Reading the Comics, October 7, 2014: Repeated Comics Edition


Since my last roundup of mathematics-themed comic strips there’s been a modest drizzle of new ones, and I’m not sure that I can find any particular themes to them, except that Zach Weinersmith and the artistic collective behind Eric the Circle apparently like my attention. Well, what the heck; that’s easy enough to give.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 29) hopes to be that guy who appears somewhere around the fourth comment of every news article ever that mentions a correlation being found between two quantities. A lot of what’s valuable about science is finding causal links between things, but it’s only in rare and, often, rather artificial circumstances that such links are easy to show. What’s more often necessary is showing that as one quantity changes so does another, which allows one to suspect a link. Then, typically, one would look for a plausible reason they might have anything to do with one another, and look for ways to experiment and prove whether there is or is not.

But just because there is a correlation doesn’t by itself mean that one thing necessarily has anything to do with another. They could be coincidence, for example, or they could be influenced by some other confounding factor. To be worth mention in a decent journal, a correlation is probably going to be strong enough that it’s hard to believe it’s just coincidence, but there could yet be some confounding factor. And even if there is a causal link, in the complicated mess that is reality it can be difficult to discern which way the link flows. This is summarized in deductive logic by saying that correlation does not imply causation, but that uses deductive logic’s definition of “imply”.

In deductive logic to say “this implies that” means it is impossible for “this” to be true and “that” false simultaneously. It is perfectly permissible for both “this” and “that” to be true, and permissible for “this” to be false and “that” false, and — this is the point where Intro to Logic students typically crash — permissible for “this” to be false and “that” true. Colloquially, though, “imply” has a different connotation, something more along the lines of “this” and “that” have to both be false or both be true together. Don’t make that mistake on your logic test.

When a logician says that correlation does not imply causation, she is saying that it is imaginable for the correlation to be true while the causation is false. She is not saying the causation is false; she is just saying that the case is not proved from the fact of a correlation being true. And that’s so; if we just knew two things were correlated we would have to experiment to find whether there is a causal link. But finding a correlation one of the ways to start finding casual links; it’d be obviously daft not to use them as the start of one’s search. Anyway, that guy in about the fourth comment of every news report about a correlation just wants you to know it’s very important he tell you he’s smarter than journalists.

Saturday Morning Breakfast Cereal pops back up again (October 1) with an easier-to-describe joke about August Ferdinand Möbius and his rather famous strip, here applied to the old gag about walking to school uphill both ways. One hates to be a spoilsport, but Möbius was educated at home until 13, so this comic is not reliable as a shorthand biography of the renowned mathematician.

Eric the Circle has had a couple strips by “Griffinetsabine”, one on October 3, and another on the 7th of October, based on the Shape Singles Bar. Both strips are jokes about two points connecting by a line, suggesting that Griffinetsabine knew the premise was good for a couple of variants. I’d have spaced out the publication of them farther but perhaps this was the best that could be done.

Mikael Wulff and Anders Morgenthaler’s Truth Facts (September 30) — a panel strip that’s often engaging in showing comic charts — gives a guide to what the number of digits you’ve memorized says about you. (For what it’s worth, I peter out at “897932”.) I’m mildly delighted to find that their marker for Isaac Newton is more or less correct: Newton did work out pi to fifteen decimal places, by using his binomial theorem and a calculation of the area within a particular wedge of the circle. (As I make it out Wulff and Morgenthaler put Newton at fourteen decimal points, but they might have read references to Newton working out “fifteen decimal points” as meaning something different to what I do.) Newton’s was not the best calculation of pi in the 1660s when he worked it out — Christoph Grienberger, an Austrian Jesuit astronomer, had calculated 38 decimal places a generation earlier — but I can’t blame Wulff and Morgenthaler for supposing Newton to be a more recognizable name than Grienberger. I imagine if Einstein or Stephen Hawking had done any particularly unique work in calculating the digits of pi they’d have appeared on the chart too.

John Graziano’s Ripley’s Believe It or Not (October 1) — and don’t tell me that attribution doesn’t look weird — shares a story about the followers of the Ancient Greek mathematician, philosopher, and mystic Pythagoras, that they were forbidden to wear wool, eat beans, or pick up things they had dropped. I have heard the beans thing before and I think I’ve heard the wool prohibition before, but I don’t remember hearing about them not being able to pick up things before.

I’m not sure I can believe it, though: Pythagoras was a strange fellow, so far as the historical record is clear. It’s hard to be sure just what is true about him and his followers, though, and what is made up, either out of devoted followers building up the figure they admire or out of critics making fun of a strange fellow with his own little cult. Perhaps it’s so, perhaps it’s not. I would like to see a primary source, and I don’t think any exist.

The Little King skates a figure 8 that requires less tricky curving.
Otto Soglow’s The Little King for 29 February 1948.

Otto Soglow’s The Little King (October 5; originally run February 29, 1948) provides its normal gentle, genial humor in the Little King working his way around the problem of doing a figure 8.

July 2014 in Mathematics Blogging


We’ve finally reached the kalends of August so I can look back at the mathematics blog statistics for June and see how they changed in July. Mostly it’s a chance to name countries that had anybody come read entries here, which is strangely popular. I don’t know why.

Since I’d had 16,174 page views total at the start of July I figured I wasn’t going to cross the symbolically totally important 17,000 by the start of August and what do you know but I was right, I didn’t. I did have a satisfying 589 page views (for a total of 16,763), which doesn’t quite reach May’s heights but is a step up from June’s 492 views. The number of unique visitors as WordPress figures it was 231, up from June’s 194. That’s not an unusually large or small number of unique visitors for this year, and it keeps the views per visitor just about unchanged, 2.55 as opposed to June’s 2.54.

July’s most popular postings were mostly mathematics comics ones — well, they have the most reader-friendly hook after all, and often include a comic or two — but I’m gratified by what proved to be the month’s most popular since I like it too:

  1. To Build A Universe, and my simple toy version of an arbitrarily old universe. This builds on In A Really Old Universe and on What’s Going On In The Old Universe, and is followed by Lewis Carroll And My Playing With Universes, also some popular posts.
  2. Reading the Comics, July 3, 2014: Wulff and Morgenthaler Edition, I suppose because WuMo is a really popular comic strip these days.
  3. Reading the Comics, July 28, 2014: Homework in an Amusement Park Edition, I suppose because everybody likes amusement parks these days.
  4. Reading the Comics, July 24, 2014: Math Is Just Hard Stuff, Right? Edition, I suppose because people like thinking mathematics is hard these days.
  5. Some Things About Joseph Nebus, because I guess I had a sudden onset of being interesting?
  6. Reading the Comics, July 18, 2014: Summer Doldrums Edition, because summer gets to us all these days.

The countries sending me the most readers this month were the United States (369 views), the United Kingdom (43 views), and the Philippines (24 views). Australia, Austria, Canada, and Singapore turned up well too. Sending just a single viewer this month were Greece, Hong Kong, Italy, Japan, Norway, Puerto Rico, and Spain; Hong Kong and Japan were the only ones who did that in June, and for that matter May also. My Switzerland reader from June had a friend this past month.

Among the search terms that brought people to me this month:

  • comics strips for differential calculus
  • nebus on starwars
  • 82 % what do i need on my finalti get a c
  • what 2 monsters on monster legends make dark nebus

  • (this seems like an ominous search query somehow)
  • the 80s cartoon character who sees mathematics equations
  • starwars nebus
    (suddenly this Star Wars/Me connection seems ominous)
  • origin is the gateway to your entire gaming universe
    (I can’t argue with that)

Reading the Comics, July 3, 2014: Wulff and Morgenthaler Edition


Sorry to bring you another page of mathematics comics so soon after the last one, but, I don’t control Comic Strip Master Command. I’m not sure who does, but it’s obviously someone who isn’t paying very close attention to Mary Worth because the current psychic-child/angel-warning-about-pool-safety storyline is really going off the rails. But I can’t think of a way to get that back to mathematical topics, so let me go to safer territories instead.

Mickey's nephew figures this is rocket science.
The Disney Corporation’s _Mickey Mouse_ comic strip rerun on the 28th of June, 2014.

The Disney Corporation’s Mickey Mouse (June 28, rerun) uses the familiar old setup of mathematics stuff — here crossbred with rocket science — as establishment that someone is just way smarter than the rest of the room.

