Reading the Comics, April 24, 2016: Mental Mathematics and Calendars Edition


Warning! I do some showing off in this installment of the Reading the Comics series. Please forgive me. I was feeling a little giddy.

Scott Hilburn’s The Argyle Sweater I had just mentioned to a friend never seems to show up in these columns anymore. And Hilburn would so reliably do strips about anthropomorphized numerals. He returns on the 20th, after a hiatus of some length I haven’t actually checked here, with a name-drop of Einstein instead. I grinned, although a good part of what amused me was the look of the guy in the lower right of the panel. Funny pictures carry a comic strip far. Formulating the theory of relativity is a tricky request. The special theory … well, to do it properly takes some sophisticated work. But it doesn’t take much beyond the Pythagorean Theorem to realize that “how long” a thing is, or a time span is, is different for different observers. That’s the most important insight, I would say, and that is easily available. General relativity, which looks at accelerations and gravity, that’s another thing. I’d be interested in a popular treatment that explained enough mathematics people could make usable estimates but that could still make sense to a lay audience. Probably it’s not possible to do this. Too bad.

Mark Tatulli’s Heart of the City just uses arithmetic because it’s a nice compact problem to give a student. It did strike me that 117 times 45 is something one could amaze people with by doing in one’s head, though. Here’s why. 117 times 100 would be easy. Multiplying by hundreds always is. 117 times 50 would be not almost as easy: that’s multiplying by 100 and dividing by two. 117 times 45 … well, that’s 117 times 50 minus 117 times 5. And if you know 117 times 50, then you know 117 times 5: it’s one-tenth that. And one-tenth of a thing is easy to find.

Therefore: 117 times 100 is 11,700. Divide that by two and that’s kind of an ugly-looking number, isn’t it? But all’s not lost. Let me use another bit of falsework: 11,700 is 12,000 minus 300. Half that is 6,000 minus 150. Therefore, half of 11,700 is 5,850. So 117 times 50 is 5,850. One-tenth of that is 585. Therefore, 117 times 45 is 5,850 minus 585. And that will be … 5,275. Ta-da!

Well, no, it isn’t. It’s 5,265. I messed up the carrying. I still think that’s doing well for multiplying ugly numbers like that without writing it down. It just won’t impress people who want the actual you know correct answer.

Mark Anderson’s Andertoons wouldn’t let me down by vanishing for a while. The 21st is not explicitly a strip about extrapolating graphs. I’ll take it as such, though. Once again the art amuses me. I like the crash-up of charted bars. Yes, I saw the Schrödinger’s Cat thing two days later.

Jef Mallett’s Frazz for the 23rd I drag into a mathematics blog because of the long historical links between calendars and mathematics. But Caulfield does talk about something that’s baffled everyone. There’s seven days to the week. There’s seven classically known heavenly bodies in the solar system, besides the Earth. Naming a day for each seems obvious now that we’ve committed to it. But why aren’t the bodies honored in order?

Geocentrism seems like, at first, a plausible reason. The ancients wouldn’t order the sky Sun-Mercury-Venus-Moon-Mars-Jupiter-Saturn. But that doesn’t help. Geocentric models of the solar system (always, so far as I’m aware) put the Moon closest, then Mercury, then Venus, the Sun, Mars, Jupiter, and Saturn.

The answer that, at least, gets repeated in histories of the calendar (for example, here, David Ewing Duncan’s The Calendar: The 5000-Year Struggle To Align The Clock And The Heavens — And What Happened To The Missing Ten Days, which was the first book I had on hand) amounts to a modular arithmetic thing. The Babylonians, if Duncan is right, named a planet-god for each hour of the day. (We treat the Moon and Sun as planets for this discussion.) The planet-gods took their hourly turn in order. If the first hour of the day is Saturn’s to rule, the next is Jupiter’s, then Mars’s, the Sun’s, Venus’s, Mercury’s, and the Moon’s. Then back to Saturn and the system keeps going like that.

So if the first hour of the day is Saturn’s, then who has the first hour of the next day? … the Sun does. If the Sun has the first hour of the day, then who has the first hour of the day after that? … the Moon. And from here you know the pattern. At least you do if you understand that English derives most of its day names from the Norse gods, matched as best they can with those of the Roman State Religion. So, Tiw matches with Mars; Woden with Mercury; Thor with Jupiter; Freya with Venus. The apparently scrambled order of days, relative to the positions of the planets, amounts to what you get if you keep adding 24 to a number by modulo 7 arithmetic.

