Mark Anderson’s Andertoons for the 15th is sort of a wordplay strip, fussing around the connotations of some numbers like 86 and 22 (as in catch-22) to get to a nonsense result.

A couple years back we needed to patch a bunch of weak spots in the roof. We found all the spots that needed shoring up and measured how long they were, and went to buy some wood and get it cut to fit. I turned over the list of sizes and the guy told us we’d have to buy more than one of the standard-size sheets of plywood to do it. I thought, wait, no, that can’t be, and sketched out possible ways to cut the wood and fit pieces together. Finally I concluded that, oh, yes, the guy whose job it was to figure out how much wood was needed for particular tasks knew what he was talking about. His secret? I don’t know. What finally convinced me was adding up the total area of the wood we’d need, and finding that it was more than what one sheet would be.

Dave Blazek’s Loose Parts for the 19th uses a whiteboard full of mathematics as visual shorthand for “some really complicated subject”. It’s a good set of mathematics symbols on the whiteboard. They don’t mean anything in the combination shown, though. It’s just meant to bewilder.

The key is the Mover’s claim that he can look at any amount of stuff and tell you whether it fits in the moving bins. Working out something like this is a version of the knapsack problem. The knapsack problem is … well, the problem you imagine it might be, if someone told you “some mathematicians study a thing called the knapsack problem”? That’s about right. Formally, it’s about selecting from a set of things of different value. How hard is it to pick a subset of things with exactly that value? Or find that there is no such subset?

Well, in a sense, not hard at all. You can just keep trying combinations. Eventually you’ll either find a set that works, or you’ll try every possibility and find none of them work. This is known as “exhaustion”, and correctly. If there are ten things, there are 3,628,800 possibilities. Then it gets really bad. If there are twenty things, there are 2,432,902,008,176,640,000 possibilities. Finding the one that works? That could take a while.

So being able to tell whether a collection of things can fit within a particular space? That’s a form of the knapsack problem. Being able to always solve that any faster than just “try out every combination until you find one that works”? That would be incredible. The problem is hard. That’s a technical term. It means what you imagine it means, but more precisely.

So why the mathematician’s response? It’s because the problem of hacking the common Internet security algorithms is also hard. (I am discussing here how difficult hacking would be if the algorithm were implemented perfectly. There are many hacking techniques available because of bugs. Programs are not written perfectly. Compilers do not translate them to computer code perfectly. Computers are not built perfectly. These and more flaws make hacking more possible than it should be.) It’s the same kind of hard as this knapsack problem. I mean “the same” more technically than you might imagine. If you had a method to quickly solve this knapsack problem, then, you could use this to break computer encryption quickly. And, it turns out, vice-versa, so at least there’s some fairness to things. So if the the Mover can, truly, always instantly tell whether a set of things fit in the moving bins, then hacking e-mails should be possible to. The Mover would have to team up with a mathematician who studies computational problems like this. I don’t know how to do it, myself. I think about the how to do this and feel lost, myself.

So is the Mover full of it? Let’s put this more nicely. Is he at least unduly optimistic about his claims?

Nah. What makes the knapsack problem hard is that you have to find a solution that quickly finds answers for every possible set of things. But the Mover doesn’t have to deal with that. Most of the stuff is in boxes. It’s in mostly simple polygonal shapes. There’s not, like, 400 million items, each the size of a Cheerio. The Mover may plausibly have never encountered a set of things to move where he couldn’t tell whether it fits.

And, yes, there’s selection bias. Suppose he declared that no, this load had to fit into two vans. But that actually a sufficiently clever arrangement would have let it fit in one. Who would ever know he was wrong? He’d only ever know his intuition was wrong if he declared something would fit in one van and, in fact, it couldn’t.

Percy Crosby’s Skippy for the 21st is a student-at-the-board problem. It’s using the punch line that “I don’t know” might be a true answer to any problem. There are many real mathematics problems for which nobody really knows an answer.

But Miss Larkin has good advice here. Maybe you don’t know the final answer. But do you know anything? Write it down. It’s good for partial credit, at least. Working out a part of the problem might also be useful, too. Often you can work out how to do a hard problem by looking at a similar but simpler problem. If Skippy is lost at 8 + 4 + 7 + 5, could he do at least 8 + 4 + 7? Could he do 8 + 4? Maybe this wouldn’t help him get to the ultimate answer. Often a difficult problem turns out to be solved by solving a circle of simple problems, that starve out the hard.

