Ian Stewart

by Professor Stewart's Hoard of Mathematical Treasures

  • On average, how many throws would it take to get the complete works of Shakespeare? You may assume that his works contain 5,000,000 characters, all included in the table. It’s not true, but assume it anyway.

  Answers on page 329

  In 2003, lecturers and students from the University of Plymouth MediaLab tried the experiment with real monkeys - six Celebes crested macaques - and a computer keyboard. The experimental subjects produced five pages of typing, mainly looking like this:

  SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

  and then trashed the keyboard comprehensively.

  The mathematical statement goes back to Émile Borel, in a 1913 paper ‘Statistical mechanics and irreversibility’, and his 1914 book Le Hasard (Chance). The Argentine writer Jorge Luis Borges traced the underlying idea back to Aristotle’s Metaphysics. The Roman orator Cicero, unimpressed by Aristotle’s views, compared the statement to believing that ‘if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the Annals of Ennius. I doubt whether fortune could make a single verse of them.’

  Well, no . . . unless you used a really great quantity.

  Monkeys Against Evolution

  The monkey on the typewriter has been used to attack the theory of evolution.43 Random mutations in DNA are like the monkey. And while it is true that eventually the monkey can type anything, it is also true that it won’t type anything remotely interesting during the lifetime of the universe. Now, a key protein like haemoglobin, which carries oxygen in our blood, is specified by more than 1,700 DNA ‘letters’ A, C, T, G. The chance of this molecule arising by random mutations is so tiny that it might as well be zero. Therefore haemoglobin cannot have evolved, Darwin was wrong, God must have created it, QED.

  This criticism turns out to be facile, and rests on several misconceptions. One is that the haemoglobin molecule is a ‘target’ at which evolution must aim. However, haemoglobin is not the only molecule that could carry oxygen and deliver it where required. Haemoglobin does that job because it has two similar but distinct forms. In one of them, oxygen atoms bind to the four iron atoms in the molecule; in the other, they don’t. The molecule ‘flexes’ slightly from one form to the other. Most of the haemoglobin molecule plays no essential role in this process, although it does provide a suitably flexible scaffolding for the bits that matter. So a huge variety of other molecules could in principle do the same job. Nature evolved one, and that was all it needed. Well, actually it evolved several variants, which if anything helps to support the point I’m making.

  That point alone doesn’t cut the odds down enough, though. The second point is that biological molecules don’t evolve from scratch every time: evolution keeps a living library of molecules, and modifies them or fits them together to build new ones. Indeed, haemoglobin is made from two copies of each of two smaller molecules, alpha and beta units. Moreover, this modular structure helps the combined molecule to flex appropriately.

  A more appropriate analogy, then, equips the monkey with a wordprocessor, not a typewriter, and the wordprocessor has ‘macro’ keys, which can be assigned to reproduce a series of keystrokes. If the monkey creates a macro every time it types a sensible word - analogous to evolution keeping anything that works - then pretty soon the monkey’s computer will build up a dictionary, and can type sequences of words with ease by concentrating on the macro keys. Repeating the process produces sequences of meaningful sentences, and so on. It might not generate Shakespeare, but in a few years, let alone billions, a monkey with macros could put together an article that you could read on the train.

  That said, evolving something to play the role of haemoglobin takes a long time, even when gigantic numbers of molecules play the game in parallel - as they do today, and presumably did in the distant past. It took about 3 billion years for haemoglobin to evolve. However, for much of that time, it wouldn’t have had any useful function - complex creatures able to survive in a toxic oxygen atmosphere did not arrive until 1.5 billion years or so had passed, and blood cells arose a lot later than that - and it turned up fairly rapidly, by geological standards, once the scene was set for it to do something useful. But it did so through a sequence of processes that combined small molecules into bigger ones, then those into bigger ones still. It didn’t just faff around at random hoping to hit the haemoglobin jackpot by choosing the right 1,700 DNA letters.

  Universal Letter of Reference

  Dear Search Committee Chair,

  I am writing this letter for Mr XXXXX, who has applied for a position in your department.

  I should start by saying that I cannot recommend him too highly.

  In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you.

  His dissertation is the sort of work you don’t expect to see these days. It definitely demonstrates his complete capabilities.

  In closing, let me say that you will be fortunate if you can get him to work for you.

  Sincerely,

  A. D. Visor (Prof.)

  From Focus Newsletter, Mathematical Association of America.

  Snakes and Adders

  This is a playable game for two or more players with topological and combinatorial features. It is a slight modification of a game that Larry Black invented in 1960, called the Black Path Game.

  Start by drawing a grid on paper; 8×8 is about right. Draw a cross at top left. Remove the diagonally opposite corner square - I’ll explain why in a moment.

  Starting position for the game.

  The first player draws one of the following symbols in the square next to the + sign, horizontally or vertically:

  Symbols to be drawn.

  Players then take turns to draw one of the three symbols - whichever they prefer - in the unique square that extends the wiggling ‘snake’ started by the first player. The snake can overlap itself at a + symbol.

  State of the game after a few moves.

  The snake is the heavy line.

  Whoever first makes the snake run into the edge of the board, including the indentation at bottom right, loses. The topology of the snake implies that it can’t stop at an interior point of the big square, and it can’t run into a closed loop. So it must eventually terminate at the edge.

