## Reading the Comics, June 26, 2017: Deluge Edition, Part 1

So this past week saw a lot of comic strips with some mathematical connection put forth. There were enough just for the 26th that I probably could have done an essay with exclusively those comics. So it’s another split-week edition, which suits me fine as I need to balance some of my writing loads the next couple weeks for convenience (mine).

Tony Cochrane’s Agnes for the 25th of June is fun as the comic strip almost always is. And it’s even about estimation, one of the things mathematicians do way more than non-mathematicians expect. Mathematics has a reputation for precision, when in my experience it’s much more about understanding and controlling error methods. Even in analysis, the study of why calculus works, the typical proof amounts to showing that the difference between what you want to prove and what you can prove is smaller than your tolerance for an error. So: how do we go about estimating something difficult, like, the number of stars? If it’s true that nobody really knows, how do we know there are some wrong answers? And the underlying answer is that we always know some things, and those let us rule out answers that are obviously low or obviously high. We can make progress.

Russell Myers’s Broom Hilda for the 25th is about one explanation given for why time keeps seeming to pass faster as one age. This is a mathematical explanation, built on the idea that the same linear unit of time is a greater proportion of a young person’s lifestyle so of course it seems to take longer. This is probably partly true. Most of our senses work by a sense of proportion: it’s easy to tell a one-kilogram from a two-kilogram weight by holding them, and easy to tell a five-kilogram from a ten-kilogram weight, but harder to tell a five from a six-kilogram weight.

As ever, though, I’m skeptical that anything really is that simple. My biggest doubt is that it seems to me time flies when we haven’t got stories to tell about our days, when they’re all more or less the same. When we’re doing new or exciting or unusual things we remember more of the days and more about the days. A kid has an easy time finding new things, and exciting or unusual things. Broom Hilda, at something like 1500-plus years old and really a dour, unsociable person, doesn’t do so much that isn’t just like she’s done before. Wouldn’t that be an influence? And I doubt that’s a complete explanation either. Real things are more complicated than that yet.

Mac and Bill King’s Magic In A Minute for the 25th features a form-a-square puzzle using some triangles. Mathematics? Well, logic anyway. Also a good reminder about open-mindedness when you’re attempting to construct something.

Norm Feuti’s Retail for the 26th of June, 2017. So, one of my retail stories that I might well have already told because I only ever really had one retail job and there’s only so many stories you get working a year and a half in a dying mall’s book store. I was a clerk at Walden Books. The customer wanted to know for this book whether the sticker’s 10 percent discount was taken before or after the state’s 6 percent sales tax was applied. I said I thought the discount taken first and then tax applied, but it didn’t matter if I were wrong as the total would be the same amount. I calculated what it would be. The customer was none too sure about this, but allowed me to ring it up. The price encoded in the UPC was wrong, something like a dollar more than the cover price, and the subtotal came out way higher. The customer declared, “See?” And wouldn’t have any of my explaining that he was hit by a freak event. I don’t remember other disagreements between the UPC price and the cover price, but that might be because we just corrected the price and didn’t get a story out of it.

Norm Feuti’s Retail for the 26th is about how you get good at arithmetic. I suspect there’s two natural paths; you either find it really interesting in your own right, or you do it often enough you want to find ways to do it quicker. Marla shows the signs of learning to do arithmetic quickly because she does it a lot: turning “30 percent off” into “subtract ten percent three times over” is definitely the easy way to go. The alternative is multiplying by seven and dividing by ten and you don’t want to multiply by seven unless the problem gives a good reason why you should. And I certainly don’t fault the customer not knowing offhand what 30 percent off \$25 would be. Why would she be in practice doing this sort of problem?

Johnny Hart’s Back To B.C. for the 26th reruns the comic from the 30th of December, 1959. In it … uh … one of the cavemen guys has found his calendar for the next year has too many days. (Think about what 1960 was.) It’s a common problem. Every calendar people have developed has too few or too many days, as the Earth’s daily rotations on its axis and annual revolution around the sun aren’t perfectly synchronized. We handle this in many different ways. Some calendars worry little about tracking solar time and just follow the moon. Some calendars would run deliberately short and leave a little stretch of un-named time before the new year started; the ancient Roman calendar, before the addition of February and January, is famous in calendar-enthusiast circles for this. We’ve now settled on a calendar which will let the nominal seasons and the actual seasons drift out of synch slowly enough that periodic changes in the Earth’s orbit will dominate the problem before the error between actual-year and calendar-year length will matter. That’s a pretty good sort of error control.

8,978,432 is not anywhere near the number of days that would be taken between 4,000 BC and the present day. It’s not a joke about Bishop Ussher’s famous research into the time it would take to fit all the Biblically recorded events into history. The time is something like 24,600 years ago, a choice which intrigues me. It would make fair sense to declare, what the heck, they lived 25,000 years ago and use that as the nominal date for the comic strip. 24,600 is a weird number of years. Since it doesn’t seem to be meaningful I suppose Hart went, simply enough, with a number that was funny just for being riotously large.

Mark Tatulli’s Heart of the City for the 26th places itself on my Grand Avenue warning board. There’s plenty of time for things to go a different way but right now it’s set up for a toxic little presentation of mathematics. Heart, after being grounded, was caught sneaking out to a slumber party and now her mother is sending her to two weeks of Math Camp. I’m supposing, from Tatulli’s general attitude about how stuff happens in Heart and in Lio that Math Camp will not be a horrible, penal experience. But it’s still ominous talk and I’m watching.

Brian Fies’s Mom’s Cancer story for the 26th is part of the strip’s rerun on GoComics. (Many comic strips that have ended their run go into eternal loops on GoComics.) This is one of the strips with mathematical content. The spatial dimension of a thing implies relationships between the volume (area, hypervolume, whatever) of a thing and its characteristic linear measure, its diameter or radius or side length. It can be disappointing.