Samson’s Dark Side of the Horse for the 21st is the Roman Numerals joke for this time around. I’m not sure this whether this is a repeat. The strip does a lot of Roman Numerals jokes, and counting-sheep jokes.

Our roof patches held up for their need, which was just to last a couple months while we contracted for a replacement roof. And, happily, the roof replacement got done speedily and during a week that did not rain. (Back in grad school the apartment I was in had its roof replaced on a day that, it turns out, would get a spontaneous downpour halfway through. My apartment was on the top floor. This made for an exciting afternoon.)

I don’t actually like it when a split week has so many more comics one day than the next, but I also don’t like splitting across a day if I can avoid it. This week, I had to do a little of both since there were so many comic strips that were relevant enough on the 8th. But they were dominated by the idea of going back to school, yet.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 8th is another back-to-school gag. And it uses arithmetic as the mathematics at its most basic. Arithmetic might not be the most fundamental mathematics, but it does seem to be one of the parts we understand first. It’s probably last to be forgotten even on a long summer break.

Mark Pett’s Mr Lowe rerun for the 8th is built on the familiar old question of why learn arithmetic when there’s computers. Quentin is unconvinced of this as motive for learning long division. I’ll grant the case could be made better. I admit I’m not sure how, though. I think long division is good as a way to teach, especially, the process of estimating and improving estimates of a calculation. There’s a lot of real mathematics in doing that.

Guy Gilchrist’s Nancy for the 8th is another back-to-school strip. Nancy’s faced with “this much math” so close to summer. Her given problem’s a bit of a mess to me. But it’s mostly teaching whether the student’s got the hang of the order of operations. And the instructor clearly hasn’t got the sense right. People can ask whether we should parse “12 divided by 3 times 4” as “(12 divided by 3) times 4” or as “12 divided by (3 times 4)”, and that does make a major difference. Multiplication commutes; you can do it in any order. Division doesn’t. Leaving ambiguous phrasing is the sort of thing you learn, instinctively, to avoid. Nancy would be justified in refusing to do the problem on the grounds that there is no unambiguous way to evaluate it, and that the instructor surely did not mean for her to evaluate it all four different plausible ways.

By the way, I’ve seen going around Normal Person Twitter this week a comment about how they just discovered the division symbol ÷, the obelus, is “just” the fraction bar with dots above and below where the unknown numbers go. I agree this is a great mnemonic for understanding what is being asked for with the symbol. But I see no evidence that this is where the symbol, historically, comes from. We first see ÷ used for division in the writings of Johann Henrich Rahn, in 1659, and the symbol gained popularity particularly when John Pell picked it up nine years later. But it’s not like Rahn invented the symbol out of nowhere; it had been used for subtraction for over 125 years at that point. There were also a good number of writers using : or / or \ for division. There were some people using a center dot before and after a / mark for this, like the % sign fell on its side. That ÷ gained popularity in English and American writing seems to be a quirk of fate, possibly augmented by it being relatively easy to produce on a standard typewriter. (Florian Cajori notes that the National Committee on Mathematical Requirements recommended dropping ÷ altogether in favor of a symbol that actually has use in non-mathematical life, the / mark. The Committee recommended this in 1923, so you see how well the form agenda is doing.)

Mark Leiknes’s Cow and Boy rerun for the 9th only mentions mathematics, and that as a course that Billy would rather be skipping. But I like the comic strip and want to promote its memory as much as possible. It’s a deeply weird thing, because it has something like 400 running jokes, and it’s hard to get into because the first couple times you see a pastoral conversation interrupted by an orca firing a bazooka at a cat-helicopter while a panda brags of blowing up the moon it seems like pure gibberish. If you can get through that, you realize why this is funny.

Dave Blazek’s Loose Parts for the 9th uses chalkboards full of stuff as the sign of a professor doing serious thinking. Mathematics is will-suited for chalkboards, at least in comic strips. It conveys a lot of thought and doesn’t need much preplanning. Although a joke about the difficulties in planning out blackboard use does take that planning. Yes, there is a particular pain that comes from having more stuff to write down in the quick yet easily collaborative medium of the chalkboard than there is board space to write.

Brian Basset’s Red and Rover for the 9th also really only casually mentions mathematics. But it’s another comic strip I like a good deal so would like to talk up. Anyway, it does show Red discovering he doesn’t mind doing mathematics when he sees the use.

