Two math anthologies.

My latest Insider post covered scouting notes on Danny Salazar, Kendall Graveman, and others from that same game. My weekly Klawchat transcript is up, and I have a new boardgame review over at Paste for the Kennerspiel des Jahres-nominated strategy game Rococo.

My friend Steve is quite familiar with my affinity for just about all things math – we first met in math class in seventh grade – and for Christmas this year bought me a pair of popular math texts, one new and one classic. (I bought him a lot of tea, as he consumes it even faster than I do.) Both were collections of short pieces, with the unevenness that comes with such an anthology, but with high points making both books well worth reading.

The new title was The Best Writing on Mathematics 2014, a book that opens with a sort of dry exhortation on the apparently declining interest in math among students, a theme revisited later in the book, although I think a large part of that is a function of how we teach math in the United States – something that is in and of itself the subject of a separate essay. The separation of abstract math from its practical uses will only sit well with students who are naturally able to deal with math’s abstractions, to hear the music in numbers and formulas, to understand topics like calculus on an intuitive level; modern American instruction tends to make the majority of students, those who don’t grasp this material as quickly, feel less able or competent in the subject. Math anxiety, the subject of so-and-so’s column, isn’t an innate medical condition like anxiety disorder; it is created by teachers and curricula that quickly tell students they’re just not good enough at this stuff.

I enjoy abstract math – one of the best books I’ve ever read was on the highly abstruse Riemann Hypothesis, called Prime Obsession, one of the great unsolved problems in mathematics and one without any apparent practical applications. Yet I also enjoy writing on the pervasive uses of math in other fields, from physics to probability. One of the best essays in the book, and unfortunately one of the shortest, is from game designer and engineer Soren Johnson, who discusses the uses of probability and controlled randomness in creating successful games, specifically citing the random component in Settlers of Catan that has diminished its standing among the most hardcore boardgaming segment that prefers the less random and more complex style of games like Puerto Rico or Agricola. (My issue with Settlers isn’t the randomness but the length of the games. It’s still a classic and one of the best light-strategy games ever created.)

There are several pieces built around randomness, including a high-level essay from Charles Seife (author of Zero, which I enjoyed, as well as Proofiness: How You’re Being Fooled by the Numbers, which is on my to-be-read shelf) on the nature of randomness and our inability to understand it, and another essay on the power of the random in musical compositions. The essay by Prakash Gorrochum titled “Errors of Probability in Historical Context” should be required reading for journalists everywhere, covering the gambler’s fallacy, reasoning on the mean, and misunderstanding statistical independence (Bayes’ theorem). One essay tackles the problem of the Jordan Curve – defined a simple closed curve in a plane or planar region, dividing it into exactly two parts, thus never crossing itself – and its applicability to some amazing works of art. I alluded to the closing essay on Twitter the other day; it discusses the proposed solution to the abc problem, for which the alleged solver had to invent a whole new kind of mathematics, which means that only a few dozen people in the world might be able to interpret his proof, let alone test or critique it.

The selection of titles seems idiosyncratic, as some have very little to do with math proper, such as the dreadful essay on various ancient tools and devices used for mathematical calculations, or the too-lengthy chapter disproving the contention that ancient Celts in modern-day Scotland knew and understood the features of the five regular polyhedra a millennium before anyone else seemed to catch on. The collection ends on several high notes, however, including Gorrochum’s essay (which you can read in its entirety online) and that abc problem/solution story, the latter of which is almost creepy because of how bizarre the whole backstory is. I’d never heard of this series before Steve bought me this book but the handful of strong essays in it made it a great read.

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The other book Steve bought for me was Martin Gardner’s collection Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi, which is the first in the series of books anthologizing Gardner’s many essays on popular math from his long-running column in Scientific American. Gardner’s writing exudes his sheer joy in math itself, yet most of these essays explore tangible questions even when they’re as useless as the hexaflexagons of the book’s title. Those peculiar shapes are formed by folding one or more strips of paper according to prescribed patterns to form regular polygons, in this specific case hexagons, that can be pushed and folded to reveal hidden sides and features, a chance discovery explored by some very famous names from math and science (Richard Feynman was among their earliest practicioners). A similar vein runs through his essays on the games Hex, invented independently by Nobel Prize-winning game theorist John Nash and Danish polymath Piet Hein, a totally nonrandom game of tile placement on a rhomboid board of hexagonal spaces where each player is trying to complete an unbroken chain from one of his sides to the side facing it. The game can’t end in a draw, and on smaller boards there are unbeatable strategies for whichever player goes first, inspiring much mathematical hand-wringing over the search for algorithms to predict perfect plays.

Other essays pose specific logic and math puzzles to the reader, many of which can be worked out in your head (and are much worse if you start putting pencil to paper). He explores the history of the “boss puzzle,” also known as the 15-14 puzzle, where the player is presented with a 4×4 grid with 15 numbered tiles on it, all in ordered rows but with the final row going 13-15-14. The player is told to use the single open space to move tiles around to get all fifteen tiles into the proper order. (The puzzle is unsolvable because it has a parity of one, meaning there’s a single tile displacement.) He also discusses several popular math and logic paradoxes, such as the division of a rectangle into several triangular pieces, then the reassembly that makes it appear that some surface area has disappeared. (It hasn’t.) They’re fun to puzzle over for their own sake, but the sleights of math used here or in the card tricks Gardner describes in another chapter expose holes in our critical thinking processes – ways we can miss obvious fallacies because something looks or sounds “right” on its face.

The chapter that might be most familiar to readers in subject matter discusses the Birthday Paradox. Given a group of 24 people selected at random, what are the odds that at least two members of the group have the same birthday? The answer is better than one half, which seems at first rather hard to believe as there are 365 days in the calendar. The odds that the first two people don’t have the same birthday are 364/365; the odds that the third person added to the group won’t have the same birthday as either of the first two are 363/365; and so on. The probability that n people won’t have the same birthday is thus a product of all of these individual probabilities (the formula is here); the 23rd person added to the group drops the probability that there is no birthday match under 0.5. It seems intuitively incorrect that just 23* people could suffice to raise the odds of a match over 50% when the number of dates is 365, and there are many methods of figuring these odds incorrectly, such as multiplying the apparent odds of a match (2/365 * 3/365 * 4/365…) or adding up the same fractions. Gardner’s explanations of such paradoxes were both clear and a pleasure to read, which is why so much of his work remains in print a half-century after he started writing. The chapter doesn’t discuss the Monty Hall problem, but describes a similar question around hands of cards that might illuminate that more famous question if you’ve struggled to understand its explanation.

* Gardner’s chapter uses 24 as the threshold, but I’m pretty sure it’s 23, using both methods to calculate the odds. If anyone can show the magic number is 24, please post it in the comments, because then I’ve got this wrong too.

Gardner also discusses magic squares, which seem to me to be the logical ancestors of the much simpler sudoku; the Tower of Hanoi problem; and some topological oddities that arise from manipulations of a Mobius strip (or two of them together). He gets a little ahead of himself, perhaps a function of the space limitations of the print world, in the chapter on fallacies by presenting two pure math fallacies without explaining exactly why they fail. Both revolve around attempts to prove that two unequal entities are equal; one fails through a disguised attempt to divide by zero, the other by treating i as a real (rather than imaginary) number, but I wouldn’t assume either fallacy was obvious by the way Gardner presents them.

I first encountered Gardner’s work in junior high school through his now out-of-print Aha! Gotcha! book, which took a similar approach to math tricks and paradoxes but was aimed at a younger audience; Hexaflexagons is the more grown-up version, aimed at math-loving kids like me who just refused to grow up.