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  • Joseph Nebus 3:00 pm on Wednesday, 10 February, 2016 Permalink | Reply
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    Proportional Dice 

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

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

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

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

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

    • Thumbup 3:09 pm on Wednesday, 10 February, 2016 Permalink | Reply


    • howardat58 3:14 pm on Wednesday, 10 February, 2016 Permalink | Reply

      Vegetarians clearly have different definitions.


    • Chiaroscuro 4:00 pm on Wednesday, 10 February, 2016 Permalink | Reply

      So, let’s make these A/B=C/D for the dice, assuming in-order rolls. 1296 possibilities.

      If A=C and B=D, it’ll always work. So that’s 36.

      Additionally: If A=B and C=D, it’ll always work (1=1). So that’s 36,. minus the 6 where A=B=C=D.

      Then 1/2=2/4 (and converse and inverse and both), 1/2=3/6 (same), 2/4=3/6 (same), 1/3=2/6 (same). 4, 4, 4, 16 total.

      36+30+16=82, unless I’ve missed some. 82/1296, which reduces to 41/648.


      • Chiaroscuro 4:02 pm on Wednesday, 10 February, 2016 Permalink | Reply

        Ooh! I missed 2/3=4/6. (and converse, and inverse, and both). So another 4, meaning 86/1296.


  • Joseph Nebus 12:44 am on Monday, 8 February, 2016 Permalink | Reply
    Tags: , , , , Super Bowl   

    Reading the Comics, February 6, 2016: Lottery Edition 

    As mentioned, the lottery was a big thing a couple of weeks ago. So there were a couple of lottery-themed comics recently. Let me group them together. Comic strips tend to be anti-lottery. It’s as though people trying to make a living drawing comics for newspapers are skeptical of wild long-shot dreams.

    T Lewis and Michael Fry’s Over The Hedge started a lottery storyline the 1st of February. Verne, the turtle, repeats the tired joke that the lottery is a tax on people bad at mathematics. Enormous jackpots, like the $1,500,000,000 payout of a couple weeks back, break one leg of the anti-lottery argument. If the expected payout is large enough then the expectation value of playing can become positive. The expectation value is one of those statistics terms that almost tells you what it is just by the name. It’s what you would expect as the average result if you could repeat some experiment arbitrarily many times. If the payout is 1.5 billion, and the chance of winning one in 250 million, then the expected value of the payout is six dollars. If a ticket costs less than six dollars, then — if you could play over and over, hundreds of millions of times — you’d expect to come out ahead each time you play.

    If you could. Of course, you can’t play the lottery hundreds of millions of times. You can play a couple of times at most. (Even if you join a pool at work and buy, oh, a thousand tickets. That’s still barely better than playing twice.) And the payout may be less than the full jackpot; multiple winners are common things in the most enormous jackpots. Still, if you’re pondering whether it’s sensible to spend two dollars on a billion-dollar lottery jackpot? You’re being fussy. You’ll spend at least that much on something more foolish and transitory — the lottery ticket can at least be used as a bookmark — I’ll bet.

    Jef Mallett’s Frazz for the 4th of February picks up the anti-lottery crusade. Caulfield does pin down that lotteries work because people figure they have a better chance of winning than they truly do. Nobody buys a ticket because they figure it’s worth losing a dollar or two. It’s because they figure the chance is worth a little money.

    Ken Cursoe’s Tiny Sepuku for the 4th of February consults the Chinese Zodiac Monkey for help on finding lucky numbers. There’s not really any finding them. Lotteries work hard to keep the winning numbers as unpredictable as possible. I have heard the lore that numbers up to 31 are picked by more people — they’re numbers that can be birthdays — so that multiple winners on the same drawing are more likely. I don’t know that this is true, though. I suspect that I could feel comfortable even with a four-way split of one and a half billions of dollars. Five-way would be out of the question, of course. Better to tear up the ticket than take that undignified split.

    Ahead of the exam, Ruthie asks, 'Instead of two number 2 pencils, can we bring one number 3 pencil and one number 1? Or one number 4 pencil or four number 1 pencils? And will there be any math on this test? I'm not good at math.'

    In Rick Detorie’s One Big Happy for the 3rd of February, 2016. The link will probably expire in early March.

    In Rick Detorie’s One Big Happy for the 3rd of February features Ruthie tossing off a confusing pile of numbers on the way to declaring herself bad at mathematics. It’s always the way.

    Breaking up a whole number like 4 into different sums of whole numbers is a mathematics problem also. Splitting up 4 into, say, ‘2 plus 1 plus 1’, is a ‘partition’ of the number. I’m not sure of important results that follow this sort of integer partition directly. But splitting up sets of things different ways runs through a lot of mathematics. Integer partitions are the ones you can do in elementary school.

    Percy Crosby’s Skippy for the 3rd of February — I believe it originally ran December 1928 — is a Roman numerals joke. The mathematical content may be low, but what the heck. It’s kind of timely. The Super Bowl, set for today, has been the most prominent use of Roman numerals we have anymore since the Star Trek movies stopped using them a quarter-century ago.

    Bill Amend’s FoxTrot for the 7th of February seems to be in agreement. And yes, I’m disappointed the Super Bowl is giving up on Roman numerals, much the way I’m disappointed they’re using a standardized and quite boring logo for each year. Part of the glory of past Super Bowls is seeing old graphic design eras preserved like fossils.

    Brian Gordon’s Fowl Language for the 5th of February shows a duck trying to explain incredibly huge numbers to his kid. It’s hard. You need to appreciate mathematics some to start appreciating real vastness. I’m not sure anyone can really have a feel for a number like 300 sextillion, the character’s estimate for the number of stars there are. You can make rationalizations for what numbers that big are like, but I suspect the mind shies back from staring directly at it.

    Infinity, and the many different sizes of infinity, might be easier to work with. One doesn’t need to imagine infinitely many things to work out the properties of infinitely large sets. You could do as well with a neatly drawn rectangle and some other, bigger, rectangles. But if you want to talk about the number 300,000,000,000,000,000,000,000 then you do want to think of something true about that number which isn’t also true about eight or about nine hundred million. But geology teaches us to ponder Deep Time. Astronomy trains us to imagine incredibly vast distances. Why not spend some time pondering huge numbers?

    And with all that said, I’d like to make one more call for any requests for my winter 2016 Mathematics A To Z glossary. There are quite a few attractive letters left unclaimed; a word or short term could be yours!

  • Joseph Nebus 3:00 pm on Friday, 5 February, 2016 Permalink | Reply
    Tags: Bob Dylan, , , ,   

    Reading the Comics, February 2, 2016: Pre-Lottery Edition 

    So a couple weeks ago one of the multi-state lotteries in the United States reached a staggering jackpot of one and a half billion dollars. And it turns out that “a couple weeks” is about the lead time most syndicated comic strip artists maintain. So there’s a rash of lottery-themed comic strips. There’s enough of them that I’m going to push those off to the next Reading the Comics installment. I’ll make do here with what Comic Strip master Command sent us before thoughts of the lottery infiltrated folks’ heads.