Last week started off going like Gangbusters, a phrase I think that’s too old-fashioned for my father to say but that I’ve picked up because I like listening to old-time radio and, you know, Gangbusters really does get going like that. Give it a try sometime, if you’re open to that old-fashioned sort of narrative style and blatant FBI agitprop. You might want to turn the volume down a little before you do. It slowed down the second half of the week, which is mostly fine as I’d had other things taking up my time. Let me finish off last week and hope there’s a good set of comics to review for next Sunday and maybe Tuesday.

Ted Shearer’s Quincy for the 4th of May, 1978 was rerun the 28th of June. It’s got the form of your student-resisting-the-word-problem joke. And mixes in a bit of percentages which is all the excuse I need to include it here. That and how Shearer uses halftone screening. It’s also a useful reminder of how many of our economic problems could be solved quickly if poor people got more money.

Olivia Walch’s Imogen Quest for the 28th features Gottfried Leibniz — missing his birthday by three days, incidentally — and speaks of the priority dispute about the invention of calculus. I’m not sure there is much serious questioning anymore about Leibniz’s contributions to mathematics. I think they might be even more strongly appreciated these days than they ever used to be, as people learn more about his work in computing machines and the attempt to automate calculation.

Mark Anderson’s Andertoons for the 28th is our soothing, familiar Andertoons for this essay. I remember in learning about equivalent forms of fractions wondering why anyone cared about reducing them. If two things have the same meaning, why do we need to go further? There are a couple answers. One is that it’s easier on us to understand a quantity if it’s a shorter, more familiar form. has a meaning that just does not. And another is that we often want to know whether two things are equivalent, or close. Is \frac{1147}{1517} more or less than ? Good luck eyeballing that.

And we learn, later on, that a lot of mathematics is about finding different ways to write the same thing. Each way has its uses. Sometimes a slightly more complicated way to write a thing makes proving something easier. There’s about two solids months of Real Analysis, for example, where you keep on writing that and this “adding zero” turns out to make proofs possible. Even easy.

Mark Tatulli’s Heart of the City remains on my watch-with-caution list as the Math Camp story continues. But the strip from the 28th tickles me with the idea of crossing mathematics camp with Pinocchio‘s Pleasure Island. I’m imagining something where Heart starts laughing at something and ends up turning into something from Donald Duck’s Mathmagic land.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 1st of July is your traditional “mathematics is something hard” joke. I have the feeling it’s a rerun, but I lack the emotional investment in whether it is a rerun to check. The joke’s comfortable and familiar as it is, anyway.

It turns out last Saturday only had the one comic strip that was even remotely on point for me. And it wasn’t very on point either, but since it’s one of the Creators.com strips I’ve got the strip to show. That’s enough for me.

Ruben Bolling’s Super-Fun-Pak Comix for the 8th is an installation of They Came From The Third Dimension. “Dimension” is one of those oft-used words that’s come loose of any technical definition. We use it in mathematics all the time, at least once we get into Introduction to Linear Algebra. That’s the course that talks about how blocks of space can be stretched and squashed and twisted into each other. You’d expect this to be a warmup act to geometry, and I guess it’s relevant. But where it really pays off is in studying differential equations and how systems of stuff changes over time. When you get introduced to dimensions in linear algebra they describe degrees of freedom, or how much information you need about a problem to pin down exactly one solution.

It does give mathematicians cause to talk about “dimensions of space”, though, and these are intuitively at least like the two- and three-dimensional spaces that, you know, stuff moves in. That there could be more dimensions of space, ordinarily inaccessible, is an old enough idea we don’t really notice it. Perhaps it’s hidden somewhere too.

Amanda El-Dweek’s Amanda the Great of the 9th started a story with the adult Becky needing to take a mathematics qualification exam. It seems to be prerequisite to enrolling in some new classes. It’s a typical set of mathematics anxiety jokes in the service of a story comic. One might tsk Becky for going through university without ever having a proper mathematics class, but then, I got through university without ever taking a philosophy class that really challenged me. Not that I didn’t take the classes seriously, but that I took stuff like Intro to Logic that I was already conversant in. We all cut corners. It’s a shame not to use chances like that, but there’s always so much to do.