That is, at least, the generally agreed-upon explanation. I am not aware of what actual researchers of Babylonian culture believe. Duncan, I must admit, takes a hit in his credibility by saying on the page after this that “recently chronobiologists have discovered that the seven-day cycle … may also have biological precedents”. I’m sorry but I just don’t believe him, or whoever he got that from.

Kevin Fagan’s Drabble for the 24th amuses me by illustrating the common phenomenon. We have all taken out the calculator (or computer) to do some calculation that really doesn’t need it. I understand and am sympathetic. It’s so obviously useful to let the calculator work out 117 times 45 and get it right instantly. It’s easy to forget sometimes it’s faster to not bother with the calculator. We are all of us a little ridiculous.

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


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

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

Continue reading “Reading the Comics, March 4, 2015: Driving Me Crazy Edition”

Reading the Comics, February 28, 2015: Calendar Reform Edition


It’s the last day of the shortest month of the year, a day that always makes me think about whether the calendar could be different. I was bit by the calendar-reform bug as a child and I’ve mostly recovered from the infection, but some things can make it flare up again and I’ve never stopped being fascinated by the problem of keeping track of days, which you’d think would not be so difficult.

That’s why I’m leading this review of comics with Jef Mallet’s Frazz (February 27) even if it’s not transparently a mathematics topic. The biggest problem with calendar reform is there really aren’t fully satisfactory ways to do it. If you want every month to be as equal as possible, yeah, 13 months of 28 days each, plus one day (in leap years, two days) that doesn’t belong to any month or week is probably the least obnoxious, if you don’t mind 13 months to the year meaning there’s no good way to make a year-at-a-glance calendar tolerably symmetric. If you don’t want the unlucky, prime number of 13 months, you can go with four blocks of months with 31-30-30 days and toss in a leap day that’s again, not in any month or week. But people don’t seem perfectly comfortable with days that belong to no month — suggest it to folks, see how they get weirded out — and a month that doesn’t belong to any week is right out. Ask them. Changing the default map projection in schools is an easier task to complete.

There are several problems with the calendar, starting with the year being more nearly 365 days than a nice, round, supremely divisible 360. Also a factor is that the calendar tries to hack together the moon-based months with the sun-based year, and those don’t fit together on any cycle that’s convenient to human use. Add to that the need for Easter to be close to the vernal equinox without being right at Passover and you have a muddle of requirements, and the best we can hope for is that the system doesn’t get too bad.

Continue reading “Reading the Comics, February 28, 2015: Calendar Reform Edition”

How My Mathematics Blog Was Read, For January 2015


And after reaching 20,000 views on the final day of December, 2014, could I reach 21,000 views by the end of January? Probably I could have, but in point of fact I did not. I am not complaining, though: I finished the month with 20,956 page views all told, after a record 944 pages got viewed by somebody, somewhere, for some reason. This is a record high for me, going well past the 831 that had been the January 2013 and December 2014 (tied) record. And likely I’ll reach 21,000 in the next couple days anyway.

According to WordPress, this was read by 438 distinct visitors, reading 2.16 views per visitor on average. That isn’t quite a record: January 2013 remains my high count for visitors, at 473, but it’s still, all told, some pretty nice numbers especially considering I don’t think I had my best month of blog-writing. I can’t wait to get some interesting new topics in here for February and see that they interest absolutely nobody.

The new WordPress statistics page is still awful, don’t get me wrong, but it has been getting a little bit better, and it does offer some new data I couldn’t gather easily before. Among them: that in January 205 I received 197 likes overall — a high for the past twelvemonth, which is as far as I can figure out how to get it, and up from 128 in December — and 51 comments, up from December 29, and also a high for the twelvemonth.

The three countries sending me viewers were, once again, the big three of the United States (594), Canada (56), and the United Kingdom (52), with Austria sending in 32 viewers, and Germany and Argentina ending 22 each. And India, for a wonder sent me a noticeable-to-me 18 readers, although on a per capita basis that still isn’t very many, I admit.

There was a bumper crop of single-reader countries, though, up from last month’s six: Belgium, Estonia, Finland, Greece, Hungary, Indonesia, Iraq, Japan, Kuwait, Libya, Mexico, Paraguay, Slovakia, and the United Arab Emirates each found only one person viewing anything around here. Greece and Mexico are repeats from December.