Wulff and Morgenthaler’s Truth Facts — a new strip from the people who do that WuMu which is replacing the strangely endless reruns of Get Fuzzy in your local newspaper (no, I don’t know why Get Fuzzy has been rerunning daily strips since November, and neither do its editors, so far as they’re admitting) — shows a little newspaper sidebar each day. The premise is sure to include a number of mathematics/statistics type jokes and on June 28th they went ahead with the joke that delivers statistics about statistics, so that’s out of the way.

Dave Whamond’s Reality Check (June 29) brings out two of the songs that prominently mention numbers.

Mel Henze’s Gentle Creatures (June 30) drops in a bit of mathematics technobabble for the sake of sounding all serious and science-y and all that. But “apply the standard Lagrangian model” is a better one than average since Joseph-Louis Lagrange was an astoundingly talented and omnipresent mathematician and physicist. Probably his most useful work is a recasting of Newton’s laws of physics in a form in which you don’t have to worry so much about forces at every moment and can instead look at the kinetic and potential energy of a system. This generally reduces the number of equations one has to work with to describe what’s going on, and that usually means it’s easier to understand them. That said I don’t know a specific “Lagrangian model” that would necessarily be relevant. The most popular “Lagrangian model” I can find talks about the flow of particles in a larger fluid and is popular in studying atmospheric pollutants, though the couple of medical citations stuggest it’s also useful for studying how things get transported by the bloodstream. Anyway, it’s nice to hear somebody besides Einstein get used as a science name.

Mary Beth figures if she works her apple-dividing and giving right she can get al the apple.
John Rose’s _Barney Google and Snuffy Smith_ for the 1st of July, 2014, featuring neither Barney Google nor Snuffy Smith.

John Rose’s Barney Google and Snuffy Smith (July 1) plays with division word problems and percentages and the way people can subvert the intentions of a problem given any chance.

Bill Watterson’s Calvin and Hobbes (July 1, rerun) lets Calvin’s Dad gently blow Calvin’s mind by pointing out that rotational motion means that different spots on the same object are moving at different speeds yet the object stays in one piece. When you think hard enough about it rotation is a very strange phenomenon (I suppose you could say that about any subject, though), and the difference in speeds within a single object is just part of it. Sometime we must talk about the spinning pail of water.

Wulff and Morgenthaler’s WuMo (July 1) — I named this edition after them for some reason, after all — returns to the potential for mischief in how loosely one uses the word “half”.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers (July 3) dips into the well of mathematics puns. I admit I had to reread the caption before noticing where the joke was. It’s been a busy week.

June 2014 In Mathematics Blogging


And with the start of July I look over how well the mathematics blog did in June and see what I can learn from that. It seems more people are willing to read when I post stuff, which is worth knowing, I guess. After May’s near-record of 751 views and 315 visitors I expected a fall, and, yeah, it came. The number of pages viewed dropped to 492, which is … well, the fourth-highest this year at least? And the number of unique visitors fell to 194, which is actually the lowest of this year. The silver lining is this means the views per visitor, 2.54, was the second-highest since WordPress started sharing those statistics with me, so, people who come around find themselves interested. I start the month at 16,174 views total and won’t cross 17,000 at that rate come July, but we’ll see what I can do. And between WordPress and Twitter I’m (as of this writing) at exactly 400 followers, which isn’t worldshaking but is a nice big round number. I admit thinking how cool it would be if that were 400 million but I’d probably get stage fright if it were.

If one thing defined June it was “good grief but there’s a lot of mathematics comics”, which I attributed to Comic Strip Master Command ordering cartoonists to clear the subject out before summer vacation. It does mean the top five posts for June are almost comically lopsided, though:

Now, that really is something neat about triangles in the post linked above so please do read it. What I’m not clear about is why the June 16th comics post was so extremely popular; it’s nearly twice as viewed as the runner-up. If I were sure what keyword is making it so popular I’d do more with that.

Now on to the international portion of this contest: what countries are sending me the most visitors? Of course the United States comes in first, at 336 views. Denmark finished second with 17, and there was a three-way tie for third as Australia, Austria, and the United Kingdom sent sixteen each. (Singapore and Canada came in next with nine each.) I had a pretty nice crop of single-reader countries this month: Argentina, Bosnia and Herzegovina, Cambodia, Egypt, Ghana, Hong Kong, Indonesia, Japan, Paraguay, Saudi Arabia, Switzerland, and Thailand. Hong Kong, Japan, and Switzerland are repeats from last month and nobody’s got a three-month streak going.

Among the interesting search terms to bring people to me:

  • names for big numbers octillion [ happy to help? ]
  • everything to need to know about trapezoids [ I’m going to be the world’s authority on trapezoids! ]
  • what does the fact that two trapezoids make a parallelogram say about tth midline [ I have some ideas but don’t want to commit to anything particular ]
  • latching onto you 80 version [ I … think I’m being asked for lyrics? ]
  • planet nebus [ I feel vaguely snarked upon, somehow ]
  • origin is the gateway to your entire gaming universe [ … thank you? ]
  • nebus student job for uae [ Um … I guess I can figure out a consulting fee or something if you ask? ]

Reading the Comics, June 27, 2014: Pretty Easy Edition


I don’t mean to complain, because it really is a lot of fun to do these comic strip roundups, but Comic Strip Master Command has been sending a flood of comics my way. I hope it’s not overwhelming readers, or me. The downside of the great number of mathematics-themed comics this past week has been that they aren’t very deep examples, but, what the heck. Many of them are interesting anyway. As usual I’m including examples of the Comics Kingdom and the Creators.com comics because I’m not yet confident how long those links remain visible to non-subscribers.

A caveman invents the 'science fictiony' number of eleven.
Mike Peters’s _Mother Goose and Grimm_ for the 23rd of June, 2014

Mike Peters’ Mother Goose and Grimm (June 23) presents the cavemen-inventing-stuff pattern and the invention of a “science-fictiony” number. This is amusing, sure, but the dynamic is historically valid: it does seem like the counting numbers (1, 2, 3, and so on) were more or less intuitive, but negative numbers? Rationals? Irrationals? Zero? They required development and some fairly sophisticated reasoning to think of. You get a hint of the suspicion with which the newly-realized numbers were viewed when you think of the connotations of terms like “complex” numbers, or “imaginary” numbers, or even “negative” numbers. For that matter, Arabic numerals required some time for Europeans — who were comfortable with Roman numerals — to feel comfortable with; histories of mathematics will mention how Arabic numerals were viewed with suspicion and sometimes banned as being too easy for merchants or bankers to use to defraud customers who didn’t know what the symbols meant or how to use them.

Thom Bluemel’s Birdbrains (June 23) also takes us to the dawn of time and the invention of the calendar. Calendars are deeply intwined with mathematics, as they typically try to reconcile several things that aren’t quite perfectly reconcilable: the changes of the season, the cycles of the moon, the position of the sun in the sky, the length of the day. But the attempt to do as well as possible, using rules easy enough for normal human beings to understand, is productive.

Mark Pett’s Lucky Cow (June 23, rerun) lets Neil do some accounting the modern old-fashioned way. I trust there are abacus applications out there; somewhere in my pile of links I had a Javascript-based slide rule simulator, after all. I never quite got abacus use myself.

Mark Parisi’s Off The Mark (June 23) shows off one of those little hazards of skywriting and mathematical symbols. I admit the context threw me; I had to look again to read the birds as the less-than sign.

Dilton Doiley and Jughead find wonder in the vast and the numerous.
Henry Scarpelli and Craig Boldman’s _Archie_ rerun on the 24th of June, 2014

Henry Scarpelli and Craig Boldman’s Archie (June 24) has resident nerd Dilton Doiley pondering the vastness of the sky and the number of stars and feel the sense of wonder that inspires. The mind being filled with ever-increasing wonder and awe isn’t a unique sentiment, and thinking hard of very large, very numerous things is one of the paths to that sensation. Jughead has a similar feeling, evidently.

The orangutan who 'probably never had a thought in his life' works out a pretty involved derivative using the product and chain rules.
Mort Walker (Addison)’s _Boner’s Ark_ rerun the 26th of June, 2014 (originally run the 31st of July, 1968).

Mort Walker (“Addison”)’s Boner’s Ark (June 26, originally run July 31, 1968) features once again the motif of “a bit of calculus proves someone is really smart”. The orangutan’s working out of a derivative starts out well, too, using the product rule correctly through the first three lines, a point at which the chain rule and the derivative of the arccotangent function conspire to make things look really complicated. I admit I’m impressed Walker went to the effort to get things right that far in and wonder where he got the derivative worked out. It’s not one of the standard formulas you’d find in every calculus textbook, although you might find it as one of the more involved homework problem for Calculus I.