Mark Anderson’s Andertoons for the 10th relieves the worry that Mark Anderson’s Andertoons might not have got in an appearance this week. It’s your common kid at the chalkboard sort of problem, this one a kid with no idea where to put the decimal. As always happens I’m sympathetic. The rules about where to move decimals in this kind of multiplication come out really weird if the last digit, or worse, digits in the product are zeroes.

Mel Henze’s Gentle Creatures is in reruns. The strip from the 10th is part of a story I’m so sure I’ve featured here before that I’m not even going to look up when it aired. But it uses your standard story problem to stand in for science-fiction gadget mathematics calculation.

Dave Blazek’s Loose Parts for the 12th is the natural extension of sleep numbers. Yes, I’m relieved to see Dave Blazek’s Loose Parts around here again too. Feels weird when it’s not.

John Deering’s Strange Brew for the 13th is a “math club” joke featuring horses. Oh, it’s a big silly one, but who doesn’t like those too?

Dan Thompson’s Brevity for the 14th is one of the small set of punning jokes you can make using mathematician names. Good for the wall of a mathematics teacher’s classroom.

Shaenon K Garrity and Jefferey C Wells’s Skin Horse for the 14th is set inside a virtual reality game. (This is why there’s talk about duplicating objects.) Within the game, the characters are playing that game where you start with a set number (in this case 20) tokens and take turn removing a couple of them. The “rigged” part of it is that the house can, by perfect play, force a win every time. It’s a bit of game theory that creeps into recreational mathematics books and that I imagine is imprinted in the minds of people who grow up to design games.

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

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

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

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

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

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

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

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

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

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

Jimmy Johnson’s Arlo and Janis for the 18th capped off a week of trying to explain some point about the compression and dilution of time in comic strips. Comic strips use space and time to suggest more complete stories than they actually tell. They’re much like every other medium in this way. So, to symbolize deep thinking on a subject we get once again a panel full of mathematics. Yes, I noticed the misquoting of “E = mc^{2}” there. I am not sure what Arlo means by “Remember the boat?” although thinking on it I think he did have a running daydream about living on a boat. Arlo and Janis isn’t a strongly story-driven comic strip, but Johnson is comfortable letting the setting evolve. Perhaps all this is forewarning that we’re going to jump ahead to a time in Arlo’s life when he has, or has had, a boat. I don’t know.

So now let me get to the other half of last week’s comics. Also, not to spoil things, but this coming week is looking pretty busy so I may have anothe split-week Reading the Comics coming up. The shocking thing this time is that the Houston Chronicle has announced it’s discontinuing its comics page. I don’t know why; I suppose because they’re fed up with people coming loyally to a daily feature. I will try finding alternate sources for the things I had still been reading there, but don’t know if I’ll make it.

I’m saddened by this. Back in the 90s comics were just coming onto the Internet. The Houston Chronicle was one of a couple newspapers that knew what to do with them. It, and the Philadelphia Inquirer and the San Jose Mercury-News, had exactly what we wanted in comics: you could make a page up of all the strips you wanted to read, and read them on a single page. You could even go backwards day by day in case you missed some. The Philadelphia Inquirer was the only page that let you put the comics in the order you wanted, as opposed to alphabetical order by title. But if you were unafraid of opening up URLs you could reorder the Houston Chronicle page you built too.

And those have all faded away. In the interests of whatever interest is served by web site redesigns all these papers did away with their user-buildable comics pages. The Chronicle was the last holdout, but even they abolished their pages a few years ago, with a promise for a while that they’d have a replacement comics-page scheme up soon. It never came and now, I suppose, never will.

Most of the newspapers’ sites had become redundant anyway. Comics Kingdom and GoComics.com offer user-customizable comics pages, with a subscription model that makes it clear that money ought to be going to the cartoonists. I still had the Chronicle for a few holdouts, like Joe Martin’s strips or the Jumble feature. And from that inertia that attaches to long-running Internet associations.

So among the other things January 2017 takes away from us, it is taking the last, faded echo of the days in the 1990s when newspapers saw comics as awesome things that could be made part of their sites.

Lorie Ransom’s The Daily Drawing for the 11th is almost but not quite the anthropomorphized-numerals joke for this installment. It’s certainly the most numerical duck content I’ve got on record.