Nicholas Gurewitch’s Perry Bible Fellowship for the 26th is a repeat of one I get on my mathematics Twitter friends now and then. Should warn, it’s kind of racy content, at least as far as my usual recommendations here go. It’s also a little baffling because while the reveal of the unclad woman is funny … what, exactly, does it mean? The symbols don’t mean anything; they’re just what fits graphically. I think the strip is getting at Dr Loring not being able to see even a woman presenting herself for sex as anything but mathematics. I guess that’s funny, but it seems like the idea isn’t quite fully developed.

Zach Weinersmith’s Saturday Morning Breakfast Cereal Again for the 26th has a mathematician snort about plotting a giraffe logarithmically. This is all about representations of figures. When we plot something we usually start with a linear graph: a couple of axes perpendicular to one another. A unit of movement in the direction of any of those axes represents a constant difference in whatever that axis measures. Something growing ten units larger, say. That’s fine for many purposes. But we may want to measure something that changes by a power law, or that grows (or shrinks) exponentially. Or something that has some region where it’s small and some region where it’s huge. Then we might switch to a logarithmic plot. Here the same difference in space along the axis represents a change that’s constant in proportion: something growing ten times as large, say. The effective result is to squash a shape down, making the higher points more nearly flat.

And to completely smother Weinersmith’s fine enough joke: I would call that plot semilogarithmically. I’d use a linear scale for the horizontal axis, the gazelle or giraffe head-to-tail. But I’d use a logarithmic scale for the vertical axis, ears-to-hooves. So, linear in one direction, logarithmic in the other. I’d be more inclined to use “logarithmic” plots to mean logarithms in both the horizontal and the vertical axes. Those are useful plots for turning up power laws, like the relationship between a planet’s orbital radius and the length of its year. Relationships like that turn into straight lines when both axes are logarithmically spaced. But I might also describe that as a “log-log plot” in the hopes of avoiding confusion.

## What Is The Most Probable Date For Easter? What Is The Least?

If I’d started pondering the question a week earlier I’d have a nice timely post. Too bad. Shouldn’t wait nearly a year to use this one, though.

My love and I got talking about early and late Easters. We know that we’re all but certainly not going to be alive to see the earliest possible Easter, at least not unless the rule for setting the date of Easter changes. Easter can be as early as the 22nd of March or as late as the 25th of April. Nobody presently alive has seen a 22nd of March Easter; the last one was in 1818. Nobody presently alive will; the next will be 2285. The last time Easter was its latest date was 1943; the next time will be 2038. I know people who’ve seen the one in 1943 and hope to make it at least through 2038.

But that invites the question: what dates are most likely to be Easter? What ones are least? In a sense the question is nonsense. The rules establishing Easter and the Gregorian calendar are known. To speak of the “chance” of a particular day being Easter is like asking the probability that Grover Cleveland was president of the United States in 1894. Technically there’s a probability distribution there. But it’s different in some way from asking the chance of rolling at least a nine on a pair of dice.

But as with the question about what day is most likely to be Thanksgiving we can make the question sensible. We have to take the question to mean “given a month and day, and no information about what year it is, what is the chance that this as Easter?” (I’m still not quite happy with that formulation. I’d be open to a more careful phrasing, if someone’s got one.)

When we’ve got that, though, we can tackle the problem. We could do as I did for working out what days are most likely to be Thanksgiving. Run through all the possible configurations of the calendar, tally how often each of the days in the range is Easter, and see what comes up most often. There’s a hassle here. Working out the date of Easter follows a rule, yes. The rule is that it’s the first Sunday after the first full moon after the spring equinox. There are wrinkles, mostly because the Moon is complicated. A notional Moon that’s a little more predictable gets used instead. There are algorithms you can use to work out when Easter is. They all look like some kind of trick being used to put something over on you. No matter. They seem to work, as far as we know. I found some Matlab code that uses the Easter-computing routine that Karl Friedrich Gauss developed and that’ll do.

Problem. The Moon and the Earth follow cycles around the sun, yes. Wait long enough and the positions of the Earth and Moon and Sun. This takes 532 years and is known as the Paschal Cycle. In the Julian calendar Easter this year is the same date it was in the year 1485, and the same it will be in 2549. It’s no particular problem to set a computer program to run a calculation, even a tedious one, 532 times. But it’s not meaningful like that either.

The problem is the Julian calendar repeats itself every 28 years, which fits nicely with the Paschal Cycle. The Gregorian calendar, with different rules about how to handle century years like 1900 and 2100, repeats itself only every 400 years. So it takes much longer to complete the cycle and get Earth, Moon, and calendar date back to the same position. To fully account for all the related cycles would take 5,700,000 years, estimates Duncan Steel in Marking Time: The Epic Quest To Invent The Perfect Calendar.

Write code to calculate Easter on a range of years and you can do that, of course. It’s no harder to calculate the dates of Easter for six million years than it is for six hundred years. It just takes longer to finish. The problem is that it is meaningless to do so. Over the course of a mere(!) 26,000 years the precession of the Earth’s axes will change the times of the seasons completely. If we still use the Gregorian calendar there will be a time that late September is the start of the Northern Hemisphere’s spring, and another time that early February is the heart of the Canadian summer. Within five thousand years we will have to change the calendar, change the rule for computing Easter, or change the idea of it as happening in Europe’s early spring. To calculate a date for Easter of the year 5,002,017 is to waste energy.

We probably don’t need it anyway, though. The differences between any blocks of 532 years are, I’m going to guess, minor things. I would be surprised if the frequency of any date’s appearance changed more than a quarter of a percent. That might scramble the rankings of dates if we have several nearly-as-common dates, but it won’t be much.

So let me do that. Here’s a table of how often each particular calendar date appears as Easter from the years 2000 to 5000, inclusive. And I don’t believe that by the year we would call 5000 we’ll still have the same calendar and Easter and expectations of Easter all together, so I’m comfortable overlooking that. Indeed, I expect we’ll have some different calendar or Easter or expectation of Easter by the year 4985 at the latest.

