I don’t ever try speaking for Comic Strip Master Command, and it almost never speaks to me, but it does seem like this week’s strips mentioning mathematical themes was trying to stick to the classic subjects: anthropomorphized numbers, word problems, ways to measure time and space, under-defined probability questions, and sudoku. It feels almost like a reunion weekend to have all these topics come together.
Dan Thompson’s Brevity (November 23) is a return to the world-of-anthropomorphic-numbers kind of joke, and a pun on the arithmetic mean, which is after all the statistic which most lends itself to puns, just edging out the “range” and the “single-factor ANOVA F-Test”.
Phil Frank Joe Troise’s The Elderberries (November 23, rerun) brings out word problem humor, using train-leaves-the-station humor as a representative of the kinds of thinking academics do. Nagging slightly at me is that I think the strip had established the Professor as one of philosophy and while it’s certainly not unreasonable for a philosopher to be interested in mathematics I wouldn’t expect this kind of mathematics to strike him as very interesting. But then there is the need to get the idea across in two panels, too.
Jonathan Lemon’s Rabbits Against Magic (November 25) brings up a way of identifying the time — “half seven” — which recalls one of my earliest essays around here, “How Many Numbers Have We Named?”, because the construction is one that I find charming and that was glad to hear was still current. “Half seven” strikes me as similar in construction to saying a number as “five and twenty” instead of “twenty-five”, although I’m ignorant as to whether the actually is any similarity.
Scott Hilburn’s The Argyle Sweater (November 26) brings out a joke that I thought had faded out back around, oh, 1978, when the United States decided it wasn’t going to try converting to metric after all, now that we had two-liter bottles of soda. The curious thing about this sort of hyperconversion (it’s surely a satiric cousin to the hypercorrection that makes people mangle a sentence in the misguided hope of perfecting it) — besides that the “yard” in Scotland Yard is obviously not a unit of measure — is the notion that it’d be necessary to update idiomatic references that contain old-fashioned units of measurement. Part of what makes idioms anything interesting is that they can be old-fashioned while still making as much sense as possible; “in for a penny, in for a pound” is a sensible thing to say in the United States, where the pound hasn’t been legal tender since 1857; why would (say) “an ounce of prevention is worth a pound of cure” be any different? Other than that it’s about the only joke easily found on the ground once you’ve decided to look for jokes in the “systems of measurement” field.
Mark Heath’s Spot the Frog (November 26, rerun) I’m not sure actually counts as a mathematics joke, although it’s got me intrigued: Surly Toad claims to have a stick in his mouth to use to give the impression of a smile, or 37 (“Sorry, 38”) other facial expressions. The stick’s shown as a bundle of maple twigs, wound tightly together and designed to take shapes easily. This seems to me the kind of thing that’s grown as an application of knot theory, the study of, well, it’s almost right there in the name. Knots, the study of how strings of things can curl over and around and cross themselves (or other strings), seemed for a very long time to be a purely theoretical playground, not least because, to be addressable by theory, the knots had to be made of an imaginary material that could be stretched arbitrarily finely, and could be pushed frictionlessly through it, which allows for good theoretical work but doesn’t act a thing like a shoelace. Then I think everyone was caught by surprise when it turned out the mathematics of these very abstract knots also describe the way proteins and other long molecules fold, and unfold; and from there it’s not too far to discovering wonderful structures that can change almost by magic with slight bits of pressure. (For my money, the most astounding thing about knots is that you can describe thermodynamics — the way heat works — on them, but I’m inclined towards thermodynamic problems.)
Henry Scarpelli and Crag Boldman’s Archie (November 28, rerun) offers an interesting problem: when Veronica was out of town for a week, Archie’s test scores improved. Is there a link? This kind of thing is awfully interesting to study, and awfully difficult to: there’s no way to run a truly controlled experiment to see whether Veronica’s presence affects Archie’s test scores. After all, he never takes the same test twice, even if he re-takes a test on the same subject (and even if the re-test were the exact same questions, he would go into it the second time with relevant experience that he didn’t have the first time). And a couple good test scores might be relevant, or might just be luck, or it might be that something else happened to change that week that we haven’t noticed yet. How can you trace down plausible causal links in a complicated system?