Tom II Wilson’s Ziggy for the 11th is an Early Pi Day joke. Cosmically there isn’t any reason we couldn’t use π in take-a-number dispensers, after all. Their purpose is to give us some certain order in which to do things. We could use any set of numbers which can be put in order. So the counting numbers work. So do the integers. And the real numbers. But practicality comes into it. Most people have probably heard that π is a bit bigger than 3 and a fair bit smaller than 4. But pity the two people who drew and figuring out who gets to go first. Still, I won’t be surprised if some mathematics-oriented place uses a gimmick like this, albeit with numbers that couldn’t be confused. At least not confused by people who go to mathematics-oriented places. That would be for fun rather than cake.

I can’t promise that the Jumble for the 11this the last one I’ll ever feature here. I might find where David L Hoyt and Jeff Knurek keep a linkable reference to their strips and point to them. But just in case of the worst here’s an abacus gag for you to work on.

Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries for the 12th is, I have to point out, a rerun. So if you’re trying to do the puzzle the reference to “the number of the last president” isn’t what you’re thinking of. It is an example of the conflation of intelligence with skill at arithmetic. It’s also an example the conflation of intelligence with a mastery of trivia. But I think it leans on arithmetic more. I am not sure when this strip first appeared. “The last president” might have been Bill Clinton (42) or George W Bush (43). But this means we’re taking the square root of either 33 or 34. And there’s no doing that in your head. The square root of a whole number is either a whole number — the way the square root of 36 is — or else it’s an irrational number. You can work out the square root of a non-perfect-square by hand. But it’s boring and it’s worse than just writing “” or “”. Except in figuring out if that number is larger than or smaller than five or six. It’s good for that.

Dave Blazek’s Loose Parts for the 13th is the actuary joke for this installment. Actuarial studies are built on one of the great wonders of statistics: that it is possible to predict how often things will happen. They can happen to a population, as in forecasts of how many people will be in traffic accidents or fires or will lose their jobs or will move to a new city. We may have no idea to whom any of these will happen, and they may have no way of guessing, but the enormous number of people and great number of things that can combine to make a predictable state of affairs. I suppose it’s imaginable that a group could study its dynamics well enough to identify who screws up the most and most seriously. So they might be able to say what the odds are it is his fault. But I imagine in practice it’s too difficult to define screw-ups or to assign fault consistently enough to get the data needed.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is another multiverse strip, echoing the Dinosaur Comics I featured here Sunday. I’ll echo my comments then. If there is a multiverse — again, there is not evidence for this — then there may be infinitely many versions of every book of the Bible. This suggests, but it does not mandate, that there should be every possible incarnation of the Bible. And a multiverse might be a spendthrift option anyway. Just allow for enough editions, and the chance that any of them will have a misprint at any word or phrase, and we can eventually get infinitely many versions of every book of the Bible. If we wait long enough.

I might turn this into a regular feature. A couple more comic strips, all this week on gocomics.com, ran nice little mathematically-linked themes, and as far as I can tell I’m the only one who reads any of them so I might spread the word some.

Grant Snider’s Incidental Comics returns again with the Triangle Circus, in his strip of the 12th of March. This strip is also noteworthy for making use of “scalene”, which is also known as “that other kind of triangle” which nobody can remember the name for. (He’s had several other math-panel comic strips, and I really enjoy how full he stuffs the panels with drawings and jokes in most strips.)

Dave Blazek’s Loose Parts from the 15th of March puts up a version of the Cretan Paradox that amused me much more than I thought it would at first glance. I kept thinking back about it and grinning. (This blurs the line between mathematics and philosophy, but those lines have always been pretty blurred, particularly in the hotly disputed territory of Logic.)

Bud Fisher’s Mutt and Jeff is in reruns, of course, and shows a random scattering of strips from the 1930s and 1940s and, really, seem to show off how far we’ve advanced in efficiency in setup-and-punchline since the early 20th century. But the rerun from the 17th of March (I can’t make out the publication date, although the figures in the article probably could be used to guess at the year) does demonstrate the sort of estimating-a-value that’s good mental exercise too.

I note that where Mutt divides 150,000,000 into 700,000,000 I would instead have divided the 150 million into 750,000,000, because that’s a much easier problem, and he just wanted an estimate anyway. It would get to the estimate of ten cents a week later in the word balloon more easily that way, too. But making estimates and approximations are in part an art. But I don’t think of anything that gives me 2/3ds of a cent as an intermediate value on the way to what I want as being a good approximation.