For this enormous date range, though, here’s the frequency of Easters on each possible date:

Date Number Of Occurrences, 2000 – 5000 Probability Of Occurence
22 March 12 0.400%
23 March 17 0.566%
24 March 41 1.366%
25 March 74 2.466%
26 March 75 2.499%
27 March 68 2.266%
28 March 90 2.999%
29 March 110 3.665%
30 March 114 3.799%
31 March 99 3.299%
1 April 87 2.899%
2 April 83 2.766%
3 April 106 3.532%
4 April 112 3.732%
5 April 110 3.665%
6 April 92 3.066%
7 April 86 2.866%
8 April 98 3.266%
9 April 112 3.732%
10 April 114 3.799%
11 April 96 3.199%
12 April 88 2.932%
13 April 90 2.999%
14 April 108 3.599%
15 April 117 3.899%
16 April 104 3.466%
17 April 90 2.999%
18 April 93 3.099%
19 April 114 3.799%
20 April 116 3.865%
21 April 93 3.099%
22 April 60 1.999%
23 April 46 1.533%
24 April 57 1.899%
25 April 29 0.966%

Dates of Easter from 2000 through 5000. Computed using Gauss’s algorithm.

If I haven’t missed anything, this indicates that the 15th of April is the most likely date for Easter, with the 20th close behind and the 10th and 14th hardly rare. The least probable date is the 22nd of March, with the 23rd of March and the 25th of April almost as unlikely.

And since the date range does affect the results, here’s a smaller sampling, one closer fit to the dates of anyone alive to read this as I publish. For the years 1925 through 2100 the appearance of each Easter date are:

Date Number Of Occurrences, 1925 – 2100 Probability Of Occurence
22 March 0 0.000%
23 March 1 0.568%
24 March 1 0.568%
25 March 3 1.705%
26 March 6 3.409%
27 March 3 1.705%
28 March 5 2.841%
29 March 6 3.409%
30 March 7 3.977%
31 March 7 3.977%
1 April 6 3.409%
2 April 4 2.273%
3 April 6 3.409%
4 April 6 3.409%
5 April 7 3.977%
6 April 7 3.977%
7 April 4 2.273%
8 April 4 2.273%
9 April 6 3.409%
10 April 7 3.977%
11 April 7 3.977%
12 April 7 3.977%
13 April 4 2.273%
14 April 6 3.409%
15 April 7 3.977%
16 April 6 3.409%
17 April 7 3.977%
18 April 6 3.409%
19 April 6 3.409%
20 April 6 3.409%
21 April 7 3.977%
22 April 5 2.841%
23 April 2 1.136%
24 April 2 1.136%
25 April 2 1.136%

Dates of Easter from 1925 through 2100. Computed using Gauss’s algorithm.

If we take this as the “working lifespan” of our common experience then the 22nd of March is the least likely Easter we’ll see, as we never do. The 23rd and 24th are the next least likely Easter. There’s a ten-way tie for the most common date of Easter, if I haven’t missed one or more. But the 30th and 31st of March, and the 5th, 6th, 10th, 11th, 12th, 15th, 17th, and 21st of April each turn up seven times in this range.

The Julian calendar Easter dates are different and perhaps I’ll look at that sometime.

• #### ksbeth 7:34 pm on Tuesday, 18 April, 2017 Permalink | Reply

Very interesting

Liked by 1 person

• #### Joseph Nebus 3:31 am on Wednesday, 19 April, 2017 Permalink | Reply

Thank you!

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• #### mx. fluffy 💜 (@fluffy) 11:51 pm on Thursday, 20 April, 2017 Permalink | Reply

I’m surprised there’s such a periodicity in the modal peaks! What happens if you extend the computations out for a few more millennia? Do they even out or get even more pronounced?

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• #### Joseph Nebus 2:59 am on Tuesday, 25 April, 2017 Permalink | Reply

I’m surprised by it too, yes. If we pretend that the current scheme for calculating Easter would be meaningful, then, extended over the full 5,700,000-year cycle … the peaks don’t disappear. The 19th of April turns up as Easter about 3.9 percent of the time. Next most likely are the 18th, 17th, 15th, 12th, and 10th of April.

I don’t know just what causes this. I suspect it’s some curious interaction between the 19-year Metonic cycle of the lunar behavior and the very slight asymmetries the Gregorian calendar. The 21st of March is a tiny bit more likely to be a Tuesday, Wednesday, or Sunday than it is any other day of the week. My hunch is these combine to make the little peaks that linger.

The 22nd of March and 25th of April are the least common Easters; the 23rd and 24th of March, then 24th of April, come slightly more commonly.

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## How To Work Out The Length Of Time Between Two Dates

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

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

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

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

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

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

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

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

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

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

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

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

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

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## When Is Thanksgiving Most Likely To Happen?

So my question from last Thursday nagged at my mind. And I learned that Octave (a Matlab clone that’s rather cheaper) has a function that calculates the day of the week for any given day. And I spent longer than I would have expected fiddling with the formatting to get what I wanted to know.

It turns out there are some days in November more likely to be the fourth Thursday than others are. (This is the current standard for Thanksgiving Day in the United States.) And as I’d suspected without being able to prove, this doesn’t quite match the breakdown of which months are more likely to have Friday the 13ths. That is, it’s more likely that an arbitrarily selected month will start on Sunday than any other day of the week. It’s least likely that an arbitrarily selected month will start on a Saturday or Monday. The difference is extremely tiny; there are only four more Sunday-starting months than there are Monday-starting months over the course of 400 years.

