Posts Tagged ‘recreational mathematics’

Front cover of The Simpsons and Their Mathematical Secrets by Simon SinghThe Simpsons and Their Mathematical Secrets, Simon Singh (Bloomsbury 2013)

I don’t like The Simpsons and I don’t think Simon Singh is a very good writer. But there is some interesting maths in this book. As the Emperor Vespasian said when criticized for taxing urinals in Rome: Pecunia non olet – “money doesn’t smell”. And simple sources can yield riches in other ways. There’s a good example of that in chapter 9 of this book, “To Infinity and Beyond”, where Singh looks at the mathematics of pancake-sorting. It was first discussed in 1975 by the geometer Jacob E. Goodman of the City College of New York. Suppose there’s a pile of pancakes of different sizes. You can insert a spatula at any point in the pile and flip the block of pancakes above it. Goodman posed this question about sorting the pancakes into order of size:

If there are n pancakes, what is the maximum number of flips (as a function of n) that I will ever have to use to rearrange them? (ch. 9, pg. 110)

It sounds simple, but isn’t. As the pile gets higher, the problem gets harder. The answer is 20 flips for 18 pancakes and 22 flips for 19. And 20 pancakes? Surprisingly, mathematicians don’t know: “nobody has been able to sidestep the brute computational approach by finding a clever equation that predicts pancake numbers”. The best mathematicians can do is find the upper limit: pancake(n) < (5n + 5)/3 flips.

This limit was proved in a paper “co-authored by William H. Gates and Christos H. Papadimitriou” in 1979 (pg. 112). The first co-author is better known now as Bill Gates of Microsoft. The Simpsons enter the story because David S. Cohen, a writer for the series, extended the problem in a mathematical paper published in 1995: the pancakes don’t just come in different sizes, they’re burnt on one side and have to be flipped both in order of size and with the burnt side down. Now the number of flips is “between 3n/2 and 2n – 2” (pg. 113). The source of the problem may seem trivial, but the maths of the solution isn’t. Pancake-flipping has important parallels with “rearranging data” in computer science.

Cohen has degrees in both computer science and physics, but his expertise isn’t unique: “the writing team of The Simpsons have equally remarkable backgrounds in mathematical subjects” (ch. 0 (sic), “The Truth about the Simpsons”, pg. 3). They have degrees and doctorates in tough subjects from colleges like Harvard, Berkeley and Princeton. And they’ve been engaged, according to Cohen, in a “decades-long conspiracy to secretly educate cartoon viewers” (back cover). They haven’t had much success with that, but they’ve succeeded in other ways: TV is no good at education, but very good at propaganda and manipulation. That’s one reason I dislike The Simpsons, which is obviously inspired by cultural Marxism, despite its occasional un-PC jokes. Another reason is that I think the characters and colours are ugly and dispiriting. Or is that cultural Marxism again? But I have to admit that the series is cleverly done. To appeal to so many people for so long takes skill, but explicit maths has been low in the mix.

It had to be, because The Simpsons wouldn’t have been successful otherwise. It has a lot of stupid fans and stupid people aren’t interested in Fermat’s Last Theorem, strategies for rock-scissors-paper or equations for pancake-numbers. That’s why you need to freeze the frame to find a lot of the explicit maths in The Simpsons. Or you did before Singh wrote this book and froze the frames for you. The implicit maths in The Simpsons is everywhere, but that’s because maths is everything, including an ugly cartoon and its science-fiction offshot. Singh discusses Futurama too and the “taxi-cab numbers” inspired by the Indian mathematician Srinivasa Ramanujan (1887-1920). I’ve never seen Futurama and I wish I could say the same of The Simpsons. I certainly hope I never see it again. But it’s an important programme and this is an interesting book.

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How Many Socks Make a Pair? Surprisingly Interesting Everyday Maths, Rob Eastaway (2008)

I’ve been returning to this book with pleasure and profit for over a year now: it doesn’t just interest, it informs and enlightens too. Unlike Ian Stewart’s The Mathematics of Life (2011), which promised much and delivered little, it seems simple but points to the profound. Maths is like that: it’s a mansion with many rooms or a mountain with gentle slopes and sheer cliffs or an ocean with shallows and abysses. Infinitely many rooms, in fact, infinitely high cliffs, and infinitely deep abysses. Maths is wider than the world because it is the foundation for all actual and possible worlds and is perhaps, at the most fundamental level, the substance of all actual and possible worlds. Some of the topics introduced here, like fractals, probability, and the Fibonacci sequence, lead on to very difficult and important mathematics, but both intelligent children and amateur adults should be able to take the first steps towards the peaks, where problems wait that are still challenging and defeating professional mathematics. It’s a book that has a P, please, Rob: it discusses puzzles, paradoxes, pranks, playfulness, penney ante, Pythagoras’ theorem, and Pascal’s triangle. Plus the palindromic performativity of 196 – or rather, the non-palindromic. If you reverse and add a number like 59 or 382, you soon arrive at a palindrome, or a number that reads the same in both directions.

Despite being a lot smaller than 382, 89 takes longer, requiring 24 reversals-and-additions. 1,186,060,307,891,929,990, on the other hand, takes 281 rev-adds. And 196? It hasn’t produced a palindrome yet, despite having a lot of computer time and power thrown at it: Eastaway notes that “it is the smallest of many numbers that are now thought to be ‘unpalindromable’” (pg. 101). In base ten, anyway: in other bases, 196 quickly produces a palindrome. That’s not something noted here, but it would be a much longer book if it stopped to follow every thread. In fact, it would be infinitely long, like the book in Jorge Luis Borges’ story “The Book of Sand” (“El libro de arena”), or would take infinitely long to write. But that’s one of the things I like about this book: it doesn’t lay down the law, it leads you down the lane and then gives you the chance to explore further for yourself. You can expand and adapt the maths here to your heart’s content and for once the hyperbole on the back-cover isn’t misleading: “a witty book that provokes the imagination” is the quote from The Times, while the London Maths Society said that it “exudes a friendly charm that is hard to resist.” I agree and I wish more young males were reading books like this and looking at less porn. But porn, like everything else, is under the sway of Mathematica, the Magistra Mundi, or Mistress of the World, and if you’re like me How Many Socks…? may even make you feel guilty about neglecting the Mistress. I know that I should put more effort into understanding some of the topics it covers, like “Calculating without a calculator” in chapter 2. But maths is like a endless box of chocolates: there’s also something else to sample. To taste the magic, try this book.

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