    Punkinhead: 'I was counting to five and couldn't remember what came after seven.' Tiger: 'If you're counting to five nothing comes after seven.' Punkinhead: 'I thought sure he would know.'

    Bud Blake’s Tiger for the 28th of January, 2016. I do like Punkinhead’s look of dismay in the second panel that Tiger has failed him.

    Bud Blake’s Tiger for the 28th of January (a rerun; Blake’s been dead a long while) is a cute one about kids not understanding numbers. And about expectations of those who know more than you, I suppose. I’d say this is my favorite of this essay’s strips. Part of that is that it reminds me of a bit in one of the lesser Wizard of Oz books. In it the characters have to count by twos to seventeen to make a successful wish. That’s the sort of problem you expect in fairy lands and quick gags.

    Mort Walker’s Beetle Bailey (Vintage) from the 7th of July, 1959 (reprinted the 28th of January) also tickles me. It uses the understanding of mathematics as stand-in for the understanding of science. I imagine it’s also meant to stand in for intelligence. It’s also a good riff on the Sisyphean nature of teaching. The equations on the board at the end almost look meaningful. At least, I can see some resemblance between them and the equations describing orbital mechanics. Camp Swampy hasn’t got any obvious purpose or role today. But the vintage strips reveal it had some role in orbital rocket launches. This was in the late 50s, before orbital rockets worked.

    General: 'How's your porject coming along to teach the men some science, Captain?' Captain: 'Wonderful, sir. Six months ago they didn't know what the square root of four was! Now they don't know what this [ blackboard full of symbols ] is!'

    Mort Walker’s Beetle Bailey (Vintage) for the 7th of July, 1959. This is possibly the brightest I’ve ever seen Beetle, and he doesn’t know what he’s looking at.

    Matt Lubchansky’s Please Listen To Me for the 28th of January is a riff on creationist “teach the controversy” nonsense. So we get some nonsense about a theological theory of numbers. Historically, especially in the western tradition, much great mathematics was done by theologians. Lazy histories of science make out religion as the relentless antagonist to scientific knowledge. It’s not so.

    The equation from the last panel, F(x) = \mathcal{L}\left\{f(t)\right\} = \int_0^{\infty} e^{-st} f(t) dt , is a legitimate one. It describes the Laplace Transform of the function f(t). It’s named for Pierre-Simon Laplace. That name might be familiar from mathematical physics, astronomy, the “nebular” hypothesis of planet formation, probability, and so on. Laplace transforms have many uses. One is in solving differential equations. They can change a differential equation, hard to solve, to a polynomial, easy to solve. Then by inverting the Laplace transform you can solve the original, hard, differential equation.

    Another major use that I’m familiar with is signal processing. Often we will have some data, a signal, that changes in time or in space. The Laplace transform lets us look at the frequency distribution. That is, what regularly rising and falling patterns go in to making up the signal (or could)? If you’ve taken a bit of differential equations this might sound like it’s just Fourier series. It’s related. (If you don’t know what a Fourier series might be, don’t worry. I bet we’ll come around to discussing it someday.) It might also remind readers here of the z-transform and yes, there’s a relationship.

    The transform also shows itself in probability. We’re often interested in the probability distribution of a quantity. That’s what the possible values it might have are, and how likely each of those values is. The Laplace transform lets us switch between the probability distribution and a thing called the moment-generating function. I’m not sure of an efficient way of describing what good that is. If you do, please, leave a comment. But it lets you switch from one description of a thing to another. And your problem might be easier in the other description.

    John McPherson’s Close To Home for the 30th of January uses mathematics as the sort of thing that can have an answer just, well, you see it. I suppose only geography would lend itself to a joke like this (“What state is Des Moines in?”)

    Wally explains to the Pointy-Haired Boss that he's in the Zeno's Paradox phase of the project, in which 'every step we take gets us halfway closer to launch', a pace that he hopes 'it will look' like he's keeping up. First week in, he is.

    Scott Adams’s Dilbert for the 31st of January. The link will probably expire around the end of February or start of March.

    Scott Adams’s Dilbert for the 31st of January mentions Zeno’s Paradox, three thousand years old and still going strong. I haven’t heard the paradox used as an excuse to put off doing work. It does remind me of the old saw that half your time is spent on the first 90 percent of the project, and half your time on the remaining 10 percent. It’s absurd but truthful, as so many things are.

    Samson’s Dark Side Of The Horse for the 2nd of February (I’m skipping some lottery strips to get here) plays on the merger of the ideas of “turn my life completely around” and “turn around 360 degrees”. A perfect 360 degree rotation would be an “identity tranformation”, leaving the thing it’s done to unchanged. But I understand why the terms merged. As with many English words or terms, “all the way around” can mean opposite things.

    But anyone playing pinball or taking time-lapse photographs or just listening to Heraclitus can tell you. Turning all the way around does not leave you quite what you were before. People aren’t perfect at rotations, and even if they were, the act of breaking focus and coming back to it changes what one’s doing.

  • Joseph Nebus 3:00 pm on Wednesday, 3 February, 2016 Permalink | Reply  

    How January 2016 Treated My Mathematics Blog 

    With no small expenditure of will I kept myself from looking at monthly statistics for my blogs through January. I didn’t look partway through the month or try to project what my readership might be; I tried to just let it be what it was and not worry.

    It all … wasn’t so bad. There were, says WordPress, 998 pages viewed here in January. That’s surely not going to make me feel pained that I didn’t log out and click refresh on my pages twice or something. Anyway, that’s up a tiny bit from December (954), and down from the November madhouse that was mostly caused by Apartment 3-G spillover (1,215 views). For what it’s worth, my 2015 average was 31 page views a day; January saw an average of 32. So, no big changes there. There were 523 unique visitors, up from December’s 449 and even November’s 519. I think this unique visitor count might be a record but WordPress doesn’t give me data more than a year old. So I can’t be sure.

    The number of likes, which has to be a measure of reader involvement, was down to 202. It had been at 245 in December and 220 in November. It was as high was 518 in June, but that was the month I posted something (nearly) every day, part of the Summer A To Z project.

    And speaking of the Summer A To Z project, I’m planning on the very different concept of a Winter A To Z project. Have requests for mathematical or mathematics-linked terms for me? Please pop over there and add one or more in the comments. (If all the requests go to the same comment thread it’ll be easier for me to lose the whole batch at once, instead of post-by-post.)