But an arbitrary month is different from an arbitrary November. It turns out Novembers are most likely to start on a Sunday, Tuesday, or Thursday. And that makes the 26th, 24th, and 22nd the most likely days to be Thanksgiving. The 23rd and 25th are the least likely days to be Thanksgiving. Here’s the full roster, if I haven’t made any serious mistakes with it:

November Will Be Thanksgiving
22 58
23 56
24 58
25 56
26 58
27 57
28 57
times in 400 years

I don’t pretend there’s any significance to this. But it is another of those interesting quirks of probability. What you would say the probability is of a month starting on the 1st — equivalently, of having a Friday the 13th, or a Fourth Thursday of the Month that’s the 26th — depends on how much you know about the month. If you know only that it’s a month on the Gregorian calendar it’s one thing (specifically, it’s 688/4800, or about 0.14333). If you know only that it’s a November than it’s another (58/400, or 0.145). If you know only that it’s a month in 2016 then it’s another yet (1/12, or about 0.08333). If you know that it’s November 2016 then the probability is 0. Information does strange things to probability questions.

## A Thanksgiving Thought Fresh From The Shower

It’s well-known, at least in calendar-appreciation circles, that the 13th of a month is more likely to be Friday than any other day of the week. That’s on the Gregorian calendar, which has some funny rules about whether a century year — 1900, 2000, 2100 — will be a leap year. Three of them aren’t in every four centuries. The result is the pattern of dates on the calendar is locked into this 400-year cycle, instead of the 28-year cycle you might imagine. And this makes some days of the week more likely for some dates than they otherwise might be.

This got me wondering. Does the 13th being slightly more likely imply that the United States Thanksgiving is more likely to be on the 26th of the month? The current rule is that Thanksgiving is the fourth Thursday of November. We’ll pretend that’s an unalterable fact of nature for the sake of having a problem we can solve. So if the 13th is more likely to be a Friday than any other day of the week, isn’t the 26th more likely to be a Thursday than any other day of the week?

And that’s so, but I’m not quite certain yet. What’s got me pondering this in the shower is that the 13th is more likely a Friday for an arbitrary month. That is, if I think of a month and don’t tell you anything about what it is, all we can say is it chance of the 13th being a Friday is such-and-such. But if I pick a particular month — say, November 2017 — things are different. The chance the 13th of November, 2017 is a Friday is zero. So the chance the 26th of December, 2017 is a Thursday is zero. Our calendar system sets rules. We’ll pretend that’s an unalterable fact of nature for the sake of having a problem we can solve, too.

So: does knowing that I am thinking of November, rather than a completely unknown month, change the probabilities? And I don’t know. My gut says “it’s plausible the dates of Novembers are different from the dates of arbitrary months”. I don’t know a way to argue this purely logically, though. It might have to be tested by going through 400 years of calendars and counting when the fourth Thursdays are. (The problem isn’t so tedious as that. There’s formulas computers are good at which can do this pretty well.)

But I would like to know if it can be argued there’s a difference, or that there isn’t.

## Reading the Comics, August 29, 2015: Unthemed Edition

I can’t think of any particular thematic link through the past week’s mathematical comic strips. This happens sometimes. I’ll make do. They’re all Gocomics.com strips this time around, too, so I haven’t included the strips. The URLs ought to be reasonably stable.

J C Duffy’s Lug Nuts (August 23) is a cute illustration of the first, second, third, and fourth dimensions. The wall-of-text might be a bit off-putting, especially the last panel. It’s worth the reading. Indeed, you almost don’t need the cartoon if you read the text.

Tom Toles’s Randolph Itch, 2 am (August 24) is an explanation of pie charts. This might be the best stilly joke of the week. I may just be an easy touch for a pie-in-the-face.

Charlie Podrebarac’s Cow Town (August 26) is about the first day of mathematics camp. It’s also every graduate students’ thesis defense anxiety dream. The zero with a slash through it popping out of Jim Smith’s mouth is known as the null sign. That comes to us from set theory, where it describes “a set that has no elements”. Null sets have many interesting properties considering they haven’t got any things. And that’s important for set theory. The symbol was introduced to mathematics in 1939 by Nicholas Bourbaki, the renowned mathematician who never existed. He was important to the course of 20th century mathematics.

Eric the Circle (August 26), this one by ‘Arys’, is a Venn diagram joke. It makes me realize the Eric the Circle project does less with Venn diagrams than I expected.

John Graziano’s Ripley’s Believe It Or Not (August 26) talks of a Akira Haraguchi. If we believe this, then, in 2006 he recited 111,700 digits of pi from memory. It’s an impressive stunt and one that makes me wonder who did the checking that he got them all right. The fact-checkers never get their names in Graziano’s Ripley’s.

Mark Parisi’s Off The Mark (August 27, rerun from 1987) mentions Monty Hall. This is worth mentioning in these parts mostly as a matter of courtesy. The Monty Hall Problem is a fine and imagination-catching probability question. It represents a scenario that never happened on the game show Let’s Make A Deal, though.

Jeff Stahler’s Moderately Confused (August 28) is a word problem joke. I do wonder if the presence of battery percentage indicators on electronic devices has helped people get a better feeling for percentages. I suppose only vaguely. The devices can be too strangely nonlinear to relate percentages of charge to anything like device lifespan. I’m thinking here of my cell phone, which will sit in my messenger bag for three weeks dropping slowly from 100% to 50%, and then die for want of electrons after thirty minutes of talking with my father. I imagine you have similar experiences, not necessarily with my father.

Thom Bluemel’s Birdbrains (August 29) is a caveman-mathematics joke. This one’s based on calendars, which have always been mathematical puzzles.

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## Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?

Well, I thought it’d be unlikely to get too many more mathematics comics before the end of the year, but Comic Strip Master Command apparently sent out orders to clear out the backlog before the new calendar year starts. I think Dark Side of the Horse is my favorite of the strips, blending a good joke with appealing artwork, although The Buckets gives me the most to talk about.