This month’s most popular articles were mostly comic strip posts, although they were a pretty popular set; none of these had fewer than 35 views per, which feels high to me. The top posts of the last 30 days, then, were:

  1. Reading the Comics, January 6, 2015: First of the Year Edition, in which I included drawing a sloppy `2′ as a snoring `Z’ as somehow connected to mathematics.
  2. Reading the Comics, January 24, 2015: Many, But Not Complicated Edition, which includes an explanation for why margins of errors on surveys are always like three or four percent.
  3. Reading the Comics, January 11, 2015: Standard Genres And Bloom County Edition, in which I reveal my best guess for Jon Bon Jovi’s shorts size in the late 80s.
  4. 20,000: My Math Blog’s Statistics, because my narcissism is apparently quite popular?
  5. Reading the Comics, January 17, 2015: Finding Your Place Edition, where, again, I can flog that thing about a watch as a compass.
  6. How Many Trapezoids I Can Draw, which also reveals how many trapeziums I think are different in interesting ways.
  7. A bit more about Thomas Hobbes, and his attempt to redefine the very nature of mathematics, which didn’t succeed in quite the way he wanted.

Among the interesting search terms that brought people here the past month have been ([sic] on all of them):

  • science fiction and trapazoids (Somebody should totally write the definitive SFnal treatment of trapezoids, I agree.)
  • food. stotagre nebus (I feel strangely threatened by this.)
  • a group of student offer at least one of mathematics,physics, and statistcs , 14 of them offer mathematics, 12 offer physics,and 16 offer statistics.7 offer statistics and maths 6 offer maths and physics, 4 offer physics and statistics only, while 5 offer all the three subject (Help?)
  • hiw to draw diffrent trameziums
  • soglow otto radio (Pretty sure I used to listen to that back on WRSU in my undergrad days.)
  • if a calendar has two consecutive months with friday the 13th which would they be (February and March, in a non-bissextile — that is, non-leap — year)
  • how to measure a christmas tree made of triangles and trapeziums (I would use a tape measure, myself)

So if I would summarize January 2015 in my readership here, I would say: tramezium?

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, May 26, 2014: Definitions Edition


The most recent bunch of mathematics-themed comics left me feeling stumped for a theme. There’s no reason they have to have one, of course; cartoonists, as far as I know, don’t actually take orders from Comic Strip Master Command regarding what to write about, but often they seem to. Some of them seem to touch on definitions, at least, including of such ideas as the value of a quantity and how long it is between two events. I’ll take that.

Jef Mallet’s Frazz (May 23) does the kid-resisting-the-question sort of joke (not a word problem, for a change of pace), although I admit I didn’t care for the joke. I needed too long to figure out how the meaning of “value” for a variable might be ambiguous. Caulfield kind of has a point about mathematics needing to use precise words, but the process of making a word precise is a great and neglected part of mathematical history. Consider, for example: contemporary (English-language, at least) mathematicians define a prime number to be a counting number (1, 2, 3, et cetera) with exactly two factors. Why exactly two factors, except to rule out 1 as a prime number? But then why rule that 1 can’t be a prime number? As an idea gets used and explored we get a better idea of what’s interesting about it, and what it’s useful for, and can start seeing whether some things should be ruled out as not fitting a concept we want to describe, or be accepted as fitting because the concept is too useful otherwise and there’s no clear way to divide what we want from what we don’t.

I still can’t buy Caulfield’s proposition there, though.

Steve Boreman’s Little Dog Lost (May 25) circles around a bunch of mathematical concepts without quite landing on any of them. The obvious thing is the counting ability of animals: the crow asserts that crows can only count as high as nine, for example, and the animals try to work out ways to deal with the very large number of 2,615. The vulture asserts he’s been waiting for 2,615 days for the Little Dog to cross the road, and wonders how many years that’s been. The first installment of the strip, from the 26th of March, 2007, did indeed feature Vulture waiting for Little Dog to cross the road, although as I make it out there’s 2,617 days between those events.

At a guess, either Boreman was not counting the first and the last days of the interval between March 26, 2007, and May 25, 2014, or maybe he forgot the leap days. Finding how long it is between dates is a couple of kinds of messes, first because it isn’t necessarily clear whether to include the end dates, and second because the Gregorian calendar is a mess of months of varying lengths plus the fun of leap years, which include an exception for century years and an exception to the exception, making it all the harder. My preferred route for finding intervals is to not even try working the time out by myself, and instead converting every date to the Julian date, a simple serial count of the number of dates since noon Universal Time on the 1st of January, 4713 BC, on the Julian calendar. Let the Navy deal with leap days. I have better things to worry about.

