained Twitter feed. But there’s a Twitter bot posting the digits of π in an enormous chained Twitter feed. If there isn’t, there is now.

    John Zakour and Scott Roberts’s Working Daze for the 14th is your basic Pi Day Wordplay panel. I think there were a few more along these lines but I didn’t record all of them. This strip will serve for them all, since it’s drawn from an appealing camera angle to give the joke life.

    Dave Blazek’s Loose Parts for the 14th is a mathematics wordplay panel but it hasn’t got anything to do with π. I suspect he lost track of what days he was working on, back six or so weeks when his deadline arrived.

    Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 15th is some sort of joke about the probability of the world being like what it seems to be. I’m not sure precisely what anyone is hoping to express here or how it ties in to world peace. But the world does seem to be extremely well described by techniques that suppose it to be random and unpredictable in detail. It is extremely well predictable in the main, which shows something weird about the workings of the world. It seems to be doing all right for itself.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is built on the staggering idea that the Earth might be the only place with life in the universe. The cosmos is a good stand-in for infinitely large things. It might be better as a way to understand the infinitely large than actual infinity would be. Somehow thinking of the number of stars (or whatnot) in the universe and writing out a representable number inspires an understanding for bigness that the word “infinity” or the symbols we have for it somehow don’t seem to, at least to me.

    Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 17th gives us valuable information about how long ahead of time the comic strips are working. Arithmetic is probably the easiest thing to use if one needs an example of a fact. But even “2 + 2 = 4” is a fact only if we accept certain ideas about what we mean by “2” and “+” and “=” and “4”. That we use those definitions instead of others is a reflection of what we find interesting or useful or attractive. There is cultural artifice behind the labelling of this equation as a fact.

    Jimmy Johnson’s Arlo and Janis for the 18th capped off a week of trying to explain some point about the compression and dilution of time in comic strips. Comic strips use space and time to suggest more complete stories than they actually tell. They’re much like every other medium in this way. So, to symbolize deep thinking on a subject we get once again a panel full of mathematics. Yes, I noticed the misquoting of “E = mc2” there. I am not sure what Arlo means by “Remember the boat?” although thinking on it I think he did have a running daydream about living on a boat. Arlo and Janis isn’t a strongly story-driven comic strip, but Johnson is comfortable letting the setting evolve. Perhaps all this is forewarning that we’re going to jump ahead to a time in Arlo’s life when he has, or has had, a boat. I don’t know.

     
  • Joseph Nebus 6:00 pm on Saturday, 18 March, 2017 Permalink | Reply
    Tags: , , , , , ,   

    How Interesting Is March Madness? 


    And now let me close the week with some other evergreen articles. A couple years back I mixed the NCAA men’s basketball tournament with information theory to produce a series of essays that fit the title I’ve given this recap. They also sprawl out into (US) football and baseball. Let me link you to them:

     
  • Joseph Nebus 6:00 pm on Thursday, 16 March, 2017 Permalink | Reply
    Tags: , , , Dustin, , Red and Rover, , weddings   

    Reading the Comics, March 11, 2017: Accountants Edition 


    And now I can wrap up last week’s delivery from Comic Strip Master Command. It’s only five strips. One certainly stars an accountant. one stars a kid that I believe is being coded to read as an accountant. The rest, I don’t know. I pick Edition titles for flimsy reasons anyway. This’ll do.

    Ryan North’s Dinosaur Comics for the 6th is about things that could go wrong. And every molecule of air zipping away from you at once is something which might possibly happen but which is indeed astronomically unlikely. This has been the stuff of nightmares since the late 19th century made probability an important part of physics. The chance all the air near you would zip away at once is impossibly unlikely. But such unlikely events challenge our intuitions about probability. An event that has zero chance of happening might still happen, given enough time and enough opportunities. But we’re not using our time well to worry about that. If nothing else, even if all the air around you did rush away at once, it would almost certainly rush back right away.

    'The new SAT multiple-choice questions have 4 answers instead of 5, with no penalty for guessing.' 'Let's see ... so if I took it now ... that would be one chance in four, which would be ... 25%?' 'Yes.' 'But back when I took it, my chances were ... let's see ... um ...' 'Remember, there's no penalty for guessing.'

    Steve Kelley and Jeff Parker’s Dustin for the 7th of March, 2017. It’s the title character doing the guessing there. Also, Kelley and Parker hate their title character with a thoroughness you rarely see outside Tom Batiuk and Funky Winkerbean. This is a mild case of it but, there we are.