Greg Cravens’s The Buckets (December 28) is about what might seem only loosely a mathematical topic: that the calendar is really a pretty screwy creation. And it is, as anyone who’s tried to program a computer to show dates has realized. The core problem, I suppose, is that the calendar tries to meet several goals simultaneously: it’s supposed to use our 24-hour days to keep track of the astronomical year, which is an approximation to the cycle of seasons of the year, and there’s not a whole number of days in a year. It’s also supposed to be used to track short-term events (weeks) and medium-term events (months and seasons). The number of days that best approximate the year, 365 and 366, aren’t numbers that lend themselves to many useful arrangements. The months try to divide that 365 or 366 reasonably uniformly, with historial artifacts that can be traced back to the Roman calendar was just an unspeakable mess; and, something rarely appreciated, the calendar also has to make sure that the date of Easter is something reasonable. And, of course, any reforming of the calendar has to be done with the agreement of a wide swath of the world simultaneously. Given all these constraints it’s probably remarkable that it’s only as messed up as it is.

To the best of my knowledge, January starts the New Year because Tarquin Priscus, King of Rome from 616 – 579 BC, found that convenient after he did some calendar-rejiggering (particularly, swapping the order of February and January), though I don’t know why he thought that particularly convenient. New Years have appeared all over the calendar year, though, with the start of January, the start of September, Christmas Day, and the 25th of March being popular options, and if you think it’s messed up to have a new year start midweek, think about having a new year start in the middle of late March. It all could be worse.

• #### ivasallay 9:01 am on Wednesday, 31 December, 2014 Permalink | Reply

There are many nights I try to go to sleep thinking about numbers so I definitely related to the Dark Side of the Horse strip.

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• #### Joseph Nebus 6:12 am on Friday, 2 January, 2015 Permalink | Reply

I tend to have nights like that more often when I haven’t got any reason to get up early the next day. It’s like my mind is trying to make sure I don’t get a good long sleep any more than absolutely necessary.

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• #### elkement 4:56 pm on Monday, 5 January, 2015 Permalink | Reply

I like your comment about making a computer show calendar dates! I still feel that whatever I am working on, it always comes down to fighting with date formats or delimiters or commas, like the German decimal comman versus the English decimal point.

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## Reading the Comics, December 14, 2014: Pictures Gone Again? Edition

I’ve got enough comics to do a mathematics-comics roundup post again, but none of them are the King Features or Creators or other miscellaneous sources that demand they be included here in pictures. I could wait a little over three hours and give the King Features Syndicate comics another chance to say anything on point, or I could shrug and go with what I’ve got. It’s a tough call. Ah, what the heck; besides, it’s been over a week since I did the last one of these.

Bill Amend’s FoxTrot (December 7) bids to get posted on mathematics teachers’ walls with a bit of play on two common uses of the term “degree”. It’s also natural to wonder why the same word “degree” should be used to represent the units of temperature and the size of an angle, to the point that they even use the same symbol of a tiny circle elevated from the baseline as a shorthand representation. As best I can make out, the use of the word degree traces back to Old French, and “degré”, meaning a step, as in a stair. In Middle English this got expanded to the notion of one of a hierarchy of steps, and if you consider the temperature of a thing, or the width of an angle, as something that can be grown or shrunk then … I’m left wondering if the Middle English folks who extended “degree” to temperatures and angles thought there were discrete steps by which either quantity could change.

As for the little degree symbol, Florian Cajori notes in A History Of Mathematical Notations that while the symbol (and the ‘ and ” for minutes and seconds) can be found in Ptolemy (!), in describing Babylonian sexagesimal fractions, this doesn’t directly lead to the modern symbols. Medieval manuscripts and early printed books would use abbreviations of Latin words describing what the numbers represented. Cajori rates as the first modern appearance of the degree symbol an appendix, composed by one Jacques Peletier, to the 1569 edition of the text Arithmeticae practicae methods facilis by Gemma Frisius (you remember him; the guy who made triangulation into something that could be used for surveying territories). Peletier was describing astronomical fractions, and used the symbol to denote that the thing before it was a whole number. By 1571 Erasmus Reinhold (whom you remember from working out the “Prutenic Tables”, updated astronomical charts that helped convince people of the use of the Copernican model of the solar system and advance the cause of calendar reform) was using the little circle to represent degrees, and Tycho Brahe followed his example, and soon … well, it took a century or so of competing symbols, including “Grad” or “Gr” or “G” to represent degree, but the little circle eventually won out. (Assume the story is more complicated than this. It always is.)

Mark Litzer’s Joe Vanilla (December 7) uses a panel of calculus to suggest something particularly deep or intellectually challenging. As it happens, the problem isn’t quite defined well enough to solve, but if you make a reasonable assumption about what’s meant, then it becomes easy to say: this expression is “some infinitely large number”. Here’s why.

The numerator is the integral $\int_{0}^{\infty} e^{\pi} + \sin^2\left(x\right) dx$. You can think of the integral of a positive-valued expression as the area underneath that expression and between the lines marked by, on the left, $x = 0$ (the number on the bottom of the integral sign), and on the right, $x = \infty$ (the number on the top of the integral sign). (You know that it’s x because the integral symbol ends with “dx”; if it ended “dy” then the integral would tell you the left and the right bounds for the variable y instead.) Now, $e^{\pi} + \sin^2\left(x\right)$ is a number that depends on x, yes, but which is never smaller than $e^{\pi}$ (about 23.14) nor bigger than $e^{\pi} + 1$ (about 24.14). So the area underneath this expression has to be at least as big as the area within a rectangle that’s got a bottom edge at y = 0, a top edge at y = 23, a left edge at x = 0, and a right edge at x infinitely far off to the right. That rectangle’s got an infinitely large area. The area underneath this expression has to be no smaller than that.

Just because the numerator’s infinitely large doesn’t mean that the fraction is, though. It’s imaginable that the denominator is also infinitely large, and more wondrously, is large in a way that makes the ratio some more familiar number like “3”. Spoiler: it isn’t.