Samson’s Dark Side Of The Horse (May 26) sees Horace trying to count sheep to get himself to sleep; different ways of denoting numbers confound him. I’m not sure if it’s known why counting sheep, or any task like that, is useful in getting to sleep. My guess would be that it just falls into the sort of activity that can be done without a natural endpoint and without demanding too much attention to keep one awake, while demanding enough attention that one isn’t thinking about the bank account or the noise inside the walls or the way the car lurches two lanes to the right every time one taps the brake at highway speeds. That’s a guess, though.

Tom Horacek’s Foolish Mortals (May 26) uses the “on a scale of one to ten” standard for something that’s not usually described so vaguely, and I like the way it teases the idea of how to measure things. The “scale of one to ten” is logically flawed, since we have no idea what the units are, how little of something one represents or how much the ten does, or even whether it’s a linear scale — the difference between “two” and “three” is the same as that between “three” and “four”, the way lengths and weight work — or a logarithmic one — the ratio between “two” and “three” equals that between “three” and “four”, the way stellar magnitudes, decibel sound readings, and Richter scale earthquake intensity measure work — or, for that matter, what normal ought to be. And yet there’s something useful in making the assessment, surely because the first step towards usefully quantifying a thing is to make a clumsy and imprecise quantification of it.

Dave Blazek’s Loose Parts (May 26) kind of piles together a couple references so a character can identify himself as a double major in mathematics and theology. Of course, the generic biography for a European mathematician, between about the end of the Western Roman Empire and the Industrial Revolution, is that he (males most often had the chance to do original mathematics) studied mathematics alongside theology and philosophy, and possibly astronomy, although that reflects more how the subjects were seen as rather intertwined, and education wasn’t as specialized and differentiated as it’s now become. (The other generic mathematician would be the shopkeeper or the exchequer, but nobody tells jokes about their mathematics.)

And, finally, Doug Savage’s Savage Chickens (May 28) brings up the famous typing monkeys (here just the one of them), and what really has to be counted as a bit of success for the project.

Reading The Comics, March 17, 2014: After The Ides Edition


Rather than wait to read today’s comics I’m just going to put in a fresh entry going over mathematical points raised in the funny pages. This one turned out to include a massive diversion into the wonders of the ancient Roman calendar, which is a mathematical topic, really, although there’s no calculations involved in it just here.

Bill Hinds’s Cleats (March 7, rerun) calls on one of the common cultural references to percentages, the idea of athletes giving 100 percent efforts. (Edith is feeling more like an 80 percent effort, or less than that.) The idea of giving 100 percent in a sport is one that invites the question, 100 percent of what; granting that there is some standard expectable effort made, then, even the sports reporting cliche of giving 110 percent is meaningful.
Cleats continued on the theme the next day, as Edith was thinking more of giving about 79 percent of 80 percent, and it’s not actually that hard to work out in your head what percent that is, if you know anything about doing arithmetic in your head.

Jef Mallet’s Frazz (March 14) was not actually the only comic strip among the roster I normally read to make a Pi Day reference, but I think it suffices as the example for the whole breed. I admit that I feel a bit curmudgeonly that I don’t actually care about Pi Day. I suppose that as a chance for people to promote the idea of learning mathematics, and maybe attach it to some of the many interesting things to be said about mathematics using Pi as the introductory note the idea is fine, but just naming a thing isn’t by itself a joke. I’m told that Facebook (I’m not on it) was thick with people posting photographs of pies, which is probably more fun when you think of it than when you notice everybody else thought of it too. Anyway, organized Pi Day events are still pretty new as Internet Pop Holidays go. Perhaps next year’s comics will be sharper.

Jenny Campbell’s Flo and Friends (March 15) comes back to useful mental arithmetic work, in this case in working out a reasonable tip. A twenty-percent tip is, mercifully, pretty easy to remember just as what’s-her-name specifies. (I can’t think of the kid’s name and there’s no meet-our-cast page on the web site. None of the commenters mention her name either, although they do make room to insult health care reform and letting students use calculators to do arithmetic, so, I’m sorry I read that far down too.) But as ever you need to make sure the process is explained clearly and understood, and Tina needed to run a sanity check on the result. Sanity checks, as suggested, won’t show that your answer is right, but they will rule out some of the wrong ones. (A fifteen percent tip is a bit annoying to calculate exactly, but dividing the original amount by six will give you a sixteen-and-two-thirds percent tip, which is surely close enough, especially if you round off to a quarter-dollar.)