    The most popular posts, for a change, weren’t dominated by the Reading the Comics series. Instead we had:

    The United States, as ever, sent me more page views — 600 — than any other country. Next, with 54, was Hong Kong, which I don’t think has ever sent me so many readers. The United Kingdom gave me 51 page views, and both Germany and Canada had 35 apiece. Austria had 28. India, Singapore, and Poland each sent twelve, but I have to say Singapore wins that on a per-capita basis.

    My single-reader countries this time were Bangladesh, Cyprus, Czech Republic, the European Union, Ireland, Israel, the Netherlands, Nigeria, Oman, Romania, Slovenia, and Sweden. Bangladesh, Czech Republic, the European Union, Ireland, and Nigeria were all there in December, too. The European Union is on a two-month streak, and Nigeria a four-month streak.

    Search terms? We all like those, right? Among the ones bringing people here were:

    • frank and ernest jan 5, 2016
    • funny spring comic
    • origin is the gateway to your entire gaming universe.
    • eighth grade math rule where y always goes up by plus 2
    • unscramble talafo
    • creative form of right angle triangle sketches

    I have no idea what that “eighth grade math rule” might mean and I’d appreciate suggestions. Maybe whoever was looking for it will come back once somebody knows.

    • scifihammy 3:19 pm on Wednesday, 3 February, 2016 Permalink | Reply

      I like your search topics! Mine are invariably Unknown! :)
      Don’t stress about the stats. They really don’t mean anything.


      • Joseph Nebus 8:17 pm on Friday, 5 February, 2016 Permalink | Reply

        Thank you. Sadly Google started doing something with its search terms that keeps WordPress from being able to tell us what sends people to WordPress. The logic of how they can find things without knowing what they are escapes me, but Google and WordPress probably know what they’re doing, and it’s awful. But that does mean most of my search terms are really Unknown. I just grab at what’s left.

        Liked by 1 person

        • scifihammy 8:20 am on Saturday, 6 February, 2016 Permalink | Reply

          I think this applies to the views too! My daughter was explaining it to me once – but I forgot! :)


          • Joseph Nebus 5:20 am on Wednesday, 10 February, 2016 Permalink | Reply

            Oh, yes, there does seem to be something mysterious about the page view count. Especially since the great collapse in viewing numbers last year. I don’t know what the answer might be, besides of course not to worry too much about the exact numbers.

            Liked by 1 person

        • FlowCoef 11:16 pm on Tuesday, 9 February, 2016 Permalink | Reply

          The idea behind Google hiding their search terms was to keep 3d parties (ISPs, the NSA) from finding out what people were searching for. I agree it makes blog statistiques less useful though.


          • Joseph Nebus 5:43 am on Wednesday, 10 February, 2016 Permalink | Reply

            I hate to seem more cynical than I am, but it does feel to me that Google’s real concern was that some other advertising service might find out what people were searching for. It takes special programming skill to make every ad on every page someone visits be for the thing they just bought two days ago.


    • FlowCoef 11:15 pm on Tuesday, 9 February, 2016 Permalink | Reply

      The “eighth grade math rule” might be something like the “Rule of 3” or one of those old terms for basic algebra.


      • Joseph Nebus 5:38 am on Wednesday, 10 February, 2016 Permalink | Reply

        That might be. I wonder also if it could be someone was looking for Common Core expectations and I just happened to have a blog in the conceptual area.


  • Joseph Nebus 3:00 pm on Monday, 1 February, 2016 Permalink | Reply
    Tags: , , ,   

    Reading the Comics, January 27, 2015: Rabbit In A Trapezoid Edition 

    So the reason I fell behind on this Reading the Comics post is that I spent more time than I should have dithering about which ones to include. I hope it’s not disillusioning to learn that I have no clearly defined rules about what comics to include and what to leave out. It depends on how clearly mathematical in content the comic strip is; but it also depends on how much stuff I have gathered. If there’s a slow week, I start getting more generous about what I might include. And last week gave me a string of comics that I could argue my way into including, but few that obviously belonged. So I had a lot of time dithering.

    To make it up to you, at the end of the post I should have our pet rabbit tucked within a trapezoid of his own construction. If that doesn’t make everything better I don’t know what will.

    Mark Pett’s Mr Lowe for the 22nd of January (a rerun from the 19th of January, 2001) is really a standardized-test-question joke. But it brings up a debate about cultural biases in standardized tests that I don’t remember hearing lately. I may just be moving in the wrong circles. I remember self-assured rich white males explaining how it’s absurd to think cultural bias could affect test results since, after all, they’re standardized tests. I’ve sulked some around these parts about how I don’t buy mathematics’ self-promoted image of being culturally neutral either. A mathematical truth may be universal, but that we care about this truth is not. Anyway, Pett uses a mathematics word problem to tell the joke. That was probably the easiest way to put a cultural bias into a panel that

    T Lewis and Michael Fry’s Over The Hedge for the 25th of January uses a bit of calculus to represent “a lot of hard thinking”. Hammy the Squirrel particularly is thinking of the Fundamental Theorem of Calculus. This particular part is the one that says the derivative of the integral of a function is the original function. It’s part of how integration and differentiation link together. And it shows part of calculus’s great appeal. It has those beautiful long-s integral signs that make this part of mathematics look like artwork.

    Leigh Rubin’s Rubes for the 25th of January is a panel showing “Schrödinger’s Job Application”. It’s referring to Schrödinger’s famous thought experiment, meant to show there are things we don’t understand about quantum mechanics. It sets up a way that a quantum phenomenon can be set up to have distinct results in the everyday environment. The mathematics suggests that a cat, poisoned or not by toxic gas released or not by the decay of one atom, would be both alive and dead until some outside observer checks and settles the matter. How can this be? For that matter, how can the cat not be a qualified judge to whether it’s alive? Well, there are things we don’t understand about quantum mechanics.

    Roy Schneider’s The Humble Stumble for the 26th of January (a rerun from the 30th of January, 2007) uses a bit of mathematics to mark Tommy, there, as a frighteningly brilliant weirdo. The equation is right, although trivial. The force it takes to keep something with a mass m moving in a circle of radius R at the linear speed v is \frac{m v^2}{R} . The radius of the Moon’s orbit around the Earth is strikingly close to sixty times the Earth’s radius. The Ancient Greeks were able to argue that from some brilliantly considered geometry. Here, RE gets used as a name for “the radius of the Earth”. So the force holding the Moon in its orbit has to be approximately \frac{m v^2}{60 R_e} . That’s if we say m is the mass of the Moon, and v is its linear speed, and if we suppose the Moon’s orbit is a circle. It nearly is, and this would give us a good approximate answer to how much force holds the Moon in its orbit. It would be only a start, though; the precise movements of the Moon are surprisingly complicated. Newton himself could not fully explain them, even with the calculus and physics tools he invented for the task.

