Reading the Comics, November 27, 2015: 30,000 Edition

By rights, if this installment has any title it should be “confident ignorance”. That state appears in many of the strips I want to talk about. But according to WordPress, my little mathematics blog here reached its 30,000th page view at long last. This is thanks largely to spillover from The Onion AV Club discovering my humor blog and its talk about the late comic strip Apartment 3-G. But a reader is a reader. And I want to celebrate reaching that big, round number. As I write this I’m at 30,162 page views, because there were a lot of AV Club-related readers.

Bob Weber Jr’s Slylock Fox for the 23rd of November maybe shouldn’t really be here. It’s just a puzzle game that depends on the reader remembering that two rectangles put against the other can be a rectangle again. It also requires deciding whether the frame of the artwork counts as one of the rectangles. The commenters at Comics Kingdom seem unsure whether to count squares as rectangles too. I don’t see any shapes that look more clearly like squares to me. But it’s late in the month and I haven’t had anything with visual appeal in these Reading the Comics installments in a while. Later we can wonder if “counting rectangles in a painting” is the most reasonable way a secret agent has to pass on a number. It reminds me of many, many puzzle mysteries Isaac Asimov wrote that were all about complicated ways secret agents could pass one bit of information on.

Bob Weber Jr’s Slylock Fox for the 23rd of November, 2015. I suppose the artist is lucky they weren’t hiding out at number 38, or she wouldn’t have been able to make such a compellingly symmetric diagram.

Ryan North’s Dinosaur Comics for the 23rd of November is a rerun from goodness knows when it first ran on Quantz.com. It features T Rex thinking about the Turing Test. The test, named for Alan Turing, says that while we may not know what exactly makes up an artificial intelligence, we will know it when we see it. That is the sort of confident ignorance that earned Socrates a living. (I joke. Actually, Socrates was a stonecutter. Who knew, besides the entire philosophy department?) But the idea seems hard to dispute. If we can converse with an entity in such a way that we can’t tell it isn’t human, then, what grounds do we have for saying it isn’t human?

T Rex has an idea that the philosophy department had long ago, of course. That’s to simply “be ready for any possible opening with a reasonable conclusion”. He calls this a matter of brute force. That is, sometimes, a reasonable way to solve problems. It’s got a long and honorable history of use in mathematics. The name suggests some disapproval; it sounds like the way you get a new washing machine through a too-small set of doors. But sometimes the easiest way to find an answer is to just try all the possible outcomes until you find the ones that work, or show that nothing can. If I want to know whether 319 is a prime number, I can try reasoning my way through it. Or I can divide it by all the prime numbers from 2 up to 17. (The square root of 319 is a bit under 18.) Or I could look it up in a table someone already made of the prime numbers less than 400. I know what’s easier, if I have a table already.

The problem with brute force — well, one problem — is that it can be longwinded. We have to break the problem down into each possible different case. Even if each case is easily disposed of, the number of different cases can grow far too fast to be manageable. The amount of working time required, and the amount of storage required, can easily become too much to deal with. Mathematicians, and computer scientists, have a couple approaches for this. One is getting bigger computers with more memory. We might consider this the brute force method to solving the limits of brute force methods.

Or we might try to reduce the number of possible cases, so that less work is needed. Perhaps we can find a line of reasoning that covers many cases. Working out specific cases, as brute force requires, can often give us a hint to what a general proof would look like. Or we can at least get a bunch of cases dealt with, even if we can’t get them all done.

Jim Unger’s Herman rerun for the 23rd of November turns confident ignorance into a running theme for this essay’s comic strips.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 24th of November has a similar confient ignorance. This time it’s of the orders of magnitude that separate billions from trillions. I wanted to try passing off some line about how there can be contexts where it doesn’t much matter whether a billion or a trillion is at stake. But I can’t think of one that makes sense for the Man At The Business Company Office setting.

Reza Farazmand’s Poorly Drawn Lines for the 25th of November is built on the same confusion about the orders of magnitude that Bottomliners is. In this case it’s ants that aren’t sure about how big millions are, so their confusion seems more natural.

The ants are also engaged in a fun sort of recreational mathematics: can you estimate something from little information? You’ve done that right, typically, if you get the size of the number about right. That it should be millions rather than thousands or hundreds of millions; that there should be something like ten rather than ten thousand. These kinds of problems are often called Fermi Problems, after Enrico Fermi. This is the same person the Fermi Paradox is named after, but that’s a different problem. The Fermi Paradox asks if there are extraterrestrial aliens, why we don’t see evidence of them. A Fermi Problem is simpler. Its the iconic example is, “how many professional piano tuners are there in New York?” It’s easy to look up how big is the population of New York. It’s possible to estimate how many pianos there should be for a population that size. Then you can guess how often a piano needs tuning, and therefore, how many full-time piano tuners would be supported by that much piano-tuning demand. And there’s probably not many more professional piano tuners than there’s demand for. (Wikipedia uses Chicago as the example city for this, and asserts the population of Chicago to be nine million people. I will suppose this to be the Chicago metropolitan region, but that still seems high. Wikipedia says that is the rough population of the Chicago metropolitan area, but it’s got a vested interest in saying so.)