Actually, as it is, the denominator isn’t quite much of anything. It’s a summation; that’s what the capital sigma designates there. By convention, the summation symbol means to evaluate whatever expression there is to the right of it — in this case, it’s $x^{\frac{1}{e}} + \cos\left(x\right)$ — for each of a series of values of some index variable. That variable is normally identified underneath the sigma, with a line such as x = 1, and (again by convention) for x = 2, x = 3, x = 4, and so on, until x equals whatever the number on top of the sigma is. In this case, the bottom doesn’t actually say what the index should be, although since “x” is the only thing that makes sense as a variable within the expression — “cos” means the cosine function, and “e” means the number that’s about 2.71828 unless it’s otherwise made explicit — we can suppose that this is a normal bit of shorthand like you use when context is clear.

With that assumption about what’s meant, then, we know the denominator is whatever number is represented by $\left(1^{\frac{1}{e}} + \cos\left(1\right)\right) + \left(2^{\frac{1}{e}} + \cos\left(2\right)\right) + \left(3^{\frac{1}{e}} + \cos\left(3\right)\right) + \left(4^{\frac{1}{e}} + \cos\left(4\right)\right) + \cdots + \left(10^{\frac{1}{e}} + \cos\left(10\right)\right)$ (and 1/e is about 0.368). That’s a number about 16.549, which falls short of being infinitely large by an infinitely large amount.

So, the original fraction shown represents an infinitely large number.

Mark Tatulli’s Lio (December 7) is another “anthropomorphic numbers” genre comic, and since it’s Lio the numbers naturally act a bit mischievously.

Greg Evans’s Luann Againn (December 7, I suppose technically a rerun) only has a bit of mathematical content, as it’s really playing more on short- and long-term memories. Normal people, it seems, have a buffer of something around eight numbers that they can remember without losing track of them, and it’s surprisingly easy to overload that. I recall reading, I think in Joseph T Hallinan’s Why We Make Mistakes: How We Look Without Seeing, Forget Things In Seconds, And Are All Pretty Sure We are Way Above Average, and don’t think I’m not aware of how funny it would be if I were getting this source wrong, that it’s possible to cheat a little bit on the size of one’s number-buffer.

Hallinan (?) gave the example of a runner who was able to remember strings of dozens of numbers, well past the norm, but apparently by the trick of parsing numbers into plausible running times. That is, the person would remember “834126120820” perfectly because it could be expressed as four numbers, “8:34, 1:26, 1:20, 8:20”, that might be credible running times for something or other and the runner was used to remembering such times. Supporting the idea that this trick was based on turning a lot of digits into a few small numbers was that the runner would be lost if the digits could not be parsed into a meaningful time, like, “489162693077”. So, in short, people are really weird in how they remember and don’t remember things.

Justin Boyd’s Invisible Bread (December 12) reveals the joy and the potential menace of charts and graphs. It’s a reassuring red dot at the end of this graph of relevant-graph-probabilities.

Several comics chose to mention the coincidence of the 13th of December being (in the United States standard for shorthand dating) 12-13-14. Chip Sansom’s The Born Loser does the joke about how yes, this sequence won’t recur in (most of our) lives, but neither will any other. Stuart Carlson and Jerry Resler’s Gray Matters takes a little imprecision in calling it “the last date this century to have a consecutive pattern”, something the Grays, if the strip is still running, will realize on 1/2/34 at the latest. And Francesco Marciuliano’s Medium Large uses the neat pattern of the dates as a dip into numerology and the kinds of manias that staring too closely into neat patterns can encourage.

• #### ivasallay 8:58 am on Monday, 15 December, 2014 Permalink | Reply

I love the Foxtrot panel.
How I would remember 49 – 61 – 32: First 7^2 – then the largest prime number less than 8^2 – Finally 2^5.
The Born Loser makes a good point.
Thanks for reading all those thousands of comics and sharing with us!

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• #### Joseph Nebus 10:42 pm on Monday, 15 December, 2014 Permalink | Reply

Oh, now, I forgot somehow to mention how Charlie Brown remembers his locker combinations. He remembers the names of baseball players whose uniform numbers are the ones he wants, which is a good scheme for people who remember names well.

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• #### fluffy 5:47 pm on Monday, 15 December, 2014 Permalink | Reply

Thanks to not-so-smart quotes, you ended up with the wrong symbols for minutes (‘) and seconds (“). Yet another case where pervasive assistive technology hurts more than it helps.

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• #### fluffy 5:47 pm on Monday, 15 December, 2014 Permalink | Reply

And ironically, WordPress decided to “help” me too by “smartifying” them. Argh.

Let’s see if HTML entities work: ' "

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• #### Joseph Nebus 10:43 pm on Monday, 15 December, 2014 Permalink | Reply

I actually did see WordPress inappropriately smarting-up my quotes when I previewed, but I figured that trying to fix it would be too much work by requiring any effort on my part. I’m sorry to have bothered you enough that you had to dig into HTML entities over it.

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• #### fluffy 3:16 pm on Tuesday, 16 December, 2014 Permalink | Reply

Eh, I know the major entities by heart. &apos;, &quot;, and, while we’re at it, &amp;.

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• #### Joseph Nebus 8:35 pm on Wednesday, 17 December, 2014 Permalink | Reply

Oh, I could hardly not remember &amp;. I just use it too much. The quote mark and apostrophes I haven’t used content-management systems long enough to need to deal with.

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• #### Aquileana 2:18 am on Tuesday, 16 December, 2014 Permalink | Reply

Great overview… So clever and enjoyable at the same time.
Sending you all my best wishes!. Aquileana :D

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• #### Joseph Nebus 8:35 pm on Wednesday, 17 December, 2014 Permalink | Reply

Thanks kindly.

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## Reading the Comics, November 20, 2014: Ancient Events Edition

I’ve got enough mathematics comics for another roundup, and this time, the subjects give me reason to dip into ancient days: one to the most famous, among mathematicians and astronomers anyway, of Greek shipwrecks, and another to some point in the midst of winter nearly seven thousand years ago.