    Dave Whamond’s Reality Check for the 26th of January isn’t quite the anthropomorphic-numerals joke for this essay. But we do get personified geometric constructs, which is close, and some silly wordplay. Much as I like the art for Over The Hedge showcasing a squirrel so burdened with thoughts that his head flops over, this might be my favorite of this bunch.

    Dave Blazek’s Loose Parts for the 27th of January is a runner-up for the silly jokes trophy this time around.

    Our pet rabbit flopped out inside of a cardboard box. The box was set up, upside-down, so he could go inside and chew on the contents. He's pulled the side flaps inward, so that the base is a trapezoidal prism.

    Cardboard boxes are normally pretty good environments for rabbits, given that they’re places the rabbits can do in and not be seen. We set the box up, but he did all the chewing.

    Now I know what you’re thinking: isn’t that actually a trapezoidal prism, underneath a rectangular prism? Yes, I suppose so. The only people who’re going to say so are trying to impress people by saying so, though. And those people won’t be impressed by it. I’m sorry. We gave him the box because rabbits generally like having cardboard boxes to go in and chew apart. He did on his own the pulling-in of the side flaps to make it stand so trapezoidal.

    • scifihammy 3:32 pm on Monday, 1 February, 2016 Permalink | Reply

      I think it’s cute he made his bed the way he liked it :)
      We used to have rabbits. They have such personalities :)


      • Joseph Nebus 12:42 am on Tuesday, 2 February, 2016 Permalink | Reply

        He’s been working hard at becoming more cute lately. I’m surprised he isn’t resting on his cute laurels considering how certain his place in the household is.

        I’m new to rabbit-keeping. As a kid I kept guinea pigs, which I liked. But they weren’t nearly so extroverted as our rabbit, and their personality was more one of “gazing out wondering if they were supposed to be invited into this meeting”. It’s a style I like, certainly, but I understand people not seeing the appeal of that.

        Liked by 1 person

        • scifihammy 10:10 am on Tuesday, 2 February, 2016 Permalink | Reply

          Aw :) We also had a guinea pig, to keep the first rabbit company. He Loved his food and would squeak loudly when he thought it was dinner time. Very different to the rabbits, who could only try the Jedi mind trick on you! :)


          • Joseph Nebus 11:40 pm on Tuesday, 2 February, 2016 Permalink | Reply

            Oh, yes, the squeaking. Guinea pigs have a knack for that. Our rabbit sneezes sometime, and I swear one time I heard him bark, but those are rare events.

            Guinea pigs also have that popcorning habit. Rabbits will jump up sometimes too, although our rabbit’s reached the point in life where he would rather not do something quite that time-consuming if he can help it.

            Liked by 1 person

    • sheldonk2014 3:48 pm on Tuesday, 2 February, 2016 Permalink | Reply

      I got this idea for a probability problem
      You take aspirin every day
      The bottle holds 250 pills
      You only take a half
      But every time you want one a whole comes out
      What will it take for the halves to start to come out
      I think I explained this rite


      • Joseph Nebus 11:47 pm on Tuesday, 2 February, 2016 Permalink | Reply

        It’s a hard problem to answer, actually. What you need to provide an answer is to know how many halves there are, and how many wholes there are, and how well-mixed they are. If you have, say, 10 half-pills and 90 whole-pills, and you’re equally likely to pick a half or a whole, then the chance of picking a half-pill is ten out of a hundred, or ten percent. (There are 10 half-pills wanted, and there are 90 + 10 or 100 things to pick from.)

        However, in a real pill bottle, the half-pills and the whole-pills aren’t going to be equally likely to come out. The entire bottle starts out as whole-pills, after all. Half-pills are added when you’ve taken out a whole pill, cut it in half, and tossed one of the halves back in. So they’re going to start out almost entirely on top, closer to the lid and presumably more likely to be shaken or picked out.

        However again — in a jumble of large and small things, that gets shaken up, the small things are likely to drift to the bottom, and the large ones to the top. You’ve seen this when it seems like all the raisins sank to the bottom and the bran to the top of the cereal box; or when all the large peanuts are at the top of the mixed-nuts jar and the crumbly little things at the bottom. Half- and whole-pills aren’t as variable in size as mixed nuts, and the bottle isn’t shaken as thoroughly, but the effect is going to hold.

        So I’m not sure the problem can be answered purely by reasoning about it. I don’t think we can count on half-pills being as likely to be pulled out as whole-pills. And without some idea of the relatively likelihood of a half versus a whole there’s not a real way to answer. We can make some assumptions that might seem reasonable. But we can’t rely on those until they’re tested by experiment.


  • Joseph Nebus 3:00 pm on Saturday, 30 January, 2016 Permalink | Reply
    Tags: , ,   

    Any Requests? 

    I’m thinking to do a second Mathematics A-To-Z Glossary. For those who missed it, last summer I had a fun string of several weeks in which I picked a mathematical term and explained it to within an inch of its life, or 950 words, whichever came first. I’m curious if there’s anything readers out there would like to see me attempt to explain. So, please, let me know of any requests. All requests must begin with a letter, although numbers might be considered.

    Meanwhile since there’s been some golden ratio talk around these parts the last few days, I thought people might like to see this neat Algebra Fact of the Day:

    People following up on the tweet pointed out that it’s technically speaking wrong. The idea can be saved, though. You can produce the golden ratio using exactly four 4’s this way:

    \phi = \frac{\cdot\left(\sqrt{4} + \sqrt{4! + 4}\right)}{4}

    If you’d like to do it with eight 4’s, here’s one approach:

    And this brings things back around to how Paul Dirac worked out a way to produce any whole number using exactly four 2’s and the normal arithmetic operations anybody knows.

    • Christopher Adamson 3:06 pm on Saturday, 30 January, 2016 Permalink | Reply

      How about A is for axiom?


    • KnotTheorist 8:48 pm on Saturday, 30 January, 2016 Permalink | Reply

      I enjoyed last year’s Mathematical A-To-Z Glossary, so I’m glad to see you’ll be doing another one!

      I’d like to see C for continued fractions.


      • Joseph Nebus 12:39 am on Tuesday, 2 February, 2016 Permalink | Reply

        Continued fractions … mm. Well, I’ll have to learn more about them, but that’s part of the fun of this. Thank you.

        Liked by 1 person

    • davekingsbury 6:29 pm on Sunday, 31 January, 2016 Permalink | Reply

      Energy = Mass times Twice the Speed of Light … or is that more like Physics?


      • Joseph Nebus 12:40 am on Tuesday, 2 February, 2016 Permalink | Reply

        E = mc^2 is physics, although it’s something that we learned from mathematical considerations. And a big swath of mathematics is the study of physics. There’s a lot to talk about in energy for mathematicians.