    Steve Kelley and Jeff Parker’s Dustin for the 7th of March talks about the SATs and the chance of picking right answers on a multiple-choice test. I haven’t heard about changes to the SAT but I’ll accept what the comic strip says about them for the purpose of discussion here. At least back when I took it the SAT awarded one point to the raw score for a correct answer, and subtracted one-quarter point for a wrong answer. (The raw scores were then converted into a 200-to-800 range.) I liked this. If you had no idea and guessed on answers you should expect to get one in five right and four in five wrong. On average then you would expect no net change to your raw score. If one or two wrong answers can be definitely ruled out then guessing from the remainder brings you a net positive. I suppose the change, if it is being done, is meant to be confident only right answers are rewarded. I’m not sure this is right; it seems to me there’s value in being able to identify certainly wrong answers even if the right one isn’t obvious. But it’s not my test and I don’t expect to need to take it again either. I can expression opinions without penalty.

    Mark Anderson’s Andertoons for the 7th is the Mark Anderson’s Andertoons for last week. It’s another kid-at-the-chalkboard panel. What gets me is that if the kid did keep one for himself then shouldn’t he have written 38?

    Brian Basset’s Red and Rover for the 8th mentions fractions. It’s just there as the sort of thing a kid doesn’t find all that naturally compelling. That’s all right I like the bug-eyed squirrel in the first panel.

    'The happy couple is about to cut the cake!' 'What kind is it?' 'A math cake.' (It has a square root of 4 sign atop it.)

    Bill Holbrook’s On The Fastrack for the 9th of March, 2017. I confess I’m surprised Holbrook didn’t think to set the climax a couple of days later and tie it in to Pi Day.

    Bill Holbrook’s On The Fastrack for the 9th concludes the wedding of accountant Fi. It uses the square root symbol so as to make the cake topper clearly mathematical as opposed to just an age.

     
  • Joseph Nebus 6:00 pm on Tuesday, 14 March, 2017 Permalink | Reply
    Tags: , ,   

    Terrible and Less-Terrible Pi 


    As the 14th of March comes around it’s the time for mathematics bloggers to put up whatever they can about π. I will stir from my traditional crankiness about Pi Day (look, we don’t write days of the year as 3.14 unless we’re doing fake stardates) to bring back my two most π-relevant posts:

    • Calculating Pi Terribly is about a probability-based way to calculate just what π’s digits are. It’s a lousy way to do it, but it works, technically.
    • Calculating Pi Less Terribly is about an analysis-based way to calculate just what π’s digits are. It’s a less bad way to do it, although we actually use better-yet ways to work out the digits of a number like this.
    • And what the heck, Normal Numbers, from an A To Z sequence. We do not actually know that π is a normal number. It’s the way I would bet, though, and here’s something about why I’d bet that way.
     
  • Joseph Nebus 6:00 pm on Sunday, 12 March, 2017 Permalink | Reply
    Tags: , Basic Instructions, , Little Iodine, Phoebe and her Unicorn, Piranha Club,   

    Reading the Comics, March 6, 2017: Blackboards Edition 


    I can’t say there’s a compelling theme to the first five mathematically-themed comics of last week. Screens full of mathematics turned up in a couple of them, so I’ll run with that. There were also just enough strips that I’m splitting the week again. It seems fair to me and gives me something to remember Wednesday night that I have to rush to complete.

    Jimmy Hatlo’s Little Iodine for the 1st of January, 1956 was rerun on the 5th of March. The setup demands Little Iodine pester her father for help with the “hard homework” and of course it’s arithmetic that gets to play hard work. It’s a word problem in terms of who has how many apples, as you might figure. Don’t worry about Iodine’s boss getting fired; Little Iodine gets her father fired every week. It’s their schtick.

    Little Iodine wakes her father early after a night at the lodge. 'You got to help me with my [hard] homework.' 'Ooh! My head! Wha'?' 'The first one is, if John has twice as many apples as Tom and Sue put together ... ' 'Huh? kay! Go on, let's get this over with.' They work through to morning. Iodine's teacher sees her asleep in class and demands she bring 'a note from your parents as to why you sleep in school instead of at home!' She goes to her father's office where her father's boss is saying, 'Well, Tremblechin, wake up! The hobo hotel is three blocks south and PS: DON'T COME BACK!'

    Jimmy Hatlo’s Little Iodine for the 1st of January, 1956. I guess class started right back up the 2nd, but it would’ve avoided so much trouble if she’d done her homework sometime during the winter break. That said, I never did.