Mark Pett’s Lucky Cow comes up again (June 26, rerun) sees Neil a little gloomy at the results of a test coming back “negative”, a joke I remember encountering on The Office (US) too. It brings up the question of why, given the connotations of the words, a “positive” test result is usually a bad thing and a “negative” one a good, and it back to the language of statistics. Normally a test — medical, engineering, or otherwise — is really checking to see how often some phenomenon occurs within a given sample. But the phenomenon will normally happen a little bit anyway, even if nothing untoward is happening. It also won’t normally happen at exactly the same rate, even if there’s nothing to worry about. What statistics asks, then, is, “is this phenomenon happening so much in this sample space that it’s not plausible for it to just be coincidence?” And in that context, yeah, everything being normal is the negative result. What happens isn’t suspicious. Of course, Neil has other issues, here.

Chip Dunham’s Overboard (June 26) plays on the fact that “half” does have a real proper meaning, but will get used pretty casually when people aren’t being careful. Or when dinner’s involved.

Percy Crosby’s Skippy (June 26, rerun) must have originally run in March sometime, and it does have Skippy and the other kid arguing about how many months it is until Christmas. Counting intervals like this does invite what’s termed a “fencepost error”, and the kids present it perfectly: do you count the month you’re in if you want to count how many months until something? There isn’t really an absolutely correct answer, though; you and the other party just have to agree on whether you mean, say, the pages on the calendar you’ll go through between today and Christmas, or whether you mean how many more times you’ll pass the 24th of the month until you get to Christmas. You will see this same dynamic in every argument about conventions ever. Two spaces after the end of the sentence.

Moose has a fair idea for how to get through the algebra book in time.
Henry Scarpelli and Craig Boldman’s _Archie_ rerun for the 27th of June, 2014.

In Henry Scarpelli and Craig Boldman’s Archie (June 27, rerun), Moose has a pretty good answer to how to get the whole algebra book read in time. It’d be nice if it quite worked that way.

Mel Henze’s Gentle Creatures (June 27) has the characters working out just what the calculations for a jump into hyperspace would be. I admit I’ve always wondered just what the calculations for that sort of thing are, but that’s a bit silly of me.

Reading the Comics, June 22, 2014: Name-Dropping Stuff Edition


Comic Strip Master Command apparently really is ordering strips to finish their mathematics jokes before the summer vacation sets in, based on how many we’ve gotten in the past week. I confess this set doesn’t give me so much to write about; it’s more a set of mathematics things getting name-dropped. And there’s always something, isn’t there?

Tom Thaves’s Frank and Ernest (June 17) showcases a particularly severe form of math anxiety. I’m sympathetic to people who’re afraid of mathematics, naturally; it’s rotten being denied a big and wonderful and beautiful part of human ingenuity. I don’t know where math anxiety comes from, although I’d imagine a lot of it comes from that mix of doing something you aren’t quite sure you’re doing correctly and being hit too severely with a sense of rejection in the case that you did it wrong. I’d like to think that recreational mathematics puzzles would help overcome that, but I have no evidence that it does, just my hunch that getting to play with numbers and pictures and logic puzzles is good for you.

Russell Myers’ Broom Hilda (June 18) taunts the schoolkid Nerwin with the way we “used to do math with our brains instead of calculators”. One hesitates to know too much about the continuity of Broom Hilda, but I believe she’s over a thousand years old and so when she was Nerwin’s age they didn’t even have Arabic numerals just yet. I’ll assume there’s some way she’d have been in school then. (Also, given how long Broom Hilda‘s been running Nerwin did used to be in classes that did mathematics without calculators.)

Hagar can't count how many beers he had, and so proposes getting a math tutor.
Chris Browne’s Hagar the Horrible for the 19th of June, 2014.

Chris Brown’s Hagar the Horrible (June 19) tries to get itself cut out and put up on the walls of math tutors’ offices. Good luck.

Crankshaft does arithmetic by counting on his fingers, including long division.
Tom Batiuk and Chuck Ayers’s Crankshaft for the 20th of June, 2014.

Tom Batiuk and Chuck Ayers’ Crankshaft (June 20) spent a couple days this week explaining how he just counts on fingers to do his arithmetic. It’s a curious echo of the storyline several years ago revealing Crankshaft suffered from Backstory Illiteracy, in which we suddenly learned he had gone all his life without knowing how to read. I hesitate to agree with him but, yeah, there’s no shame in counting on your fingers if that does all the mathematics you need to do and you get the answers you want reliably. I don’t know what his long division thing is; if it weren’t for Tom Batiuk writing the comic strip I’d call it whimsy.

Keith Knight’s The Knight Life carried on with the story of the personal statistician this week. I think the entry from the 20th is most representative. It’s fine, and fun, to gather all kinds of data about whatever you encounter, but if you aren’t going to study the data and then act on its advice you’re wasting your time. The personal statistician ends up quitting the job.

Steve McGarry’s kid-activity feature KidTown (June 22) promotes the idea of numbers as a thing to notice in the newspapers, and includes a couple of activities, one featuring a maze to be navigated by way of multiples of seven. It also has one of those math tricks where you let someone else pick a number, give him a set of mathematical operations to do, and then you can tell them what the result is. It seems to me working out why that scheme works is a good bit of practice for someone learning algebra, and developing your own mathematics trick that works along this line is further good practice.

Reading the Comics, June 16, 2014: Cleaning Out Before Summer, I Guess, Edition


I had thought the folks at Comic Strip Master Command got most of their mathematics-themed comics cleaned out ahead of the end of the school year (United States time zones) by last week, and then over the course of the weekend they went and published about a hundred million of them, so let me try catching up on that before the long dry spell of summer sets in. (And yet none of them mentioned monkeys writing Shakespeare; go figure.) I’m kind of expecting an all-mathematics-strips series tomorrow morning.

Jason Chatfield’s Ginger Meggs (June 12) puns a bit on negative numbers as also meaning downbeat or pessimistic ones. Negative numbers tend to make people uneasy, when they’re first encountered. It took western mathematics several centuries to be quite fully comfortable with them and that even with the good example of debts serving as a mental model of what negative numbers might mean. Descartes, for example, apparently used four separate quadrants, giving points their positions to the right and up, to the left and up, to the left and down, or to the right and down, from the origin point, rather than deal with negative numbers; and the Fahrenheit temperature scale was pretty much designed around the constraint that Daniel Fahrenheit shouldn’t have to deal with negative numbers in measuring the temperature in his hometown of the Netherlands. I have seen references to Immanuel Kant writing about the theoretical foundation of negative numbers, but not a clear explanation of just what he did, alas. And skepticism of exotic number constructs would last; they’re not called imaginary numbers because people appreciated the imaginative leaps that working with the square roots of negative numbers inspired.

Steve Breen and Mike Thompson’s Grand Avenue (June 12) served notice that, just like last summer, Grandma is going to make sure the kids experience mathematics as a series of chores they have to endure through an otherwise pleasant summer break.

Mike Twohy’s That’s Life (June 12) might be a marginal inclusion here, but it does refer to a lab mouse that’s gone from merely counting food pellets to cost-averaging them. The mathematics abilities of animals are pretty amazing things, certainly, and I’d also be impressed by an animal that was so skilled in abstract mathematics that it was aware “how much does a thing cost?” is a pretty tricky question when you look hard at it.

Jim Scancarelli’s Gasoline Alley (June 13) features a punch line that’s familiar to me — it’s what you get by putting a parrot and the subject of geometry together — although the setup seems clumsy to me. I think that’s because the kid has to bring up geometry out of nowhere in the first panel. Usually the setup as I see it is more along the lines of “what geometric figure is drawn by a parrot that then leaves the room”, which I suppose also brings geometry up out of nowhere to start off, really. I guess the setup feels clumsy to me because I’m trying to imagine the dialogue as following right after the previous day’s, so the flow of the conversation feels odd.

Eric the Circle (June 14), this one signed “andel”, riffs on the popular bit of mathematics trivia that in a randomly selected group of 22 people there’s about a fifty percent chance that some pair of them will share a birthday; that there’s a coincidental use for 22 in estimating π is, believe it or not, something I hadn’t noticed before.