        • elkement (Elke Stangl) 7:38 am on Monday, 8 February, 2016 Permalink | Reply

          Of course I second that :-) What about explaining a Lagrangian in layman’s terms? ;-)


          • Joseph Nebus 5:27 am on Wednesday, 10 February, 2016 Permalink | Reply

            You know, I think I’ve got a hook on how to explain that. It might even get to include a bit from my high school physics class.

            Liked by 1 person

    • davekingsbury 9:14 am on Tuesday, 2 February, 2016 Permalink | Reply

      Is the equation based on theory or is there a practical mathematics behind it?


      • Joseph Nebus 12:03 am on Wednesday, 3 February, 2016 Permalink | Reply

        I’m not sure what you mean by theory versus practical mathematics. The energy-mass equivalence does follow, mathematically, from some remarkably simple principles. Those amount to uncontroversial things like the speed of light being a constant, independent of the observer, and that momentum and energy are conserved.

        It is experimentally verified, though. We can, for example, measure the mass of atoms before and after they fuse, or fission, and measure the amount of energy released or absorbed as light in the process. The amounts match up as expected. (That’s not the only test to run, of course, but it’s an easy one to understand.) So the reasoning isn’t just good, but matches what we see in the real world.


        • davekingsbury 10:36 am on Wednesday, 3 February, 2016 Permalink | Reply

          Thanks for your clear explanation. I’m not a scientist. Theory wasn’t the right word, then – I was thinking of empirically verifiable which your 2nd paragraph shows. Are the ‘uncontroversial things in your first paragraph also measurable in the real world?


          • Joseph Nebus 8:34 pm on Friday, 5 February, 2016 Permalink | Reply

            OK. Well, these are measurable things, in that experiments give results that are what we would expect from the assumptions, and that are inconsistent with what we’d expect from alternate assumptions. For example, we now assume the speed of light (in a vacuum) to be constant. That followed a century of experimentation that finds it does appear to always be constant, and it’s consistent with tests that look to see if there might be something surprising now that we have a new effect to measure or a new tool to measure with. Assumptions about, for example, the way that velocities have to add together in order for this constant-speed-of-light to work have implications for how, say, moving electric charges will produce magnetic fields, and we see magnetic fields induced by moving electric charges consistently with that.

            We can imagine our current understanding to be incomplete, and that the real world has subtleties we haven’t yet detected. But I’m not aware of any outstanding mysteries that suggest strongly that we’re near that point.

            So, given assumptions that seem straightforward enough, and that match experiment as well as we’re able to measure, physicists and mathematicians are generally inclined to say that these assumptions are correct. Or at least correct enough for the context in which they’re used. This is starting to get into the philosophy of science and the concept of experimental proof and gets, I admit, beyond what I’m competent to discuss with authority.


            • davekingsbury 8:43 pm on Friday, 5 February, 2016 Permalink | Reply

              Thanks for taking the time (and space) to explain this so clearly and enjoyably to a rookie. No more questions, promise … for now!


    • Gillian B 5:39 am on Wednesday, 3 February, 2016 Permalink | Reply



    • gaurish 7:00 am on Monday, 8 February, 2016 Permalink | Reply

      Normal subgroup (easy one) or Number (difficult one, Bertrand Russell tried it once).


      • Joseph Nebus 5:24 am on Wednesday, 10 February, 2016 Permalink | Reply

        Oh, number is easy. Three, for example, is the thing that’s in common among Marx Brothers, blind mice, tricycle wheels, penny operas, and balls in the Midnight Multiball of the pinball game FunHouse. Normal subgroup, now that’s hard.


    • gaurish 7:10 am on Monday, 8 February, 2016 Permalink | Reply

      Transcendental numbers; Dedikind Domain; matrix; polynomial; quartenions; subjective map; vector.


      • Joseph Nebus 5:26 am on Wednesday, 10 February, 2016 Permalink | Reply

        There’s some good challenges here! My first reaction was to say I didn’t even know what a Dedekind domain was, although in looking it up I realize that I must have learned of them. I just haven’t thought of one in obviously too long, and I like the chance to learn something just in time to explain it.


    • elkement (Elke Stangl) 7:40 am on Monday, 8 February, 2016 Permalink | Reply

      C as Conjecture. More of a history of science question: When is an ‘unproven idea’ honored by being called a conjecture?


      • Joseph Nebus 5:35 am on Wednesday, 10 February, 2016 Permalink | Reply

        Conjecture may work, yes, and fit neatly against axiom trusting that I use that.

        I’m not sure there is a clear guide to when an unproven idea gets elevated to the status of conjecture. I suspect it would defy any rationally describable process. I mean about getting regarded as a name-worthy conjecture. There’s conjectures in much mathematical literature and those tend to mean the person writing the paper got a hunch that something might be so, but didn’t have the time or ability to prove it and is happy to let someone else try.

        But to be, let’s say, the Stangl Conjecture takes more. I suspect part is that it has to be something that feels likely to be true, and which has some obviously interesting consequence if true (or false). That can’t be all, though. The Collatz Conjecture, as I’ve mentioned, seems to be nothing but an amusing trifle. But then that’s also a conjecture that’s very easy for anyone to understand, and it has some beauty to it. The low importance of it might be balanced by how much fun it seems to be and how everyone can be in on the fun.

        I’ll have to do some more poking around famous conjectures, though, and see if I can better characterize what they have in common.

        Liked by 1 person

  • Joseph Nebus 11:00 pm on Wednesday, 27 January, 2016 Permalink | Reply
    Tags: , , ,   

    Silver-Leafed Numbers 

    In a comment on my “Gilded Ratios” essay fluffy wondered about a variation on the Golden and Golden-like ratios. What’s interesting about the Golden Ratio and similar numbers is that their reciprocal — one divided by them — is a whole number less than the original number. That is, 1 divided by 1.618(etc) is 0.618(etc), which is 1 less than the original number. 1 divided by 2.414(etc) is 0.414(etc), exactly 2 less than the original 2.414(etc). 1 divided by 3.302(etc) is 0.302(etc), exactly 3 less than the original 3.302(etc).

    fluffy wondered about a variation. Is there some number x that’s exactly 2 less than 2 divided by x? Or a (presumably) differently number that’s exactly 3 less than 3 divided by it? Yes, there is.

    Let me call the whole number difference — the 1 or 2 or 3 or so on, referred to above — by the name b. And let me call the other number — the one that’s b less than b divided by it — by the name x. Then a number x, for which b divided by x is exactly b less than itself, makes true the equation \frac{b}{x} = x - b . This is slightly different from the equation used last time, but not very different. Multiply both sides by x, which we know not to be zero, and we get a polynomial.