Steve Breen and Mike Thompson’s Grand Avenue (March 15) has the kids wonder what are the ides of March; besides that they’re the 15th of the month and they’re used for some memorable writing about Julius Caesar it’s a fair thing not to know. They derive from calendar-keeping, one of the oldest useful applications of mathematics and astronomy. The ancient Roman scheme set three special dates in the month: the kalends, which seem to have started as the day of the new moon as observed in Rome; the nones, when the moon was at its first quarter; and the ides, when the moon was full.

But by the time of Numa Pompilius, the second (traditional) King of Rome, who reformed the calendar around 713 BC, the lunar link was snapped, partly so that the calendar year could more nearly fit the length of the time it takes to go from one spring to another. (Among other things the pre-Numa calendar had only ten months, with the days between December and March not belonging to any month; since Romans were rather agricultural at the time and there wasn’t much happening in winter, this wasn’t really absurd, even if I find it hard to imagine living by this sort of standard. After Numa there were only about eleven days of the year unaccounted for, with the time made up, when it needed to be, by inserting an extra month, Mercedonius, in the middle of February.) Months then had, February excepted, either 29 or 31 days, with the ides being on the fifteenth day of the 31-day months (March, May, July, and October) and the thirteenth day of the 29-day months.

For reasons that surely made sense if you were an ancient Roman the day was specified as the number of days until the next kalend, none, or ide; so, for example, while the 13th of March would be the 2nd day before the ides of March, II Id Mar, the 19th of March would be recorded as the the the 14th day before the kalend of April, or, XIV Kal Apr. I admit I could probably warm up to counting down to the next month event, but the idea of having half the month of March written down on the calendar as a date with “April” in it leaves me deeply unsettled. And that’s before we even get into how an extra month might get slipped into the middle of February (between the 23rd and the 24th of the month, the trace of which can still be observed in the dominical letters of February in leap years, on Roman Catholic and Anglican calendars, and in the obscure term “bissextile year” for leap year). But now that you see that, you know why (a) the ancient Romans had so much trouble getting their database software to do dates correctly and (b) you get to be all smugly superior to anyone who tries making a crack about the United States Federal Income Tax deadline being on the Ides of April, since they never are.

(Warning: absolutely no one ever will be impressed by your knowledge of the Ides of April and their inapplicability to discussions of the United States Federal Income Tax. However, you might use this as a way to appear like you’re making friendly small talk while actually encouraging people to leave you alone.)

Tom Horacek’s Foolish Mortals (March 17), an erratically-published panel strip, calls on the legend of how mathematicians “usually” peak in their twenties. It’s certainly said of mathematicians that they do their most important work while young — note that the Fields Medal is explicitly given to mathematicians for work done when they were under forty years old — although I’m not aware of anyone who’s actually studied this, and the number of great mathematicians who insist on doing brilliant work into their old age is pretty impressive.

Certainly, for example, Newton began work on calculus (and optics and gravitation) when he was about 23, but he didn’t publish until he was about fifty. (Leibniz, meanwhile, started publishing calculus his way at about age 38.) It’s probably impossible to say what Leonhard Euler’s most important work was, but (for example) his equations describing inviscid fluids — which would be the masterpiece for anybody not Euler — he published when he was fifty. Carl Friedrich Gauss didn’t start serious work in electromagnetism until he was about 55 years old, too. The law of electric flux which Gauss worked out for that — which, again, would have been the career achievement if Gauss weren’t overflowing with them — he published when he was 58.

I guess that I’m saying is that great minds, at least, don’t necessarily peak in their twenties, or at least they have some impressive peaks afterwards too.

The Rare Days


The subject doesn’t quite feel right for my occasional roundups of mathematics-themed comic strips, but I noticed this month that the bit about “what is so rare as a day in June?” is coming up … well, twice, so it’s silly to call that “a lot” just yet, but it’s coming up at all. First was back on June 10th, with Jef Mallet’s Frazz (which actually enlightened me as I didn’t know where the line came from, and yes, it’s the Lowell family that also produced Percival), and then John Rose’s Barney Google and Snuffy Smith repeated the question on the 13th.

The question made me immediately think of an installment of Walt Kelly’s Pogo, where Pogo (I believe) asked the question and Porky Pine immediately answered “any day in February”. But it got me wondering whether the question could be answered more subtly, that is, more counter-intuitively.