One approach is an experimental design that, at least in the psychology textbooks I’ve read, gets called A-B-A, or A-B-A-B, experiment design: measure whatever it is you’re interested in during a normal time, “A”, before whatever it is whose influence you want to see has taken hold. Then measure it for a time “B” where something has changed, like, Veronica being out of town. Then go back as best as possible to the normal situation, “A” again; and, if your time and research budget allow, going back to another stretch of “B” (and, hey, maybe even “A” again) helps. If there is an influence, it ought to appear sometime after “B” starts, and fade out again after the return to “A”. The more you’re able to replicate this the sounder the evidence for a link is.
(We’re actually in the midst of something like this around home: our pet rabbit was diagnosed with a touch of arthritis in his last checkup, but mildly enough and in a strange place, so we couldn’t tell whether it’s worth putting him on medication. So we got a ten-day prescription and let that run its course and have tried to evaluate whether it’s affected his behavior. This has proved difficult to say because we don’t really have a clear way of measuring his behavior, although we can say that the arthritis medicine is apparently his favorite thing in the world, based on his racing up to take the liquid and his trying to grab it if we don’t feed it to him fast enough.)
Ralph Hagen’s The Barn (November 28) has Rory the sheep wonder about the chances he and Stan the bull should be together in the pasture, given how incredibly vast the universe is. That’s a subtly tricky question to ask, though. If you want to show that everything that ever existed is impossibly unlikely you can work out, say, how many pastures there are on Earth multiply it by an estimate of how many Earth-like planets there likely are in the universe, and take one divided by that number and marvel at Rory’s incredible luck. But that number’s fairly meaningless: among other obvious objections, wouldn’t Rory wonder the same thing if he were in a pasture with Dan the bull instead? And Rory wouldn’t be wondering anything at all if it weren’t for the accident by which he happened to be born; how impossibly unlikely was that? And that Stan was born too? (And, obviously, that all Rory and Stan’s ancestors were born and survived to the age of reproducing?)
Except that in this sort of question we seem to take it for granted, for instance, that all Stan’s ancestors would have done their part by existing and doing their part to bringing Stan around. And we’d take it for granted that the pasture should exist, rather than be a farmhouse or an outlet mall or a rocket base. To come up with odds that mean anything we have to work out what the probability space of all possible relevant outcomes is, and what the set of all conditions that satisfy the concept of “we’re stuck here together in this pasture” is.
Mark Pett’s Lucky Cow (November 28) brings up sudoku puzzles and the mystery of where they come from, exactly. This prompted me to wonder about the mechanics of making sudoku puzzles and while it certainly seems they could be automated pretty well, making your own amounts to just writing the digits one through nine nine times over, and then blanking out squares until the puzzle is hard. A casual search of the net suggests the most popular way of making sure you haven’t blanking out squares so that the puzzle becomes unsolvable (in this case, that there’s two or more puzzles that fit the revealed information) is to let an automated sudoku solver tell you. That’s true enough but I don’t see any mention of any algorithms by which one could check if you’re blanking out a solution-foiling set of squares. I don’t know whether that reflects there being no algorithm for this that’s more efficient than “try out possible solutions”, or just no algorithm being more practical. It’s relatively easy to make a computer try out possible solutions, after all.
A paper published by Mária Ercsey-Ravasz and Zoltán Toroczkai in Nature Scientific Reports in 2012 describes the recasting of the problem of solving sudoku into a deterministic, dynamical system, and matches the difficulty of a sudoku puzzle to chaotic behavior of that system. (If you’re looking at the article and despairing, don’t worry. Go to the ‘Puzzle hardness as transient chaotic dynamics’ section, and read the parts of the sentence that aren’t technical terms.) Ercsey-Ravasz and Toroczkai point out their chaos-theory-based definition of hardness matches pretty well, though not perfectly, the estimates of difficulty provided by sudoku editors and solvers. The most interesting (to me) result they report is that sudoku puzzles which give you the minimum information — 17 or 18 non-blank numbers to start — are generally not the hardest puzzles. 21 or 22 non-blank numbers seem to match the hardest of puzzles, though they point out that difficulty has got to depend on the positioning of the non-blank numbers and not just how many there are.