Pab Sungenis’s New Adventures of Queen Victoria (June 14) plays with infinities, and whether the phrase “forever and a day” could actually mean anything, or at least anything more than “forever” does. This requires having a clear idea what you mean by “forever” and, for that matter, by “more”. Normally we compare infinitely large sets by working out whether it’s possible to form pairs which match one element of the first set to one element of the second, and seeing whether elements from either set have to be left out. That sort of work lets us realize that there are just as many prime numbers as there are counting numbers, and just as many counting numbers as there are rational numbers (positive and negative), but that there are more irrational numbers than there are rational numbers. And, yes, “forever and a day” would be the same length of time as “forever”, but I suppose the Innamorati (I tried to find his character’s name, but I can’t, so, Pab Sungenis can come in and correct me) wouldn’t do very well if he promised love for the “power set of forever”, which would be a bigger infinity than “forever”.

Mark Anderson’s Andertoons (June 15) is actually roughly the same joke as the Ginger Meggs from the 12th, students mourning their grades with what’s really a correct and appropriate use of mathematics-mentioning terminology.

Keith Knight’s The Knight Life (June 16) introduces a “personal statistician”, which is probably inspired by the measuring of just everything possible that modern sports has gotten around to doing. But the notion of keeping track of just what one is doing, and how effectively, is old and, at least in principle, sensible. It’s implicit in budgeting (time, money, or other resources) that you are going to study what you do, and what you want to do, and what’s required by what you want to do, and what you can do. And careful tracking of what one’s doing leads to what’s got to be a version of the paradox of Achilles and the tortoise, in which the time (and money) spent on recording the fact of one’s recordings starts to spin out of control. I’m looking forward to that. Don’t read the comments.

Max Garcia’s Sunny Street (June 16) shows what happens when anthropomorphized numerals don’t appear in Scott Hilburn’s The Argyle Sweater for too long a time.

Autocorrected Monkeys and Pulled Tea


The Twop Twips account on Twitter — I’m not sure how to characterize what it is exactly, but friends retweet it often enough — had the above advice about the infinite monkeys problem, and what seems to me correct advice that turning on autocorrect will get them to write the works of Shakespeare more quickly. And then John Kovaleski’s monkey-featuring comic strip Bo Nanas featured the infinite monkey problem today, so obviously I have to spend more time thinking of it.

It seems fair that monkeys with autocorrect will be more likely to hit a word than a monkey without will be. Let’s try something simpler than Shakespeare and just consider the chance of typing the word “the”, and to keep the numbers friendly let’s imagine that the keyboard has just the letters and a space bar. We’ll not care about punctuation or numbers; that’s what copy editors would be for, if anyone had been employed as a copy editor since 1996, when someone in the budgeting office discovered there was autocorrect.

Anyway, there’s 27 characters on this truncated keyboard, and if the monkeys were equally likely to hit any one of them, then, there’d be 27 times 27 times 27 — that is, 19,683 — different three-character strings they might hit. Exactly one of them is the desired word “the”. So, roughly, we would expect the monkey to get the word right one time in each 19,683 attempts at a three-character string. (We wouldn’t have to wait quite so long if we’ll accept the monkey as writing continuously and pluck out three characters in a row wherever they appear, but that’s more work than I feel like doing, and I doubt it would significantly change the qualitative results, of how much faster it’d be if autocorrect were on.)

But how many tries would be needed to hit a word that gets autocorrected to “the”? And here we get into the mysteries of the English language. I’d be surprised by a spell checker that couldn’t figure out “teh” probably means “the”. Similarly “hte” should get back to “the”. So we can suppose the five other permutations of the letters in “the” will be autocorrected. So there’s six different strings of the 19,683 possibilities that will get fixed to “the”. The monkey has one chance in 3280.5 of getting one of them and so, on average, the monkey can be expected to be right once in every 3281 attempts.

But there’s other typos possible: “thw” is probably just my finger slipping, and “ghe” isn’t too implausible either. At least my spell checker recognizes both as most likely meant to be “the”. Let’s suppose that a spell checker can get to the right word if any one letter is mistaken. This means that there are some 78 other three-character strings that would get fixed to “the”, for a total of 84 possible three-character strings which are either “the” or would get autocorrected to “the”. With that many, there’s one chance in a touch more than 234 that a three-character string will get corrected to “the”, and we have to wait, considering, not very long at all.

It gets better if two-character errors are allowed, but I can’t make myself believe that the spell check will turn “yje” into “the”, and that’s something which might be typed if you just had the right hand on the wrong keys. My checker hasn’t got any idea what “yje” is supposed to be anyway, so, one wrong letter is probably the limit.

Except. “tie” is one character wrong for “the” and no spell checker will protest “tie”. Similarly “she” and “thy” and a couple of other words. And it’d be a bit much to expect “t e” or “ he” to be turned back into “the” even though both are just the one keystroke off. And a spell checker would probably suppose that “tht” is a typo for “that”. It’s hard to guess how many of the one-character-off words will not actually be caught. Let’s say that maybe half the one-character-off words will be corrected to “the”; that’s still a pretty good 39 one-character misspellings, plus five permutations, plus the correct spelling or 45 candidate three-character strings for autocorrect to get. So our monkey has something like one chance in 450 of getting “the” in banging on the keyboard three times.

For four-letter words there are many more combinations — 531,441, if we just list the strings of our 27 allowed characters — but then there are more strings which would get autocorrected. Let’s say we want the string “thus”; there are 23 ways to arrange those letters in addition to the correct one. And there are 104 one-character-off strings; supposing that half of them will get us to “thus”, then, there’s 76 strings that get one to the desired “thus”. That’s a pretty dismal one chance in about 7,000 of typing one of them, unfortunately. Things get a little better if we suppose that some two-character errors are going to be corrected, although I can’t find one which my spell checker will accept right now, and if a single error and a transposition are viable.

With longer words yet there’s more chances for spell checker forgiveness: you can get pretty far off “accommodate” or “aneurysm” and still be saved by the spell checker, which is good for me as I last spelled “accommodate” correctly sometime in 1992, and I thought it looked wrong then.

So the conclusion has to be: you’ll get a bit of an improvement in speed by turning on autocorrect, for the obvious reason that you’re more likely to get one right out of 450 than you are to get one right out of 19,000. But it’s not going to help you very much; the number of ways to spell things so completely wrong that not even spell check can find you just grows far too rapidly to be helped. If I get a little bored I might work out the chance of getting a permutation-or-one-off for strings of different lengths.

And your monkey might be ill-served by autocorrect anyway. When I lived in Singapore I’d occasionally have teh tarik (“pulled tea”), black tea with sugar and milk tossed back and forth until it’s nice and frothy. It’s a fine drink but hard to write back home about because even if you get past the spell checker, the reader assumes the “teh” is a typo and mentally corrects for it. When this came up I’d include a ritual emphasis that I actually meant what I wrote, but you see the problem. Fortunately Shakespeare wrote relatively little about southeast Asian teas, but if you wanted to expand the infinite monkey problem to the problem of guiding tourists through Singapore, you’d have to turn the autocorrect off to have any hope of success.

15,000 And A Half


I’d failed to mention the day it happened but I reached my 15,000th page view, just a couple days past the end of April. (If I haven’t added wrong, it was somebody who read something on the 5th of May.) So I like that my middling popularity is continuing, and, as I said in the review of April’s statistics, the blog-writing has felt particularly rich for me of late, for reasons I don’t consciously know. Meanwhile I’m already about a sixth of the way to 16,000, again, a gratifying touch. It’s horribly easy for a personality like mine to get worried about readership statistics; the flip side is when I’m not worried it feels so contented.

To cover the other half of my title, my dear love mentioned tripping over something in the tangent-plane article: “imagine the sphere sliced into a big and a small half by a plane. Imagine moving the plane in the direction of the smaller slice; this produces a smaller slice yet.” And how can there be a big and a small half?

Well, because I was sloppy in writing, is all. I should’ve said something like “a big and a small piece”. I failed to spend enough time editing and rereading before publishing. All I can say is this made me notice that apparently one can speak of two unequal halves of something without noticing that one is defying the literal meaning of the word. Maybe the ability to do so reflects an idea that a division of something might be equal in one way and unequal in others and the word “half” has to allow either sense. Maybe it just reflects that English is a supremely flexible language in that any word can mean pretty much anything, at any time, without any warning. Or I was just being sloppy.

Reading the Comics, May 4, 2014: Summing the Series Edition


Before I get to today’s round of mathematics comics, a legend-or-joke, traditionally starring John Von Neumann as the mathematician.