    Dana Simpson’s Phoebe and her Unicorn for the 5th mentions the “most remarkable of unicorn confections”, a sugar dodecahedron. Dodecahedrons have long captured human imaginations, as one of the Platonic Solids. The Platonic Solids are one of the ways we can make a solid-geometry analogue to a regular polygon. Phoebe’s other mentioned shape of cubes is another of the Platonic Solids, but that one’s common enough to encourage no sense of mystery or wonder. The cube’s the only one of the Platonic Solids that will fill space, though, that you can put into stacks that don’t leave gaps between them. Sugar cubes, Wikipedia tells me, have been made only since the 19th century; the Moravian sugar factory director Jakub Kryštof Rad got a patent for cutting block sugar into uniform pieces in 1843. I can’t dispute the fun of “dodecahedron” as a word to say. Many solid-geometric shapes have names that are merely descriptive, but which are rendered with Greek or Latin syllables so as to sound magical.

    Bud Grace’s Piranha Club for the 6th started a sequence in which the Future Disgraced Former President needs the most brilliant person in the world, Bud Grace. A word balloon full of mathematics is used as symbol for this genius. I feel compelled to point out Bud Grace was a physics major. But while Grace could as easily have used something from the physics department to show his deep thinking abilities, that would all but certainly have been rendered as equation and graphs, the stuff of mathematics again.

    At the White Supremacist House: 'I have the smartest people I could find to help me run this soon-to-be-great-again country, but I'm worried that they're NOT SMART ENOUGH! I want the WORLD'S SMARTEST GENIUS to be my SPECIAL ADVISOR!' Meanwhile, cartoonist Bud Grace thinks of stuff like A = pi*r^2 and a^2 + b^2 = c^2 and tries working out 241 times 365, 'carry the one ... hmmmm ... '

    Bud Grace’s Piranha Club for the 6th of March, 2017. 241 times 635 is 153,035 by the way. I wouldn’t work that out in my head if I needed the number. I might work out an estimate of how big it was, in which case I’d do this: 241 is about 250, which is one-quarter of a thousand. One-quarter of 635 is something like 150, which times a thousand is 150,000. If I needed it exactly I’d get a calculator. Unless I just needed something to occupy my mind without having any particular emotional charge.

    Scott Meyer’s Basic Instructions rerun for the 6th is aptly titled, “How To Unify Newtonian Physics And Quantum Mechanics”. Meyer’s advice is not bad, really, although generic enough it applies to any attempts to reconcile two different models of a phenomenon. Also there’s not particularly a problem reconciling Newtonian physics with quantum mechanics. It’s general relativity and quantum mechanics that are so hard to reconcile.

    Still, Basic Instructions is about how you can do a thing, or learn to do a thing. It’s not about how to allow anything to be done for the first time. And it’s true that, per quantum mechanics, we can’t predict exactly what any one particle will do at any time. We can say what possible things it might do and how relatively probable they are. But big stuff, the stuff for which Newtonian physics is relevant, involve so many particles that the unpredictability becomes too small to notice. We can see this as the Law of Large Numbers. That’s the probability rule that tells us we can’t predict any coin flip, but we know that a million fair tosses of a coin will not turn up 800,000 tails. There’s more to it than that (there’s always more to it), but that’s a starting point.

    Michael Fry’s Committed rerun for the 6th features Albert Einstein as the icon of genius. Natural enough. And it reinforces this with the blackboard full of mathematics. I’m not sure if that blackboard note of “E = md3” is supposed to be a reference to the famous Far Side panel of Einstein hearing the maid talk about everything being squared away. I’ll take it as such.

     
  • Joseph Nebus 6:00 pm on Thursday, 9 March, 2017 Permalink | Reply
    Tags: , ,   

    Words About A Wordless Induction Proof 


    This pair of tweets came across my feed. And who doesn’t like a good visual proof of a mathematical fact? I hope you enjoy.

    So here’s the proposition.

    This is the sort of identity we normally try proving by induction. Induction is a great scheme for proving identities like this. It works by finding some index on the formula. Then show that if the formula is true for one value of the index, then it’s true for the next-higher value of the index. Finally, find some value of the index for which it’s easy to check that the formula’s true. And that proves it’s true for all the values of that index above that base.

    In this case the index is ‘n’. It’s really easy to prove the base case, since 13 is equal to 12 what with ‘1’ being the number everybody likes to raise to powers. Going from proving that if it’s true in one case — 1^3 + 2^3 + 3^3 + \cdots + n^3 — then it’s true for the next — 1^3 + 2^3 + 3^3 + \cdots + n^3 + (n + 1)^3 — is work. But you can get it done.

    And then there’s this, done visually:

    It took me a bit to read fully until I was confident in what it was showing. But it is all there.

    As often happens with these wordless proofs you can ask whether it is properly speaking a proof. A proof is an argument and to be complete it has to contain every step needed to deduce the conclusion from the premises, following one of the rules of inference each step. Thing is basically no proof is complete that way, because it takes forever. We elide stuff that seems obvious, confident that if we had to we could fill in the intermediate steps. A wordless proof like trusts that if we try to describe what is in the picture then we are constructing the argument.