    Yes, quadratic formula, I see you waving your hand in the back there. And you’re right. There are two x’s that will make that equation true. The positive one is x = \frac12\left( b + \sqrt{b^2 + 4b} \right) . The negative one you get by changing the + sign, just before the square root, to a – sign, but who cares about that root? Here’s the first several of the (positive) silver-leaf ratios:

    Some More Numbers With Cute Reciprocals
    Number Silver-Leaf
    1 1.618033989
    2 2.732050808
    3 3.791287847
    4 4.828427125
    5 5.854101966
    6 6.872983346
    7 7.887482194
    8 8.898979486
    9 9.908326913
    10 10.916079783
    11 11.922616289
    12 12.928203230
    13 13.933034374
    14 14.937253933
    15 15.940971508
    16 16.944271910
    17 17.947221814
    18 18.949874371
    19 19.952272480
    20 20.954451150

    Looking over those hypnotic rows of digits past the decimal inspires thoughts. The part beyond the decimal keeps rising, closer and closer to 1. Does it ever get past 1? That is, might (say) the silver-leaf number that’s 2,038 more than its reciprocal be 2,039.11111 (or something)?

    No, it never does. There are a couple of ways to prove that, if you feel like. We can take the approach that’s easiest (to my eyes) to imagine. It takes a little algebraic grinding to complete. That is to look for the smallest number b for which the silver-leaf number, \frac12\left(b + \sqrt{b^2 + 4b}\right) , is larger than b + 1 . Follow that out and you realize that it’s any value of b for which 0 is greater than 4. Logically, therefore, we need to take b into a private room and have a serious talk about its job performance, what with it not existing.

    A harder proof to imagine working out, but that takes no symbol manipulation, comes from thinking about these reciprocals. Let’s imagine we had some b for which its corresponding silver-leaf number x is more than b + 1. Then, x – b has to be greater than 1. But if x is greater than 1, then its reciprocal has to be less than 1. We again have to talk with b about how its nonexistence is keeping it from doing its job.

    Are there other proofs? Most likely. I was satisfied by this point, and resolved not to work on it more until the shower. Updates after breakfast, I suppose.

  • Joseph Nebus 11:30 pm on Sunday, 24 January, 2016 Permalink | Reply
    Tags: , ,   

    Gilded Ratios 

    I may have mentioned that I regard the Golden Ratio as a lot of bunk. If I haven’t, allow me to mention: the Golden Ratio is a lot of bunk. I concede it’s a cute number. I found it compelling when I first had a calculator that let me use the last answer for a new operation. You can pretty quickly find that 1.618033 (etc, and the next digit is a 9 by the way) has a reciprocal that’s 0.618033 (etc).

    There’s no denying that. And there’s no denying that’s a neat pattern. But it is not some aesthetic ideal. When people evaluate rectangles that “look best” they go to stuff that’s a fair but not too much wider in one direction than the other. But people aren’t drawn to 1.618 (etc) any more reliably than they like 1.6, or 1.8, or 1.5, or other possible ratios. And it is not any kind of law of nature that the Golden Ratio will turn up. It’s often found within the error bars of a measurement, but so are a lot of numbers.

    The Golden Ratio is an irrational number, but basically all real numbers are irrational except for a few peculiar ones. Those peculiar ones happen to be the whole numbers and the rational numbers, which we find interesting, but which are the rare exception. It’s not a “transcendental number”, which is a kind of real number I don’t want to describe here. That’s a bit unusual, since basically all real numbers are transcendental numbers except for a few peculiar ones. Those peculiar ones include whole and rational numbers, and square roots and such, which we use so much we think they’re common. But not being transcendental isn’t that outstanding a feature. The Golden Ratio is one of those strange celebrities who’s famous for being a celebrity, and not for any actual accomplishment worth celebrating.

    I started wondering: are there other Golden-Ratio-like numbers, though? The title of this essay gives what I suppose is the best name for this set. The Golden Ratio is interesting because its reciprocal — 1 divided by it — is equal to it minus 1. Is there another number whose reciprocal is equal to it minus 2? Another number yet whose reciprocal is equal to it minus 3?

    So I looked. Is there a number between 2 and 3 whose reciprocal is it minus 2? Certainly there is. How do I know this?

    Let me call this number, if it exists, x. The reciprocal of x is the number 1/x. The number x minus 2 is the number x – 2. We’ll pick up the pace in a little bit. Now imagine trying out every single number from 2 to 3, in order. The reciprocals 1/x start out at 1/2 and drop to 1/3. The subtracted numbers start out at 0 and grow to 1. There’s no gaps or sudden jumps or anything in either the reciprocals or the subtracted numbers. So there must be some x for which 1/x and x – 2 are the same number.

    In the trade we call that an existence proof. It shows there’s got to be some answer. It doesn’t tell us much about what the answer is. Often it’s worth looking for an existence proof first. In this case, it’s probably overkill. But you can go from this to reasoning that there have to be Golden-Like-Ratio numbers between any two counting numbers. So, yes, there’s some number between 2,038 and 2,039 whose reciprocal is that number minus 2,038. That’s nice to know.

    So what is the number that’s two more than its reciprocal? That’s whatever number or numbers make true the equation \frac{1}{x} = x - 2 . That’s straightforward to solve. Multiply both sides by x, which won’t change whether the equation is true unless x is zero. (And x can’t be zero, or else we wouldn’t talk of 1/x except in hushed, embarrassed whispers.) This gets an equivalent equation 1 = x^2 - 2x . Subtract 1 from both sides, and we get 0 = x^2 - 2x - 1 and we’re set up to use the quadratic formula. The answer will be x = \left(\frac{1}{2}\right)\cdot\left(2 + \sqrt{2^2 + 4}\right) . The answer is about 2.414213562373095 (and on). (No, \left(\frac{1}{2}\right)\cdot\left(2 - \sqrt{2^2 + 4}\right) is not an answer; it’s not between 2 and 3.)

    The number that’s three more than its reciprocal? We’ll call that x again, trusting that we remember this is a different number with the same name. For that we need to solve \frac{1}{x} = x - 3 and that turns into the equation 0 = x^2 - 3x - 1 . And so x = \left(\frac{1}{2}\right)\cdot\left(3 + \sqrt{3^2 + 4}\right) and so it’s about 3.30277563773200. Yes, there’s another possible answer we rule out because it isn’t between 3 and 4.

    We can do the same thing to find another number, named x, that’s four more than its reciprocal. That starts with \frac{1}{x} = x - 4 and gets eventually to x = \left(\frac{1}{2}\right)\cdot\left(4 + \sqrt{4^2 + 4}\right) or about 4.23606797749979. We could go on like this. The number x that’s 2,038 more than its reciprocal is x = \left(\frac{1}{2}\right)\cdot\left(2038 + \sqrt{2038^2 + 4}\right) about 2038.00049082160.