The recreational word problem goes like this: two bicyclists, twenty miles apart, are pedaling toward each other, each at a steady ten miles an hour. A fly takes off from the first bicyclist, heading straight for the second at fifteen miles per hour (ground speed); when it touches the second bicyclist it instantly turns around and returns to the first at again fifteen miles per hour, at which point it turns around again and head for the second, and back to the first, and so on. By the time the bicyclists reach one another, the fly — having made, incidentally, infinitely many trips between them — has travelled some distance. What is it?

And this is not hard problem to set up, inherently: each leg of the fly’s trip is going to be a certain ratio of the previous leg, which means that formulas for a geometric infinite series can be used. You just need to work out what the lengths of those legs are to start with, and what that ratio is, and then work out the formula in your head. This is a bit tedious and people given the problem may need some time and a couple sheets of paper to make it work.

Von Neumann, who was an expert in pretty much every field of mathematics and a good number of those in physics, allegedly heard the problem and immediately answered: 15 miles! And the problem-giver said, oh, he saw the trick. (Since the bicyclists will spend one hour pedaling before meeting, and the fly is travelling fifteen miles per hour all that time, it travels a total of a fifteen miles. Most people don’t think of that, and try to sum the infinite series instead.) And von Neumann said, “What trick? All I did was sum the infinite series.”

Did this charming story of a mathematician being all mathematicky happen? Wikipedia’s description of the event credits Paul Halmos’s recounting of Nicholas Metropolis’s recounting of the story, which as a source seems only marginally better than “I heard it on the Internet somewhere”. (Other versions of the story give different distances for the bicyclists and different speeds for the fly.) But it’s a wonderful legend and can be linked to a Herb and Jamaal comic strip from this past week.

Paul Trap’s Thatababy (April 29) has the baby “blame entropy”, which fits as a mathematical concept, it seems to me. Entropy as a concept was developed in the mid-19th century as a thermodynamical concept, and it’s one of those rare mathematical constructs which becomes a superstar of pop culture. It’s become something of a fancy word for disorder or chaos or just plain messes, and the notion that the entropy of a system is ever-increasing is probably the only bit of statistical mechanics an average person can be expected to know. (And the situation is more complicated than that; for example, it’s just more probable that the entropy is increasing in time.)

Entropy is a great concept, though, as besides capturing very well an idea that’s almost universally present, it also turns out to be meaningful in surprising new places. The most powerful of those is in information theory, which is just what the label suggests; the field grew out of the problem of making messages understandable even though the telegraph or telephone lines or radio beams on which they were sent would garble the messages some, even if people sent or received the messages perfectly, which they would not. The most captivating (to my mind) new place is in black holes: the event horizon of a black hole has a surface area which is (proportional to) its entropy, and consideration of such things as the conservation of energy and the link between entropy and surface area allow one to understand something of the way black holes ought to interact with matter and with one another, without the mathematics involved being nearly as complicated as I might have imagined a priori.

Meanwhile, Lincoln Pierce’s Big Nate (April 30) mentions how Nate’s Earned Run Average has changed over the course of two innings. Baseball is maybe the archetypical record-keeping statistics-driven sport; Alan Schwarz’s The Numbers Game: Baseball’s Lifelong Fascination With Statistics notes that the keeping of some statistical records were required at least as far back as 1837 (in the Constitution of the Olympic Ball Club of Philadelphia). Earned runs — along with nearly every other baseball statistic the non-stathead has heard of other than batting averages — were developed as a concept by the baseball evangelist and reporter Henry Chadwick, who presented them from 1867 as an attempt to measure the effectiveness of batting and fielding. (The idea of the pitcher as an active player, as opposed to a convenient way to get the ball into play, was still developing.) But — and isn’t this typical? — he would come to oppose the earned run average as a measure of pitching performance, because things that were really outside the pitcher’s control, such as stolen bases, contributed to it.

It seems to me there must be some connection between the record-keeping of baseball and the development of statistics as a concept in the 19th century. Granted the 19th century was a century of statistics, starting with nation-states measuring their populations, their demographics, their economies, and projecting what this would imply for future needs; and then with science, as statistical mechanics found it possible to quite well understand the behavior of millions of particles despite it being impossible to perfectly understand four; and in business, as manufacturing and money were made less individual and more standard. There was plenty to drive the field without an amusing game, but, I can’t help thinking of sports as a gateway into the field.

Creator.com's _Donald Duck_ for 2 May 2014: Ludwig von Drake orders his computer to stop with the thinking.

The Disney Company’s Donald Duck (May 2, rerun) suggests that Ludwig von Drake is continuing to have problems with his computing machine. Indeed, he’s apparently having the same problem yet. I’d like to know when these strips originally ran, but the host site of creators.com doesn’t give any hint.

Stephen Bentley’s Herb and Jamaal (May 3) has the kid whose name I don’t really know fret how he spent “so much time” on an equation which would’ve been easy if he’d used “common sense” instead. But that’s not a rare phenomenon mathematically: it’s quite possible to set up an equation, or a process, or a something which does indeed inevitably get you to a correct answer but which demands a lot of time and effort to finish, when a stroke of insight or recasting of the problem would remove that effort, as in the von Neumann legend. The commenter Dartpaw86, on the Comics Curmudgeon site, brought up another excellent example, from Katie Tiedrich’s Awkward Zombie web comic. (I didn’t use the insight shown in the comic to solve it, but I’m happy to say, I did get it right without going to pages of calculations, whether or not you believe me.)

However, having insights is hard. You can learn many of the tricks people use for different problems, but, say, no amount of studying the Awkward Zombie puzzle about a square inscribed in a circle inscribed in a square inscribed in a circle inscribed in a square will help you in working out the area left behind when a cylindrical tube is drilled out of a sphere. Setting up an approach that will, given enough work, get you a correct solution is worth knowing how to do, especially if you can give the boring part of actually doing the calculations to a computer, which is indefatigable and, certain duck-based operating systems aside, pretty reliable. That doesn’t mean you don’t feel dumb for missing the recasting.

Rick Detorie's _One Big Happy_ for 3 May 2014: Joe names the whole numbers.

Rick DeTorie’s One Big Happy (May 3) puns a little on the meaning of whole numbers. It might sound a little silly to have a name for only a handful of numbers, but, there’s no reason not to if the group is interesting enough. It’s possible (although I’d be surprised if it were the case) that there are only 47 Mersenne primes (a number, such as 7 or 31, that is one less than a whole power of 2), and we have the concept of the “odd perfect number”, when there might well not be any such thing.

The Most Unlikely NHL Playoff Upsets of the Last Five Years


Nick Emptage, writing for puckprediction.com, has the sort of post which I can’t resist: it’s built on the application of statistics to sports. In this case it’s National Hockey League playoffs, and itself builds on an earlier post about the conditional probabilities of the home-team-advantaged winning a best-of-seven series, to look at the most unlikely playoff wins of the last several years. Since I’m from New Jersey I feel a little irrational pride at the New Jersey Devils being two of the most improbable winners, not least because I remember the Devils in the 1980s when the could lose as many as 200 games per eighty-game season, so seeing them in the playoffs at all is a wondrous thing.

Reading the Comics, March 26, 2014: Kitchen Science Department


It turns out that three of the comic strips to be included in this roundup of mathematics-themed strips mentioned things that could reasonably be found in kitchens, so that’s why I’ve added that as a subtitle. I can’t figure a way to contort the other entries to being things that might be in kitchens, but, given that I don’t get to decide what cartoonists write about I think I’m doing well to find any running themes.

Ralph Hagen’s The Barn (March 19) is built around a possibly accurate bit of trivia which tries to stagger the mind by considering the numinous: how many stars are there? This evokes, to me at least, one of the famous bits of ancient Greek calculations (for which they get much less attention than the geometers and logicians did), as Archimedes made an effort to estimate how many grains of sand could fit inside the universe. Archimedes had apparently little fear of enormous numbers, and had to strain the Greek system for representing numbers to get at such enormous quantities. But he was an ingenious reasoner: he was able to estimate, for example, the sizes and distances to the Moon and the Sun based on observing, with the naked eye, the half-moon; and his work on problems like finding the value of pi get surprisingly close to integral calculus and would probably be a better introduction to the subject than pre-calculus courses are. It’s quite easy in considering how big (and how old) the universe is to get to numbers that are really difficult to envision, so, trying to reduce that by imagining stars as grains of salt might help, if you can imagine a ball of salt eight miles across.