Eric the Circle (November 15) returns “Griffinetsabine” to the writer’s role and gives another “Shape Single’s Bar” scene. I’m amused by Eric appearing with his ex: x is practically the icon denoting “this is an algebraic expression”, while geometry … well, circles are good for denoting that, although I suspect that triangles or maybe parallelograms are the ways to denote “this is a geometric expression”. Maybe it’s the little symbol for a right angle.

Jim Meddick’s Monty (November 17) presents Monty trying to work out just how many days there are to Christmas. This is a problem fraught with difficulties, starting with the obvious: does “today” count as a shopping day until Christmas? That is, if it were the 24th, would you say there are zero or one shopping days left? Also, is there even a difference between a “shopping day” and a “day” anymore now that nobody shops downtown so it’s only the stores nobody cares about that close on Sundays? Sort all that out and there’s the perpetual problem in working out intervals between dates on the Gregorian calendar, which is that you have to be daft to try working out intervals between dates on the Gregorian calendar. The only worse thing is trying to work out the intervals between Easters on it. My own habit for this kind of problem is to use the United States Navy’s Julian Date conversion page. The Julian date is a straight serial number, counting the number of days that have elapsed since noon Universal Time at what’s called the 1st of January, 4713 BCE, on the proleptic Julian calendar (“proleptic” because nobody around at the time was using, or even imagined, the calendar, but we can project back to what date that would have been), a year picked because it’s the start of several astronomical cycles, and it’s way before any specific recordable dates in human history, so any day you might have to particularly deal with has a positive number. Of course, to do this, we’re transforming the problem of “counting the number of days between two dates” to “counting the number of days between a date and January 1, 4713 BCE, twice”, but the advantage of that is, the United States Navy (and other people) have worked out how to do that and we can use their work.

Bill Hind’s kids-sports comic Cleats (November 19, rerun) presents Michael offering basketball advice that verges into logic and set theory problems: making the ball not go to a place outside the net is equivalent to making the ball go inside the net (if we decide that the edge of the net counts as either inside or outside the net, at least), and depending on the problem we want to solve, it might be more convenient to think about putting the ball into the net, or not putting the ball outside the net. We see this, in logic, in a set of relations called De Morgan’s Laws (named for Augustus De Morgan, who put these ideas in modern mathematical form), which describe what kinds of descriptions — “something is outside both sets A and B at one” or “something is not inside set A or set B”, or so on — represent the same relationship between the thing and the sets.

Tom Thaves’s Frank and Ernest (November 19) is set in the classic caveman era, with prehistoric Frank and Ernest and someone else discovering mathematics and working out whether a negative number times a negative number might be positive. It’s not obvious right away that they should, as you realize when you try teaching someone the multiplication rules including negative numbers, and it’s worth pointing out, a negative times a negative equals a positive because that’s the way we, the users of mathematics, have chosen to define negative numbers and multiplication. We could, in principle, have decided that a negative times a negative should give us a negative number. This would be a different “multiplication” (or a different “negative”) than we use, but as long as we had logically self-consistent rules we could do that. We don’t, because it turns out negative-times-negative-is-positive is convenient for problems we like to do. Mathematics may be universal — something following the same rules we do has to get the same results we do — but it’s also something of a construct, and the multiplication of negative numbers is a signal of that.

The Mickey Mouse comic rerun the 20th of November, 2014.

Mickey Mouse (November 20, rerun) — I don’t know who wrote or draw this, but Walt Disney’s name was plastered onto it — sees messages appearing in alphabet soup. In one sense, such messages are inevitable: jumble and swirl letters around and eventually, surely, any message there are enough letters for will appear. This is very similar to the problem of infinite monkeys at typewriters, although with the special constraint that if, say, the bowl has only two letters “L”, it’s impossible to get the word “parallel”, unless one of the I’s is doing an impersonation. Here, Goofy has the message “buried treasure in back yard” appear in his soup; assuming those are all the letters in his soup then there’s something like 44,881,973,505,008,615,424 different arrangements of letters that could come up. There are several legitimate messages you could make out of that (“treasure buried in back yard”, “in back yard buried treasure”), not to mention shorter messages that don’t use all those letters (“run back”), but I think it’s safe to say the number of possible sentences that make sense are pretty few and it’s remarkable to get something like that. Maybe the cook was trying to tell Goofy something after all.

Mark Anderson’s Andertoons (November 20) is a cute gag about the dangers of having too many axes on your plot.

Gary Delainey and Gerry Rasmussen’s Betty (November 20) mentions the Antikythera Mechanism, one of the most famous analog computers out there, and that’s close enough to pure mathematics for me to feel comfortable including it here. The machine was found in April 1900, in ancient shipwreck, and at first seemed to be just a strange lump of bronze and wood. By 1902 the archeologist Valerios Stais noticed a gear in the mechanism, but since it was believed the wreck far, far predated any gear mechanisms, the machine languished in that strange obscurity that a thing which can’t be explained sometimes suffers. The mechanism appears to be designed to be an astronomical computer, tracking the positions of the Sun and the Moon — tracking the actual moon rather than an approximate mean lunar motion — the rising and etting of some constellations, solar eclipses, several astronomical cycles, and even the Olympic Games. It’s an astounding mechanism, it’s mysterious: who made it? How? Are there others? What happened to them? How was the mechanical engineering needed for this developed, and what other projects did the people who created this also do? Any answers to these questions, if we ever know them, seem sure to be at least as amazing as the questions are.

## George Berkeley’s 329th Birthday

The stream of mathematics-trivia tweets brought to my attention that the 12th of March, 1685 [1], was the birthday of George Berkeley, who’d become the Bishop of Cloyne and be an important philosopher, and who’s gotten a bit of mathematical immortality for complaining about calculus. Granted everyone who takes it complains about calculus, but Berkeley had the good sorts of complaints, the ones that force people to think harder and more clearly about what they’re doing.

Berkeley — whose name I’m told by people I consider reliable was pronounced “barkley” — particularly protested the “fluxions” of calculus as it was practiced in the day in his 1734 tract The Analyst: Or A Discourse Addressed To An Infidel Mathematician, which as far as I know nobody I went to grad school with ever read either, so maybe you shouldn’t bother reading what I have to say about them.