    If your eyes haven’t just slid gently past the equations you noticed the pattern. Suppose instead of saying 2 or 3 or 4 or 2038 we say the number b. b is some whole number, any that we like. The number whose reciprocal is exactly b less than it is the number x that makes true the equation \frac{1}{x} = x - b . And that leads to the finding the number that makes the equation x = \left(\frac{1}{2}\right)\cdot\left(b + \sqrt{b^2 + 4}\right) true.

    And, what the heck. Here’s the first twenty or so gilded numbers. You can read this either as a list of the numbers I’ve been calling x — 1.618034, 2.414214, 3.302776 — or as an ordered list of the reciprocals of x — 0.618034, 0.414214, 0.302276 — as you like. I’ll call that the gilt; you add it to the whole number to its left to get that a number that, cutely, has a reciprocal that’s the same after the decimal.

    I did think about including a graph of these numbers, but the appeal of them is that you can take the reciprocal and see digits not changing. A graph doesn’t give you that.

    Some Numbers With Cute Reciprocals
    Number Gilt
    1 .618033989
    2 .414213562
    3 .302775638
    4 .236067977
    5 .192582404
    6 .162277660
    7 .140054945
    8 .123105626
    9 .109772229
    10 .099019514
    11 .090169944
    12 .082762530
    13 .076473219
    14 .071067812
    15 .066372975
    16 .062257748
    17 .058621384
    18 .055385138
    19 .052486587
    20 .049875621

    None of these are important numbers. But they are pretty, and that can be enough on a quiet day.

    • John Friedrich 1:08 am on Monday, 25 January, 2016 Permalink | Reply

      The Golden Ratio is neat for determining the nth Fibonacci sequence number at least.

      Liked by 1 person

      • Joseph Nebus 10:22 pm on Tuesday, 26 January, 2016 Permalink | Reply

        This is true, although the Fibonacci sequence has a similar problem in being pretty but not all that useful. It’s a bit more useful than the Golden Ratio, I’ll grant, and it would be out of character for me to complain about corners of mathematics that are just fun. But fun and important are different things.


    • fluffy 7:06 am on Monday, 25 January, 2016 Permalink | Reply

      So what about finding an arbitrary $x_N$ such that $N/x = x-N$? Is that solvable?


      • fluffy 7:06 am on Monday, 25 January, 2016 Permalink | Reply

        And what of the problem of inline LaTeX in a comment? Is that solvable?


        • Joseph Nebus 10:27 pm on Tuesday, 26 January, 2016 Permalink | Reply

          I’m not sure, although I wonder if you didn’t forget to include the word ‘latex’ after the first $ in the comment. I think inline LaTeX is supposed to be enabled in these parts.


          • fluffy 7:01 pm on Thursday, 28 January, 2016 Permalink | Reply

            I didn’t know that was necessary. So I do like a^3 + b^3 = c^3?

            A preview function would be nice.


            • Joseph Nebus 10:36 pm on Thursday, 28 January, 2016 Permalink | Reply

              That’s the way, yes. WordPress’s commenting system certainly needs a preview function and an edit button.

              (There might be other themes that have preview functions. The one I’m using here is a bit old-fashioned and it might predate a preview. I know it isn’t really mobile-friendly.)


      • Joseph Nebus 10:32 pm on Tuesday, 26 January, 2016 Permalink | Reply

        It’s certainly possible. Starting from \frac{N}{x} = x - N we get the equation 0 = x^2 - Nx - N and that has solutions x = \frac{1}{2} \left( N \pm \sqrt{N^2 + 4N}\right).

        I don’t seem able to include the table that would list the first couple of these without breaking the commenting system. But it’s easy to generate from that start. The x_N for a given N gets to be quite close to N+1.


        • Eric Mann 11:55 am on Wednesday, 27 January, 2016 Permalink | Reply

          These x_N’s are nice little multiples of the (dare I say) golden ratio, yes? They will arise from a rectangle with dimensions of x and N with an embedded N by N square.


          • Joseph Nebus 10:35 pm on Thursday, 28 January, 2016 Permalink | Reply

            I don’t see any link between these x_N’s or the golden ratio. Could you tell me what you see, please?


    • Eric Mann 11:51 am on Wednesday, 27 January, 2016 Permalink | Reply

      Well done. I love the inquiry. What do the rectangles look like? Or, is there still a geometric interpretation?

      I admit I still find the sequence of rectangles with Fibonacci dimensions and an embedded spiral attractive. I appreciate it in the limit. I am not attached to the ratio being golden with a capitol g, but is it just a curiosity? An attraction? I expect more out of the ratio, rectangle, and spiral.


      • Joseph Nebus 10:32 pm on Thursday, 28 January, 2016 Permalink | Reply

        Well, rectangles in these gilt-ratio proportions would be longer and skinnier things. For example, you might see a rectangle that’s one inch wide and 20.049(etc) inches long. I doubt anyone could tell the difference between that and a rectangle that’s one inch wide, 20 inches long, though.

        I do think it’s just a curiosity, an attractive-looking number. Or family of numbers, if you open up to these sorts of variations. There’s nothing wrong with looking at something that’s just attractive, though. It’s fun, for one thing. And the thinking done about one problem surely helps one practice for other problems. I was writing recently about the Collatz Conjecture. As far as I know nothing interesting depends on the conjecture being true or false, but it’s still enjoyable.


  • Joseph Nebus 8:00 pm on Friday, 22 January, 2016 Permalink | Reply
    Tags: blackbody radiation, , , standardized tests   

    Reading the Comics, January 21, 2016: Andertoons Edition 

    It’s been a relatively sleepy week from Comic Strip Master Command. Fortunately, Mark Anderson is always there to save me.

    In the Andertoons department for the 17th of January, Mark Anderson gives us a rounding joke. It amuses me and reminds me of the strip about rounding up the 196 cows to 200 (or whatever it was). But one of the commenters was right: 800 would be an even rounder number. If the teacher’s sharp he thought of that next.

    Andertoons is back the 21st of January, with a clash-of-media-expectations style joke. Since there’s not much to say of that, I am drawn to wondering what the teacher was getting to with this diagram. The obvious-to-me thing to talk about two lines intersecting would be which sets of angles are equal to one another, and how to prove it. But to talk about that easily requires giving names to the diagram. Giving the intersection point the name Q is a good start, and P and R are good names for the lines. But without points on the lines identified, and named, it’s hard to talk about any of the four angles there. If the lesson isn’t about angles, if it’s just about the lines and their one point of intersection, then what’s being addressed? Of course other points, and labels, could be added later. But I’m curious if there’s an obvious and sensible lesson to be given just from this starting point. If you have one, write in and let me know, please.

    'I'm losing my faith in the law of averages.' 'How come, man?' 'Because of how long it's been since a teacher asked me something I knew.'

    Ted Shearer’s Quincy for the 19th of January (originally the 4th of November, 1976).