Continue reading “Reading the Comics, March 26, 2014: Kitchen Science Department”

January 2014’s Statistics


So how does the first month of 2014 compare to the last month of 2013, in terms of popularity? The raw numbers are looking up: I went from 176 unique visitors looking at 352 pages in December up to 283 unique visitors looking at 498 pages. If WordPress’s statistics are to be believed that’s my greatest number of page views since June of 2013, and the greatest number of visitors since February. This hurt the ratio of views per visitor a little, which dropped from 2.00 to 1.76, but we can’t have everything unless I write stuff that lots of people want to read and they figure they want to read a lot more based on that, which is just crazy talk. The most popular articles, though, were:

  1. Something Neat About Triangles, this delightful thing about forming an equilateral triangle starting from any old triangle.
  2. How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me).
  3. Factor Finding, linking over to IvaSallay’s quite interesting blog with a great recreational mathematics puzzle (or educational puzzle, depending on how you came into it) that drove me and a friend crazy with this week’s puzzles.
  4. What’s The Worst Way To Pack? in which I go looking for the least-efficient packing of spheres and show off these neat Mystery Science Theater 3000 foam balls I got.
  5. Reading The Comics, December 29, 2013, the old year’s last bunch of mathematics-themed comic strips.

The countries sending me readers the most often were the United States (281), Canada (52), the United Kingdom (25), and Austria (23). Sending me just a single reader each this past month were a pretty good list:
Bulgaria, France, Greece, Israel, Morocco, the Netherlands, Norway, Portugal, Romania, Russia, Serbia, Singapore, South Korea, Spain, and Viet Nam. Returning on that list from last month are Norway, Romania, Spain, and Viet Nam, and none of those were single-country viewers back in November 2013.

2013 in review


Oh, yes, and I shouldn’t have forgotten but WordPress did put together its summary of how my little effort here did for 2013.

Here’s an excerpt:

A New York City subway train holds 1,200 people. This blog was viewed about 5,700 times in 2013. If it were a NYC subway train, it would take about 5 trips to carry that many people.

Click here to see the complete report.

December 2013’s Statistics


There’s a hopeful trend in my readership statistics for December 2013 around these parts: according to WordPress, my number of readers grew from 308 in November to 352 and the number of unique visitors grew from 158 to 176. Even the number of views per visitor grew, from 1.95 to 2.00. None of these are records, but the fact of improvement is a good one.

I can’t figure exactly how to get the report on most popular articles for the exact month of December, and was too busy with other things to check the past-30-day report on New Year’s Eve, but at least the most popular articles for the 30 days ending today were:

The countries sending me the most readers were the United States, Canada, Denmark and Austria (tied, and hi again, Elke), and the United Kingdom. Sending me just one viewer each were a slew of nations: Bangladesh, Cambodia, India, Japan, Jordan, Malaysia, Norway, Romania, Slovenia, South Africa, Spain, Sweden, Turkey, and Viet Nam. On that list last month were Jordan and Slovenia, so I’m also marginally interesting to a different group of people this time around.

This has all caused me to realize that I failed to promote my string of articles inspired by Arthur Christmas and getting to the recurrence theorem and the existential dread of the universe’s end during the Christmas season. Maybe next year, then.

Reading the Comics, December 3, 2013


It’s been long enough for a couple more mathematics-themed comics to gather, so, let me share them with you. The comics easily available to me may be increasing, too, as dailyink.com has indicated they’re looking to make it easier for people who aren’t subscribers to their service to look at the daily strips. I’d be glad to include them back in my roundup of mathematics strips, at least when I see them making mathematics jokes; there’ve been surprisingly few of them lately. Maybe the King Features Syndicate artists know it’s generally too much effort for me to feature them for a joke about how silly word problems are and have been saving us both the trouble.

Frank Page’s Bob the Squirrel began a sequence November 20 with the kid Lauren doing her math homework and Bob the Squirrel, one of multiple imaginary squirrels which I follow on Twitter, helping. It starts with percentages, a concept I admit that other people find harder than I ever did, probably because the “per cent” just made it clear to me at a young age what the whole thing was about. On the 21st Bob claims to have known a squirrel named Algebra, which wouldn’t be the strangest name for a squirrel. “Algebra”, the word, isn’t drawn from anyone’s name; it’s instead drawn from the title of the book Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala (“The Compendious Book On Calculation By Completion and Balancing”), written by Muḥammad ibn Mūsā al-Khwārizmī, whose name did give us the word “algorithm”, which is the kind of successful word-generating power that you usually expect only from obscure Swedish towns. Bob closes things off with your standard breaking-the-word-problem sort of joke.

Continue reading “Reading the Comics, December 3, 2013”

October 2013’s Statistics


It’s been a month since I last looked over precisely how not-staggeringly-popular I am, so it’s time again.
For October 2013 I had 440 views, down from September’s 2013. These came from 220 distinct viewers, down again from the 237 that September gave me. This does mean there was a slender improvement in views per visitor, from 1.97 up to 2.00. Neither of these are records, although given that I had a poor updating record again this month that’s all tolerable.

The most popular articles from the past month are … well, mostly the comics, and the trapezoids come back again. I’ve clearly got to start categorizing the other kinds of polygons. Or else plunge directly into dynamical systems as that’s the other thing people liked. October 2013’s top hits were:

  1. Reading the Comics, October 8, 2013
  2. How Many Trapezoids I Can Draw
  3. Reading the Comics, September 11, 2013
  4. From ElKement: On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace
  5. Reading the Comics, September 21, 2013
  6. The Mathematics of a Pricing Game

The country sending me the most readers again was the United States (226 of them), with the United Kingdom coming up second (37). Austria popped into third for, I think, the first time (25 views), followed by Denmark (21) and at long last Canada (18). I hope they still like me in Canada.

Sending just the lone reader each were a bunch of countries: Bermuda, Chile, Colombia, Costa Rica, Finland, Guatemala, Hong Kong, Laos, Lebanon, Malta, Mexico, the Netherlands, Oman, Romania, Saudi Arabia, Slovenia, Sweden, Turkey, and Ukraine. Finland and the Netherlands are repeats from last month, and the Netherlands is going on at least three months like this.

Why I Say 1/e About This Roller Coaster


The Leap-The-Dips at Lakemont Park, Altoona, Pennsylvania, as photographed by Joseph Nebus in July 2013 from the edge of the launch platform.

So in my head I worked out an estimate of about one in three that any particular board would have remained from the Leap-The-Dips’ original, 1902, configuration, even though I didn’t really believe it. Here’s how I got that figure.

First, you have to take a guess as to how likely it is that any board is going to be replaced in any particular stretch of time. Guessing that one percent of boards need replacing per year sounded plausible, what with how neatly a chance of one-in-a-hundred fits with our base ten numbering system, and how it’s been about a hundred years in operation. So any particular board would have about a 99 percent chance of making it through any particular year. If we suppose that the chance of a board making it through the year is independent — it doesn’t change with the board’s age, or the condition of neighboring boards, or anything but the fact that a year has passed — then the chance of any particular board lasting a hundred years is going to be 0.99^{100} . That takes a little thought to work out if you haven’t got a calculator on hand.

Continue reading “Why I Say 1/e About This Roller Coaster”

My July 2013 Statistics


As I’ve started keeping track of my blog statistics here where it’s all public information, let me continue.

WordPress says that in July 2013 I had 341 pages read, which is down rather catastrophically from the June score of 713. The number of distinct visitors also dropped, though less alarmingly, from 246 down to 156; this also implies the number of pages each visitor viewed dropped from 2.90 down to 2.19. That’s still the second-highest number of pages-per-visitor that I’ve had recorded since WordPress started sharing that information with me, so, I’m going to suppose that the combination of school letting out (so fewer people are looking for help about trapezoids) and my relatively fewer posts this month hit me. There are presently 215 people following the blog, if my Twitter followers are counted among them. They hear about new posts, anyway.

My most popular posts over the past 30 days have been:

  1. John Dee, the ‘Mathematicall Praeface’ and the English School of Mathematics, which is primarily a pointer to the excellent mathematics history blog The Renaissance Mathematicus, and about the really quite fascinating Doctor John Dee, advisor to England’s Queen Elizabeth I.
  2. Counting From 52 To 11,108, some further work from Professor Inder J Taneja on a lovely bit of recreational mathematics. (Professor Taneja even pops in for the comments.)
  3. Geometry The Old-Fashioned Way, pointing to a fun little web page in which you can work out geometric constructions using straightedge and compass live and direct on the web.
  4. Reading the Comics, July 5, 2013, and finally; I was wondering if people actually still liked these posts.
  5. On Exact And Inexact Differentials, another “reblog” style pointer, this time to Carnot Cycle, a thermodynamics-oriented blog.
  6. And The $64 Question Was, in which I learned something about a classic game show and started to think about how it might be used educationally.