Fluxions were meant to represent infinitesimally small quantities, which could be added to or subtracted from a number without changing the number, but which could be divided by one another to produce a meaningful answer. That’s a hard set of properties to quite rationalize — if you can add something to a number without changing the number, you’re adding zero; and if you’re dividing zero by zero you’re not doing division anymore — and yet calculus was doing just that. For example, if you want to find the slope of a curve at a single point on the curve you’d take the x- and y-coordinates of that point, and add an infinitesimally small number to the x-coordinate, and see how much the y-coordinate has to change to still be on the curve, and then divide those changes, which are too small to even be numbers, and get something out of it.

It works, at least if you’re doing the calculations right, and Berkeley supposed that it was the result of multiple logical errors cancelling one another out that they did work; but he termed these fluxions with spectacularly good phrasing “ghosts of departed quantities”, and it would take better than a century to put all his criticisms quite to rest. The result we know as differential calculus.

I should point out that it’s not as if mathematicians playing with their shiny new calculus tools were being irresponsible in using differentials and integrals despite Berkeley’s criticisms. Mathematical concepts work a good deal like inventions, in that it’s not clear what is really good about them until they’re used, and it’s not clear what has to be made better until there’s a body of experience working with them and seeing where the flaws. And Berkeley was hardly being unreasonable for insisting on logical rigor in mathematics.

[1] Berkeley was born in Ireland. I have found it surprisingly hard to get a clear answer about when Ireland switched from the Julian to the Gregorian calendar, so I have no idea whether this birthdate is old style or new style, and for that matter whether the 1685 represents the civil year or the historical year. Perhaps it suffices to say that Berkeley was born sometime around this time of year, a long while ago.

• #### BunnyHugger 11:09 pm on Wednesday, 12 March, 2014 Permalink | Reply

This has nothing to do with mathematics, but Berkeley is also notable in that he was one of the very few married philosophers of the traditional canon.

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• #### Joseph Nebus 4:31 am on Saturday, 15 March, 2014 Permalink | Reply

And that’s a neat biographical tidbit. It’s just inspired me to wonder about the marriage status of the greats of mathematical history (though there’s obviously going to be a lot of overlap with the greats of philosophical history, at least up to Descartes’s era).

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## Golden Days

I haven’t been able to avoid people on my Twitter feed pointing out today’s a Pythagorean Triple, if you write out the month and day as digits and only use the last two digits of the year. There aren’t many such days; if I haven’t missed one there’s only fourteen per century, and we’ve just burned through the tenth of them. But if you want to have a little fun you might try working out whether I’m correct, and when the next one is going to be.

I don’t know of an efficient way of doing this, the sort of thing where you set up a couple of equations and let your favorite version of Mathematica grind away a bit and spit out an array of dates. This seems like the sort of problem best done by working out sets of integers a, b, and c, where $a^2 + b^2 = c^2$, and figure out what sets of those numbers can plausibly even be arranged as dates.

The more mysterious thing to me is that I don’t remember this being so much pointed out when we had the same Pythagorean Triple day in May, and not at all when we were really rich with them back nearly a decade ago. But I wasn’t on Twitter back then; maybe that’s the problem. I also haven’t seen people complaining that it’s a trivial thing not worth pointing out; it may be trivial, but if we aren’t going to enjoy pretty alignments of numbers, what are we supposed to do?

• #### elkement 6:54 pm on Thursday, 5 December, 2013 Permalink | Reply

Thanks – I didn’t notice!! I am always late to recognizing such things – I had missed 11/12/13 as well recently.

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• #### Joseph Nebus 11:22 pm on Thursday, 5 December, 2013 Permalink | Reply

I was late on 11/12/13 myself. But if you’re willing to take dates in the British style, day-month-year, there’s enough time to get ready for it yet.

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• #### elkement 6:04 pm on Monday, 9 December, 2013 Permalink | Reply

I think this was the reason I missed it :-) We use the British style, too! So I am still looking forward to 09:10 11.12.13

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• #### Joseph Nebus 5:15 pm on Wednesday, 11 December, 2013 Permalink | Reply

So … how was it?

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• #### elkement 6:18 pm on Wednesday, 11 December, 2013 Permalink | Reply

Thanks for the reminder – I would have missed it again :-) One hour to go in my timezone!

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• #### elkement 6:20 pm on Wednesday, 11 December, 2013 Permalink | Reply

I made a simple math mistake – it is two hours to go ;-)

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• #### LFFL 10:56 pm on Thursday, 5 December, 2013 Permalink | Reply

Lol. Interesting is all I can say. I hate math!

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• #### Joseph Nebus 11:38 pm on Thursday, 5 December, 2013 Permalink | Reply

Aw, math has a lot of fun stuff to play around with, though.

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## Reading the Comics, September 21, 2013

It must have been the summer vacation making comic strip artists take time off from mathematics-themed jokes: there’s a fresh batch of them a mere ten days after my last roundup.

John Zakour and Scott Roberts’s Maria’s Day (September 12) tells the basic “not understanding fractions” joke. I suspect that Zakour and Roberts — who’re pretty well-steeped in nerd culture, as their panel strip Working Daze shows — were summoning one of those warmly familiar old jokes. Well, Sydney Harris got away with the same punch line; why not them?

Brett Koth’s Diamond Lil (September 14) also mentions fractions, but as an example of one of those inexplicably complicated mathematics things that’ll haunt you rather than be useful or interesting or even understandable. I choose not to be offended by this insult of my preferred profession and won’t even point out that Koth totally redrew the panel three times over so it’s not a static shot of immobile talking heads.

• #### elkement 10:30 am on Sunday, 22 September, 2013 Permalink | Reply

My favorite is the weather joke – as I had once really been asked ‘as a science’ expert what those probabilities quoted on forecast websites do mean.

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