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One Explanation For Friday the 13th’s Chance


So to give one answer to my calendar puzzle, which you may recall as this: for any given month and year, we know with certainty whether there’s a Friday the 13th in it. And yet, we can say that “Friday the 13ths are more likely than any other day of the week”, and mean something by it, and even mean something true by it. Thanks to the patterns of the Gregorian calendar we are more likely to see a Friday the 13th than we are a Thursday the 13th, or Tuesday the 13th, or so on. (We’re also more likely to see a Saturday the 14th than the 14th being any other day of the week, but somehow that’s not so interesting.)

Here’s one way to look at it. In December 2011 there’s zero chance of encountering a Friday the 13th. As it happens, 2011 has only one month with a Friday the 13th in it, the lowest case which happens. In January 2012 there’s a probability of one of encountering a Friday the 13th; it’s right there on the schedule. There’ll also be Fridays the 13th in April and July of 2012. For the other months of 2012, there’s zero probability of encountering a Friday the 13th.

Imagine that I pick one of the months in either 2011 or 2012. What is the chance that it has a Friday the 13th? If I tell you which month it is, you know right away the chance is zero or one; or, at least, you can tell as soon as you find a calendar. Or you might work out from various formulas what day of the week the 13th of that month should be, but you’re more likely to find a calendar before you are to find that formula, much less work it out.

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How Did Friday The 13th Get A Chance?


Here’s a little puzzle in probability which, in a slightly different form, I gave to my students to work out. I get the papers back tomorrow. To brace myself against that I’m curious what my readers here would make of it.

Possibly you’ve encountered a bit of calendrical folklore which says that Friday the 13ths are more likely than any other day of the week’s 13th. That’s not that there are more Fridays the 13th than all the other days of the week combined, but rather that a Friday the 13th is more likely to happen than a Thursday the 13th, or a Sunday, or what have you. And this is true; one is slightly more likely to see a Friday the 13th than any other specific day of the week being that 13.

And yet … there’s a problem in talking about the probability of any month having a Friday the 13th. Arguably, no month has any probability of holding a Friday the 13th. Consider.

Is there a Friday the 13th this month? For the month of this writing, December 2011, the answer is no; the 13th is a Tuesday; the Fridays are the 2nd, 9th, 16th, 23rd, and 30th. But were this January 2012, the answer would be yes. For February 2012, the answer is no again, as the 13th comes on a Monday. But altogether, every month has a Friday the 13th or it hasn’t. Technically, we might say that a month which definitely has a Friday the 13th has a probability of 1, or 100%; and a month which definitely doesn’t has a probability of 0, or 0%, but we tend to think of those as chances in the same way we think of white or black as colors, mostly when we want to divert an argument into nitpicking over definitions.

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What Are Numbers Made Of?


To return to my second major theme: my Dearly Beloved told me that I must explain that trick where one adds up the digits of a number and finds out from that whether it’s divisible by 9. I wanted to anyway, but a request like that is irresistible. The answer can be given quickly — and several of my hopefully faithful readers did, in comments, last Friday — but I’d like to take the long way around because I do that and because it lets a lot of other interesting divisibility properties show themselves.

We use ten numerals and the place where we write them to express all the counting numbers out there. We put one of the numerals, such as `2′, in a place which denotes whether we mean to say two tens, or two hundreds, or two millions. That’s a clever tool, and not one inherent to the idea of numbers. We could as easily use different symbols for different magnitudes; the only familiar example of this (in the west) is Roman numerals, where we use I, X, C, and M for increasing powers of ten, and then notice we aren’t really quite sure what to do past M.

The Romans were not very sure either, and individual variations developed when someone found they needed to express an M of M very often. The system has fewer numerals, symbols representing numbers, than ours does, with V and L and D the only additional numerals reasonably common. By the Middle Ages some symbols were improvised to allow for extremely large numbers such as the hundred thousands, and some extra symbols were pulled in for numbers such as 7 or 40, but they have faded to the point of obscurity. This is a numbering system which runs out when the numbers get too large, which seems impossibly limited at first glance. But we haven’t changed much from these times: while we have a numbering system that can, in principle, work with arbitrarily big or tiny numbers, in practice we only use a small range of them. When we turn over arithmetic to computers, in fact, we accept numbering systems which have limits on how big (positive or negative) a number may be, or how close to zero one may work. We accept those limits because of their convenience and are only sometimes annoyed to find, for example, that the spreadsheet trying to calculate a bill has decided we want 0.9999999 of a penny.

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