My all-time most popular post remains How Many Trapezoids I Can Draw, because I think there are people out there who worry about how many different kinds of trapezoids there are. I hope I can bring a little peace to their minds. (I make the answer out at six.)

The countries sending me the most viewers the past month have been the United States (165), then Denmark (32), Australia (24), India (18), and the United Kingdom and Brazil (12 each). Sorry, Canada (11). Sending me a single viewer each were Estonia, Slovenia, South Africa, the Netherlands, Argentina, Pakistan, Angola, France, and Switzerland. Argentina and Slovenia did the same for me last month too.

Distribution of the batting order slot that ends a baseball game


The God Plays Dice blog has a nice piece attempting to model a baseball question. Baseball is wonderful for all kinds of mathematics questions, partly because the game has since its creation kept data about the plays made, partly because the game breaks its action neatly into discrete units with well-defined outcomes.

Here, Dr Michael Lugo ponders whether games are more likely to end at any particular spot in the batting order. Lugo points out that certainly we could just count where games actually end, since baseball records are enough to make an estimate from that route possible. But that’s tedious, and it’s easier to work out a simple model and see what that suggests. Lupo also uses the number of perfect games as a test of whether the model is remotely plausible, and a test like this — a simple check to whether the scheme could possibly tell us something meaningful — is worth doing whenever one builds a model of something interesting.

God plays dice

Tom Tango, while writing about lineup construction in baseball, pointed out that batters batting closer to the top of the batting order have a greater chance of setting records that are based on counting something – for example, Chris Davis’ chase for 62 home runs. (It’s interesting that enough people see Roger Maris’ 61 as the “real” record that 62 is a big deal.) He observes that over a 162-game season, each slot further down in the batting order (of 9) means 18 fewer plate appearances.

Implicitly this means that every slot in the batting order is equally likely to end the game — that is, that the number of plate appearances for a team in a game, mod 9, is uniformly distributed over {0, 1, …, 8}.

Can we check this? There are two ways to check it:

  • 1. find the number of plate appearances in every game…

View original post 652 more words

My June 2013 Statistics


I don’t understand why, but an awful lot of the advice I see about blogging says that it’s important not just to keep track of how your blog is doing, but also to share it, so that … numbers will like you more? I don’t know. But I can give it a try, anyway.

For June 2013, according to WordPress, I had some 713 page views, out of 246 unique visitors. That’s the second-highest number of page views I’ve had in any month this year (January had 831 views), and the third-highest I’ve had for all time (there were 790 in March 2012). The number of unique visitors isn’t so impressive; since WordPress started giving me that information in December 2012, I’ve had more unique visitors … actually, in every month but May 2013. On the other hand, the pages-per-viewer count of 2.90 is the best I’ve had; the implication seems to be that I’m engaging my audience.

The most popular posts for the past month were Counting From 52 to 11,108, which I believe reflects it getting picked for a class assignment somehow; A Cedar Point Follow-Up, which hasn’t got much mathematics in it but has got pretty pictures of an amusement park, and Solving The Price Is Right’s “Any Number” Game, which has got some original mathematics but also a pretty picture.

My all-time most popular posts are from the series about Trapezoids — working out how to find their area, and how many kinds of trapezoids there are — with such catchy titles as How Many Trapezoids I Can Draw, or How Do You Make A Trapezoid Right?, or Setting Out To Trap A Zoid, which should be recognized as a Dave Barry reference.

My most frequent commenters, “recent”, whatever that means, are Chiaroscuro and BunnyHugger (virtually tied), with fluffy, elkelement, MJ Howard, and Geoffrey Brent rounding out the top six.

The most common source of page clicks the past month was from the United States (468), with Brazil (51) and Canada (23) taking silver and bronze. And WordPress recorded one click each from Portugal, Serbia, Hungary, Macedonia (the Former Yugoslav Republic), Indonesia, Argentina, Poland, Slovenia, and Viet Nam. I’ve been to just one of those countries.

Solving The Price Is Right’s “Any Number” Game


A friend who’s also into The Price Is Right claimed to have noticed something peculiar about the “Any Number” game. Let me give context before the peculiarity.

This pricing game is the show’s oldest — it was actually the first one played when the current series began in 1972, and also the first pricing game won — and it’s got a wonderful simplicity: four digits from the price of a car (the first digit, nearly invariably a 1 or a 2, is given to the contestant and not part of the game), three digits from the price of a decent but mid-range prize, and three digits from a “piggy bank” worth up to $9.87 are concealed. The contestant guesses digits from zero through nine inclusive, and they’re revealed in the three prices. The contestant wins whichever prize has its price fully revealed first. This is a steadily popular game, and one of the rare Price games which guarantees the contestant wins something.

A couple things probably stand out. The first is that if you’re very lucky (or unlucky) you can win with as few as three digits called, although it might be the piggy bank for a measly twelve cents. (Past producers have said they’d never let the piggy bank hold less than $1.02, which still qualifies as “technically something”.) The other is that no matter how bad you are, you can’t take more than eight digits to win something, though it might still be the piggy bank.

What my friend claimed to notice was that these “Any Number” games went on to the last possible digit “all the time”, and he wanted to know, why?

My first reaction was: “all” the time? Well, at least it happened an awful lot of the time. But I couldn’t think of a particular reason that they should so often take the full eight digits needed, or whether they actually did; it’s extremely easy to fool yourself about how often events happen when there’s a complicated possibile set of events. But stipulating that eight digits were often needed, then, why should they be needed? (For that matter, trusting the game not to be rigged — and United States televised game shows are by legend extremely sensitive to charges of rigging — how could they be needed?) Could I explain why this happened? And he asked again, enough times that I got curious myself.

Continue reading “Solving The Price Is Right’s “Any Number” Game”

Reading the Comics, March 12, 2013


I’ve got my seven further comic strips with mentions of mathematical topics, so I can preface that a bit with my surprise that at least some of the Gocomics.com comics didn’t bother to mention Pi Day, March 14. It might still be a slightly too much of a This Is Something People Do On The Web observance to be quite sensible for the newspaper comic strips. But there are quite a few strips on Gocomics.com that only appear online, and I thought one of them might.

(I admit I’m a bit of a Pi Day grouch, on the flimsy grounds that 3/14 is roughly 0.214, which is a rotten approximation to π. But American-style date-writing never gets very good at approximating π. The day-month format used in most of the world offers 22/7 as a less strained Pi Day candidate, except that there’s few schools in session then, wiping out whatever use the day has as a playfully educational event.)

Gene Weingarten, Dan Weingarten and David Clark’s Barney and Clyde (March 4) introduces a character which I believe is new to the strip, “Norman the math fanatic”. (He hasn’t returned since, as of this writing.) The setup is about the hypothetical and honestly somewhat silly argument about learning math being more important than learning English. I’m not sure I could rate either mathematics or English (or, at least, the understanding of one’s own language) as more important. The panel ends with the traditional scrawl of symbols as shorthand for “this is complicated mathematics stuff”, although it’s not so many symbols and it doesn’t look like much of a problem to me. Perhaps Norman is fanatic about math but doesn’t actually do it very well, which is not something he should be embarrassed about.

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Reblog: Parrondo’s Paradox


Ad Nihil here presents an interesting-looking game demonstrating something I hadn’t heard of before, Parrondo’s Paradox, which apparently is a phenomenon in which a combination of losing strategies becomes a winning strategy. I do want to think about this more, so I offer the link to that blog’s report so that I hopefully will go back and consider it more when I’m able.

ad nihil

My inspiration with my daughter’s 8th grade probability problems continues. In a previous post I worked on a hypothetical story of monitoring all communications for security with a Bayesian analysis approach. This time when I saw the spinning wheel problems in her text book, I was yet again inspired to create a game system to demonstrate Parrondo’s Paradox.

“Parrondo’s paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996.” – Wikipedia.org

Simply put, with certain (not all) combinations, you may create an overall winning strategy by playing different losing scenarios alternatively in the long run. Here’s the game system I came up with this (simpler than the original I believe):

Let’s imagine a spinning wheel like below, divided into eight equal parts with 6 parts red…

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Can Rex Morgan Be Made Plausible?


'And ... My batting average of predicting the [unborn child's] sex is 97% accurate!'