Mark Anderson’s Andertoons finally appears on the 27th. Here we combine the rational division of labor with resisting mathematics problems.

Reading the Comics, March 22, 2015: Word Problems Edition

After the flurry of comic strips that did Pi Day jokes last time around, and that one had worked in a March Madness joke, I’d expected there to be at least a couple of mathematically-mind college basketball tournament strips coming up this week. If they did, they didn’t appear on the comics sites I normally read, though. This time around turned out to be much more about word problems and the problem-answerer resisting the actual answering of the word problems. It’s possible that Comic Strip Master Command didn’t notice that this would be the weekend that United States readers would spend the most of their time complaining about how their bracket picks weren’t working right.

Phil Frank and Joe Troise’s The Elderberries (March 17, rerun) mentions sudoku, and how to play it, and also shows off how explaining things really is a pleasure, at least as long as you have someone who wants to know listening to the explanation. The strip’s also made me realize I don’t remember what the Professor’s background was. Certainly anyone of any background might enjoy sudoku puzzles, or at least know them well enough to explain how to do them, though I wonder if there’s not a use of the motif here that “professors are smart people, mathematics-or-logic puzzles require smartness, so professors are skilled at mathematics-or-logic puzzles”. (For what it’s worth, I’m not much on this sort of puzzle, though I believe that just reflects that I don’t care to do them very much, so I don’t have the experience needed to do them impressively well.)

Dan Thompson’s Rip Haywire (March 17) features a word problem as part of an aptitude test. Interesting to me is that the test is a multiple-choice, which means one should be able to pick the right answer without doing the whole multiplication of “3.29 times 6.5”: 3.29 is pretty near 3.30, so the answer will be about 3 times 6.5 plus a tenth of 3 times 6.5. And 3 times 6.5 is going to be 3 times 6 plus 3 times a half, or 18 plus 1.5. So, look for the answer that’s about 19.5 plus 1.95, which will be around 21.45. In particular, look for an answer a little bit less than that (to be exact, 0.01 times 6.5 less than that.) Of course, if the exam-writer was clever, 21.45 was included as a plausible yet incorrect answer, but at least the problem can be worked out in one’s head.

Reading the Comics, July 14, 2012

I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.

A Planet Is Not A Dot

A lot of what I said in describing how we might fall into the Moon, if you and I were in the same room and suddenly the rest of the world stopped existing, was incorrect. That isn’t to say it was wrong or even bad to consider; it just means that the equations that I produced and the numbers that came out from them aren’t exactly what would happen if the sudden-failure-of-planet-Earth case were to happen. I knew the wouldn’t be exactly right going in, which leaves us the question of what I thought I was doing and why I bothered doing it.

The first reason, and the reason why it wasn’t a waste of time to consider these simple approximations of how strongly the Moon is attracting us — how fast we are falling into it, and how fast we would be falling if the Earth weren’t falling into the Moon along with us — is thanks to something which Isaac Asimov perfectly described. In an essay called “The Relativity Of Wrong”, he wrote about — well, the title says it. Ideas are not just right or wrong; they can be wrong by differing amounts, and can be wrong by such a tiny amount that it isn’t worth the complications to get it exactly right. Probably the most familiar example is the flatness of the Earth. To model the globe, or a large nation, the idea that the Earth is nearly flat is sufficiently wrong as to produce measurable, important errors where plots of land are justifiably claimed by multiple owners, maybe from multiple governments, or aren’t claimed at all and form the basis for nowhere towns in which mild fantasy or comic stories can be set. But if one wants to draw a map of the town, or of one’s own property, the curvature of the Earth is not worth considering. We can pretend the Earth is flat and get our work done a lot sooner. Other sources of error will mess up the precise result before that does.

Bases For Comparison

Back to the theme of divisibility of numbers. Since we have the idea of writing numbers with a small set of digits, and with the place of those digits carrying information about how big the number is, we can think about what’s implied by that information.

In the number 222, the first two is matched to blocks (hundreds) that are ten times as large as those for the second two (tens), and the second two is matched to units (tens) which are ten times as large as those for the third two (units). It is now extremely rare to have the size of those blocks differ from one place to the next; that is, a number before the initial two here we take without needing it made explicit to represent ten times that hundreds unit, and a number after the final two (and therefore after the decimal point) would represent units which are one-tenth that of the final two’s size.

It has also become extremely rare for the relationship between blocks to be anything but a factor of ten, with two exceptions which I’ll mention next paragraph. The only block other than those with common use which comes to my mind is the sixty-to-one division of hours or degrees into minutes, and then of minutes into seconds. Even there the division of degrees of arc into minutes and seconds might be obsolete, as it’s so much easier on the computer to enter a latitude and longitude with decimals instead. So blocks of ten, decimals, it is, or in the way actual people speak of such things, a number written in base ten.