Today’s installment of Reading the Comics has a bunch of strips that seem to touch on human psychology. That properly could always be said; what we know of mathematics is what humans have thought about. But sometimes the link between a mathematical topic and human psychology is more obvious.
Wes Molebash’s Molebashed (August 5) is a reminder that one can find interesting mental arithmetic problems anywhere. This does not mean they’re always welcome. But they can still be fun to do. For example while walking through a parking lot I noticed another state’s license plate and wondered how many six-letter combinations you could get. Well, that’s 266, obviously, but how big a number is that? Working out that sort of thing is why people have to repeat what they’re saying to me.
Mark Pett’s Mr Lowe (August 6, rerun from sometime in 2000) has a student complaining the mathematics books are two years old. The complaint is absurd but also kind of sensible. Mathematical truths are immortal, or at least they are once they’re proven. Whether something is proven is, to an extent, a cultural construct: it takes an incredible load of work to actually prove something rigorously with every step in place. We usually are content if we show enough reasoning to be confident that every step could be filled in if need be. More a matter of taste, though, is whether these truths are interesting. As an example, I mentioned just a few posts ago the versine function. There are computations which, if you’re doing them by hand, are best done with the versine function or a table of values of the versine function. But we don’t need to do that sort of work anymore, and the versine function has plunged into obscurity. Nothing that we knew about versines has stopped being true. But we’d be eccentric, at least, to make it a part of a trigonometry course in the way someone 150 years ago might have. Mathematics is not culturally neutral. Few interesting things are.
Kieran Meehan’s Pros and Cons (August 7) is a probability joke. As often happens, the probability joke is built on the gambler’s fallacy. The fallacy in this case is the supposition that if one hasn’t had an accident in an unusually long while, then one must be due. Properly, though, we should ask whether accidents are independent events. If they are independent — if the chance of having an accident does not change based on whether you had an accident yesterday, or in the past week, or in the past year, or so on — then it’s silly to say you’re “due” for one. If your rate of accidents is lower than expected, you’re just having a lucky streak is all. However, I can imagine the chance of having an accident not being independent. I can imagine going a long time without accidents making someone careless about normal risks, or inexperienced in judging new ones, and that might make an accident more likely that one expects. It’s difficult to answer a probability question without understanding human psychology.
John Graziano’s Ripley’s Believe It or Not (August 7) claims there are over 26,000 possible outcomes of tic-tac-toe. I think the claim is poorly worded, though. If by an “outcome” of a tic-tac-toe game we mean the arrangement of X and O marks then there are at most 19,683 outcomes — each of the nine cells contains an X, an O, or is left blank. That’s an overestimate, though. A grid of nine X’s can’t be a legal outcome of a game, after all; nor can one that has two X’s, one O, and six blank spaces. There have to be at least three X’s and at least two O’s, and at most four blank spaces. The number of X’s can be equal to or one greater than the number of O’s. This removes a lot of possibilities.
I think what Graziano’s Ripley’s wants to claim is there are over 26,000 different tic-tac-toe games. This I can more readily believe. There are 9 possible spaces the first player can take on the first turn; there are 8 choices for the second player on the first turn. There are 7 choices for the first player on the second turn; there are 6 choices for the second player on the second turn. And so on. So there are at most 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 possible ways to play out the game; that’s a total of 362,880 possibilities. But not all those possibilities are needed. If a game’s won after two and a half turns, it stops, and a lot of possible continuations are voided. I don’t have a good estimate of how many those are. And we might choose to rule out symmetries. The game in which X fills out the top row while O tries the center isn’t really different to the game in which X fills out the bottom row while O takes the center. For that matter, it’s not different to the one where X fills in the right column while O fills in the center column. If you don’t count symmetries like this as different games, then we have fewer games altogether. So if that is what Graziano means, then 26,000 may be a fair estimate of tic-tac-toe games.
That, by the way, is the strip that gave me the most to think about of this set.
Rick Kirkman and Jerry Scott’s Baby Blues (August 8) is another installment of Kids Doing Mathematics During Summer Vacation. This is almost the theme of the summer in mathematics comics. Possibly it’s the theme of every summer.
Bill Amend’s FoxTrot (August 9) is one of those odd jokes that also is a pretty good business opportunity. Jason Fox proposes some of the many shapes that could, in principle, hold ice cream. I believe hemispheres at least are available, actually, at least to restaurants. But some of these shapes, such as pyramids or dodecahedrons or such, seem like they could be made and just happen not to have been. (Well, half-dodecahedrons, anyway.) That probably reflects that a cone or similarly narrow-based shape forces more of a given amount of ice cream to overflow the top of the cone, suggesting abundance. Geometric possibilities must give way to making the product look bigger.