The comic strip Rex Morgan, MD, put up an interesting bit of nonsense in its current ridiculous story. (Rex and June are investigating a condo where nobody’s been paying rent; the residents haven’t because everyone living there is strippers who’re raising money for a cancer-stricken compatriot; the details are dopier, and much more slowly told, than this makes it sound.) But on the 7th this month it put up one of those things that caught me. Never mind the claim that Delores here (the cancer-stricken woman) puts up about being able to sense pregnancy. She claims she can predict the sex of the unborn child with 97 percent accuracy.

Is that plausible? Well, she may be just making the number up, since putting a decimal or a percentage into a number carries connotations of “only a fool would dare question me” similar to those of holding a clipboard and glaring at it while walking purposefully around. If she’s doing ordinary human-style rounding off, that could mean that she’s guessed five of six pregnancies correctly. I could believe a person thinking that makes her 97 percent accurate, but I wouldn’t be convinced by the claim and I doubt you would either.

So here’s a little recreational puzzle for you: how many pregnancies would Delores have to have predicted, and how many called accurately, for the claim of 97 percent accuracy to be hard to dismiss? How many until it isn’t clearly just luck or a small sample size?

Reading the Comics, January 29, 2013


I’ve got enough mathematics-themed comic strips for a fresh installment of my comic tracking. I also want to mention the January 29th Jumble has a math teacher joke in it, but I don’t know a reasonably archivable way to point to that. Jumble.com, which I think is the official web site, is suffering some kind of database glitch so there could be anything there. Also, from working it out, “rimpet” may not be a word but it does look like it ought to be.

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2012 in Review


I should maybe close out the Christmas/New Year season with the report of statistics which WordPress prepared about my little blog here. Of course they keep statistics; one of the big changes in human thought in the 20th century was that pretty near everything could not only be measured but that they could be measured statistically: what are the mean, the mode, the variances, how do things correlate, what can be done to maximize the desired and minimize the unwanted?

I don’t do quite that much tracking myself, as it’s a little too much work when all I’m doing is pointing out how Cow and Boy mentioned frustum volume formulas or something, but I do like watching the counter flicker as people find that, mostly, they want to see me talking about the area of a trapezoid. That’s by far the most popular thing I wrote in 2012, and all based on my fumbling the middle of a class. Had I not attempted to improvise in class, I would be less popular on the Internet. There’s a lesson here for our times and I don’t know what it is.

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What Is The Most Common Jeopardy! Response?


Happy New Year!

I want to bring a pretty remarkable project to people’s attention. Dan Slimmon here has taken the archive of Jeopardy! responses (you know, the answers, only the ones given in the form of a question) from the whole Jeopardy! fan archive, http://www.j-archive.com, and analyzed them. He was interested not just in the most common response — which turns out to be “What is Australia?” — but in the expectation value of the responses.

Expectation value I’ve talked about before, and for that matter, everyone mentioning probability or statistics has. Slimmon works out approximately what the expectation value would be for each clue. That is, imagine this: if you ignored the answer on the board entirely and just guessed to every answer either responded absolutely nothing or else responded “What is Australia?”, some of the time you’d be right, and you’d get whatever that clue was worth. How much would you expect to get if you just guessed that answer? Responses that turn up often, such as “Australia”, or that turn up more often in higher-value squares, are worth more. Responses that turn up rarely, or only in low-value squares, have a lower expectation value.

Simmons goes on to list, based on his data, what the 1000 most frequent Jeopardy! responses are, and what the 1000 responses with the highest expectation value are. I’m so delighted to discover this work that I want to bring folks’ attention to it. (I do have a reservation about his calculations, but I need some time to convince myself that I understand exactly his calculation, and my reservation, before I bother anyone with it.)

The comments at his page include a discussion of a technical point about the expectation value calculation which has an interesting point about the approximations often useful, or inevitable, in this kind of work, but that’ll take a separate essay to quite explain that I haven’t the time for just today.

[ Edit: I initially misunderstood Slimmon’s method and have amended the article to reflect the calculation’s details. Specifically I misunderstood him at first to have calculated the expectation value of giving a particular response, and either having it be right or wrong. Slimmon assumed that one would either give the response or not at all; getting the answer wrong costs the contestant money and so has a negative value, while not answering has no value. ]

Reading the Comics, October 13, 2012


I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.

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Reading the Comics, September 26, 2012


I haven’t time to write a short piece today so let me go through a fresh batch of math-themed comic strips instead. There might be a change coming to these features soon, both in the strips I read and in how I present them, since Comics Kingdom, which provides the King Features Syndicate comic strips, has shown signs that they’re tightening up web access to their strips.

I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.

For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though:

Continue reading “Reading the Comics, September 26, 2012”

Create a Graph: Something To Play With


I realize that I’m probably about eight years behind the curve on this one, but I only just learned that the National Center for Education Statistics has this cute little create-a-graph application (Flash required), good for whipping up several of the name-brand graphs out there.

The pie chart is particularly nice since I’ve never figured out how to make pie charts in Octave (it’s probably some tricky command like ‘pieplot’), and they have a neat historical resonance because they were turned from one of the many obscure tools of the then-obscure field of statistics into the first rank of mathematical tools by Florence Nightingale. She used them as a way to communicate, efficiently, what the causes of death in the Crimean War were.

This gives me use a fine narrative hook in class: I can ask them if they know who first made pie charts famous, and then insist they do indeed, they just never had any idea about it.

Finding, and Starting to Understand, the Answer


If the probability of having one or fewer clean sweep episodes of The Price Is Right out of 6,000 aired shows is a little over one and a half percent — and it is — and we consider outcomes whose probability is less than five percent to be so unlikely that we can rule them out as happening by chance — and, last time, we did — then there are improbably few episodes where all six contestants came from the same seat in Contestants Row, and we can usefully start looking for possible explanations as to why there are so few clean sweeps. At least, that’s the conclusion at our significance level, that five percent.

But there’s no law dictating that we pick that five percent significance level. If we picked a one percent significance level, which is still common enough and not too stringent, then we would say this might be fewer clean sweeps than we expected, but it isn’t so drastically few as to raise our eyebrows yet. And we would be correct to do so. Depending on the significance level, what we saw is either so few clean sweeps as to be suspicious, or it’s not. This is why it’s better form to choose the significance level before we know the outcome; it feels like drawing the bullseye after shooting the arrow the other way around.

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Illicitly Counted Coins


The past month I’ve had the joy of teaching a real, proper class again, after a hiatus of a few years. The hiatus has given me the chance to notice some things that I would do because that was the way I had done them, and made it easier to spot things that I could do differently.

To get a collection of data about which we could calculate statistics, I had everyone in the class flip a coin twenty times. Besides giving everyone something to do besides figure out which of my strange mutterings should be written down in case they turn out to be on the test, the result would give me a bunch of numbers, centered around ten, once they reported the number of heads which turned up. Counting the number of heads out of a set of coin flips is one of the traditional exercises to generate probability-and-statistics numbers.

Good examples are some of the most precious and needed things for teaching mathematics. It’s never enough to learn a formula; one needs to learn how to look at a problem, think of what one wants to know as a result of its posing, identify what one needs to get those results, and pick out which bits of information in the problem and which formulas allow the result to be found. It’s all the better if an example resembles something normal people would find to raise a plausible question. Here, we may not be all that interested in how many times a coin comes up heads or tails, but we can imagine being interested in how often something happens given a number of chances for it to happen, and how much that count of happenings can vary if we watch several different runs.

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Did King George III pay too little for astronomers or too much for tea?


In the opening pages of his 1998 biography George III: A Personal History, Christopher Hibbert tosses a remarkable statement into a footnote just after describing the allowance of Frederick, Prince of Wales, at George III’s birth:

Because of the fluctuating rate of inflation and other reasons it is not really practicable to translate eighteen-century sums into present-day equivalents. Multiplying the figures in this book by about sixty should give a very rough guide for the years before 1793. For the years of war between 1793 and 1815 the reader should multiply by about thirty, and thereafter by about forty.

“Not really practical” is wonderful understatement: it’s barely possible to compare the prices of things today to those of a half-century ago, and the modern economy at least existed in cartoon back then. I could conceivably have been paid for programming computers back then, but it would be harder for me to get into the field. To go back 250 years — before electricity, mass markets, public education, mass production, general incorporation laws, and nearly every form of transportation not muscle or wind-powered — and try to compare prices is nonsense. We may as well ask how many haikus it would take to tell Homer’s Odyssey, or how many limericks Ovid’s Metamorphoses would be.
Continue reading “Did King George III pay too little for astronomers or too much for tea?”