    That’s surely enough of my words.

     
  • Joseph Nebus 6:00 pm on Tuesday, 7 March, 2017 Permalink | Reply
    Tags: , , , ,   

    How February 2017 Treated My Mathematics Blog 


    It was another pretty busy month around these parts. According to WordPress’s statistics page there were 1,063 page views from 680 unique visitors. That’s slightly up from January’s 1,031 page views and 586 unique visitors, and pretty substantially up from December 2016’s 956 page views an 589 unique visitors. And that for what was a pretty easy month of writing. Most of my posts were Reading the Comics essays, for which I don’t have to think about what to write. I just have to write it. That’s way easier.

    If it was one of the most popular months I’ve had i a while, it was also one of the least popular months I’ve had in a while. There were only 77 posts given ‘likes’ in February, compared to 97 in January and 136 in December. Indeed, this was the lowest number of likes in a month in the past two years. Comments were down too, to 18, the lowest count since August 2016. January had had 34 comments and December 29. The Reading the Comics posts don’t give a lot to discuss, I suppose.

    According to the Insights tab, the most popular day for reading was Monday, with 16 percent of page views. In January it had ben Thursdays, also with 16 percent of page views; in December it was Sundays. Sunday makes sense because that’s when Reading the Comics post go up. Monday? I don’t know.

    The most popular hour was 6:00 pm, which got 11 percent of page views. The hour’s stayed consistent for the last several months, although in January it saw only 10 percent of page views. 6:00 pm Universal Time is when I put up most of my posts, so that makes sense.

    There were 64 countries in this month’s roster of country views, up from January’s 53. 22 of them were single-viewer countries, up from 13 too. My “European Union” audience is back and in force.

    Country Views
    United States 544
    United Kingdom 84
    India 52
    Canada 40
    Hong Kong SAR China 27
    Singapore 26
    Philippines 25
    Germany 19
    Puerto Rico 19
    Australia 16
    Brazil 13
    France 13
    US Virgin Islands 12
    Netherlands 10
    Slovenia 9
    Israel 8
    Thailand 8
    Czech Republic 5
    Spain 5
    Sweden 5
    Switzerland 5
    Croatia 4
    Italy 4
    New Zealand 4
    Oman 4
    South Africa 4
    Argentina 3
    Austria 3
    Belgium 3
    Colombia 3
    European Union 3
    Greece 3
    Jamaica 3
    Poland 3
    Bulgaria 2
    Denmark 2
    Estonia 2
    Finland 2
    Indonesia 2
    Mexico 2
    Morocco 2
    Ukraine 2
    Albania 1
    Algeria 1
    Armenia 1
    Bahrain 1
    Bermuda 1
    Cyprus 1
    Hungary 1
    Ireland 1
    Japan 1
    Luxembourg 1
    Macedonia 1
    Nepal 1
    Norway 1
    Romania 1
    Saudi Arabia 1
    South Korea 1 (*)
    Sri Lanka 1
    Taiwan 1
    Uganda 1
    United Arab Emirates 1
    Venezuela 1
    Vietnam 1

    South Korea is the only country that was single-reader two months in a row. I think that’s the closest to a complete turnover I’ve gotten since I started tracking this.

    The most popular posts this month were three of the Immortals and then one that just captured people’s imagination:

    Clearly I need to do more how-to mathematics posts.

    Among the search terms bringing people here:

    • what do you think would a trapezoid look like we rotate it by quarter-turn?
    • comic strip about statistics and probability
    • comic strip about velocity and scalar
    • origin is the gateway to your entire gaming universe
    • comics about gay-lussac law
    • comics about liquefaction
    • comics of pythagoras ideas about model of the universe

    I hesitate to swipe Math With Bad Drawings’ schtick, but this does suggest I ought to be commissioning some comic strips for here.

    WordPress thinks I started the month with 642 followers on WordPress. You can be among them by using the link in the upper-right corner of this theme. There’s also the chance to follow by e-mail, which a couple of people do. The advantage of following by e-mail is you get the blog by e-mail, so that I don’t have the chance to fix typos and clumsy word choices before you can see it. And I’m on Twitter, as @nebusj, if you want to see that. You get some hints of it from one of the panels on the right.

    March 2017 starts with my page here having got 46,198 views from something like 20,155 recorded unique visitors. (The blog started before WordPress counted unique visitors in any way they tell us about.) So my humor blog’s overtaken this one in both counts, but that’s all right. I post more stuff over there.

     
  • 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, , Super-Fun-Pak Comix, 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.”

      Like

      • 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.

     
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