    Ted Shearer’s Quincy for the 19th of January (originally the 4th of November, 1976) sees a loss of faith in the Law of Averages. We all sympathize. There are several different ways to state the Law of Averages. These different forms get at the same idea: on average, things are average. More, if we go through a stretch when things are not average, then, we shouldn’t expect that to continue. Things should be closer to average next time.

    For example. Let’s suppose in a typical week Quincy’s teacher calls on him ten times, and he’s got a 50-50 chance of knowing the answer for each question. So normally he’s right five times. If he had a lousy week in which he knew the right answer just once, yes, that’s dismal-feeling. We can be confident that next week, though, he’s likely to put in a better performance.

    That doesn’t mean he’s due for a good stretch, though. He’s as likely next week to get three questions right as he is to get eight right. Eight feels fantastic. But three is only a bit less dismal-feeling than one. The Gambler’s Fallacy, which is one of those things everyone wishes to believe in when they feel they’re due, is that eight right answers should be more likely than three. After all, that’ll make his two-week average closer to normal. But if Quincy’s as likely to get any question right or wrong, regardless of what came before, then he can’t be more likely to get eight right than to get three right. All we can say is he’s more likely to get three or eight right than he is to get one (or nine) right the next week. He’d better study.

    (I don’t talk about this much, because it isn’t an art blog. But I would like folks to notice the line art, the shading, and the grey halftone screening. Shearer puts in some nicely expressive and active artwork for a joke that doesn’t need any setting whatsoever. I like a strip that’s pleasant to look at.)

    Tom Toles’s Randolph Itch, 2 am for the 19th of January (a rerun from the 18th of April, 2000) has got almost no mathematical content. But it’s funny, so, here. The tag also mentions Max Planck, one of the founders of quantum mechanics. He developed the idea that there was a smallest possible change in energy as a way to make the mathematics of black-body radiation work out. A black-body is just what it sounds like: get something that absorbs all light cast on it, and shine light on it. The thing will heat up. This is expressed by radiating light back out into the world. And if it doesn’t give you that chill of wonder to consider that a perfectly black thing will glow, then I don’t think you’ve pondered that quite enough.

    Mark Pett’s Mister Lowe for the 21st of January (a rerun from the 18th of January, 2001) is a kid-resisting-the-word-problem joke. It’s meant to be a joke about Quentin overthinking the situation until he gets the wrong answer. Were this not a standardized test, though, I’d agree with Quentin. The given answers suppose that Tommy and Suzie are always going to have the same number of apples. But is inferring that a fair thing to expect from the test-takers? Why couldn’t Suzie get four more apples and Tommy none?

    Probably the assumption that Tommy and Suzie get the same number of apples was left out because Pett had to get the whole question in within one panel. And I may be overthinking it no less than Quentin is. I can’t help doing that. I do like that the confounding answers make sense: I can understand exactly why someone making a mistake would make those. Coming up with plausible wrong answers for a multiple-choice test is no less difficult in mathematics than it is in other fields. It might be harder. It takes effort to remember the ways a student might plausibly misunderstand what to do. Test-writing is no less a craft than is test-taking.

    • tkflor 4:46 am on Saturday, 23 January, 2016 Permalink | Reply

      About black body radiation – for most practical purposes, a black body at room temperature does not emit radiation in the visible range. (
      So, we won’t see “light” or “glow”.


      • Joseph Nebus 5:03 am on Sunday, 24 January, 2016 Permalink | Reply

        This is true, and I should have been clear about that. It glows in the sense that if you could look at the right part of the spectrum something would be detectable. It’s nevertheless an amazing thought.


    • Barb Knowles 6:20 pm on Saturday, 23 January, 2016 Permalink | Reply

      This cartoon is GREAT! I’m going to show it to my math colleagues..


  • Joseph Nebus 10:00 pm on Wednesday, 20 January, 2016 Permalink | Reply
    Tags: , , , , Peter Guthrie Tait,   

    Some More Mathematics Stuff To Read 

    And some more reasy reading, because, why not? First up is a new Twitter account from Chris Lusto (Lustomatical), a high school teacher with interest in Mathematical Twitter. He’s constructed the Math-Twitter-Blog-o-Sphere Bot, which retweets postings of mathematics blogs. They’re drawn from his blogroll, and a set of posts comes up a couple of times per day. (I believe he’s running the bot manually, in case it starts malfunctioning, for now.) It could be a useful way to find something interesting to read, or if you’ve got your own mathematics blog, a way to let other folks know you want to be found interesting.

    Also possibly of interest is Gregory Taylor’s Any ~Qs comic strip blog. Taylor is a high school teacher and an amateur cartoonist. He’s chosen the difficult task of drawing a comic about “math equations as people”. It’s always hard to do a narrowly focused web comic. You can see Taylor working out the challenges of writing and drawing so that both story and teaching purposes are clear. I would imagine, for example, people to giggle at least at “tangent pants” even if they’re not sure what a domain restriction would have to do with anything, or even necessarily mean. But it is neat to see someone trying to go beyond anthropomorphized numerals in a web comic. And, after all, Math With Bad Drawings has got the hang of it.

    Finally, an article published in Notices of the American Mathematical Society, and which I found by some reference now lost to me. The essay, “Knots in the Nursery:(Cats) Cradle Song of James Clerk Maxwell”, is by Professor Daniel S Silver. It’s about the origins of knot theory, and particularly of a poem composed by James Clerk Maxwell. Knot theory was pioneered in the late 19th century by Peter Guthrie Tait. Maxwell is the fellow behind Maxwell’s Equations, the description of how electricity and magnetism propagate and affect one another. Maxwell’s also renowned in statistical mechanics circles for explaining, among other things, how the rings of Saturn could work. And it turns out he could write nice bits of doggerel, with references Silver usefully decodes. It’s worth reading for the mathematical-history content.

    • elkement (Elke Stangl) 1:55 pm on Friday, 22 January, 2016 Permalink | Reply

      Your blog is really an awesome resource for all things math, no doubt!!


      • Joseph Nebus 5:02 am on Sunday, 24 January, 2016 Permalink | Reply

        That’s awfully kind of you to say. I’ve really just been grabbing the occasional thing that comes across my desk and passing that along, though, part of the great chain of vaguely sourced references.

        Liked by 1 person

        • elkement (Elke Stangl) 8:48 am on Sunday, 24 January, 2016 Permalink | Reply

          But ‘curating’ as they say today is an art, too, and after all you manage to make things accessible, e.g. by summarizing posts you reblog so neatly…. and manage to do so without much images!!


          • Joseph Nebus 10:12 pm on Tuesday, 26 January, 2016 Permalink | Reply

            Well, thank you again. I do feel like if I’m pointing to or reblogging someone else’s work I should provide a bit of context and original writing. It’s too easy to just pass around a link and say “here’s a good link”, which I wouldn’t blame anyone for doubting.

            Liked by 1 person

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