  This game is fun to play, and you may wonder what that excised corner square is all about. If you don’t cut out the corner square, but use the full 8×8 board, there is a simple winning strategy for one of the players.

  Who should win, and how?

  Answer on page 330

  Powerful Crossnumber

  Fill in the eight powers.

  Here’s a crossnumber with a difference - I’m not going to give you the clues. But I will tell you that each of the answers (2, 5, 6, 7 across; 1, 2, 3, 4 down) is a power of a whole number, and the answers comprise two squares, one cube, one fifth power, one sixth power, one seventh power, one ninth power and one twelfth power.

  Now, a sixth power is also a cube and a square, because x6 = (x2)3 = (x3)2. To avoid ambiguity, when I say that a solution is some specific power, I mean that it is not also some higher power. And there should be no leading zeros - so 0008, for instance, does not count as the cube of 2.

  Answer on page 331

  Magic Handkerchiefs

  A professional magician like the Great Whodunni is never without a handkerchief or ten, and can produce them indefinitely from a top hat, a sealed and empty box, or a volunteer’s pockets. Sometimes the odd pigeon turns up too, but to emulate this particular trick (which Whodunni learned from the American magician Edwin Tabor) all you need is two handkerchiefs - preferably of different colours. Roll up each along its diagonal to make a thick roll of cloth about a foot (30 cm) long.

  Now follow the instructions and pictures.

  Handkerchief trick.

  1. Cross the handkerchiefs with the dark one u
nderneath.

  2. Reach under the dark handkerchief, grab end A of the light handkerchief, pull it behind the dark handkerchief, and wrap it over the front of the dark handkerchief.

  3. Reach under the light handkerchief, grab end B of the dark handkerchief, pull it behind the light handkerchief, and wrap it over the front of the light handkerchief.

  4. Bring ends B and D together by swinging them underneath the rest of the handkerchief. Bring ends A and C together by swinging them over the top of the rest of the handkerchief.

  Now the two handkerchiefs are all tangled together. Hold ends A and C together in one hand, and B and D together in the other hand. Now pull your hands quickly apart.

  What happens?

  Answer on page 331

  A Bluffer’s Guide to Symmetry

  The word ‘symmetry’ is often bandied about, but in mathematics it has a precise - and very important - meaning. In everyday language, we say that an object is symmetrical if it has an elegant shape, or is well proportioned, or (getting technical) the left and right sides of the object look the same. The human figure, for instance, looks much the same when reflected in a mirror.

  The mathematical usage of the word ‘symmetry’ is significantly different and much broader: mathematicians talk of ‘a symmetry’ of an object, or ‘many symmetries’. To mathematicians, a symmetry is not a number, or a shape, but a transformation. It is a way to move an object, so that when you’ve finished, the object appears not to have changed.

  The cat (far left) looks different if you rotate it . . .

  . . . or reflect it . . .

  ... so it has no symmetries. No, that’s a lie: it has one symmetry: leave it alone. This is the trivial symmetry, and all shapes have it.

  A cat with two tails looks the same when you reflect it, so it has an axis of reflectional symmetry (grey line).

  The cat’s body has two axes of reflectional symmetry, and it also looks the same when you rotate it through 180°.

  Four cats sitting in a square are symmetric under rotations of 0° (trivial), 90°, 180° and 270°. This is 4-fold rotational symmetry.

  The same goes when you throw away the cats . . .

  ... but now there are four new axes of reflectional symmetry. So a square has eight different symmetries.

  A cube has 48 symmetries . . .

  . . . and a dodecahedron has 120.

  A circle has infinitely many rotational symmetries (any angle) and infinitely many reflectional symmetries (any diameter as axis).

  If this line of cats went on infinitely far, it would have translational symmetries: slide the cats an integer number of spaces right or left.

  A cat crystal has translational symmetries in two different directions.

  Symmetries need not be motions. Shuffling a pack of cards is a transformation . . .

  . . . and if some cards are identical, some shuffles just swap identical cards - these are permutational symmetries of the pack.

  Symmetries have come to dominate huge areas of mathematics. They are very general - it’s not only shapes that have symmetries. So do number systems, equations, and processes of all kinds. The symmetries of a mathematical ‘thing’ tell us a lot about it. For instance, Galois proved that you can’t solve the general equation of the fifth degree by an algebraic formula, and the main point of his proof is that the general equation of the fifth degree has the wrong kind of symmetries.

  Symmetries are vital in physics, too. They classify the atomic lattices of crystals - there are 230 different symmetry types, or 219 if you consider mirror images to be the same. The ‘laws of nature’ turn out to be highly symmetric, mainly because the same laws operate at all points of space and all instants of time. The symmetries of the laws tell us a lot about the solutions. Quantum physics and relativity are both based on symmetry principles.

  Front-back symmetry of a pacing giraffe. The front and back legs on each side hit the ground together.

  Symmetries are even turning up in biology. Many important biological molecules are symmetric, and the symmetries affect how they work. But you can find symmetries in the shapes of animals, in their markings, and even in how they move. For example, when a giraffe paces, it moves both left legs together, then both right legs together. So the front legs do the same as the back legs, like two people walking one behind the other, in step with each other. The symmetry here is a permutation: swap front and back.

  Perform this only in the abstract, please, or the giraffe will get upset.

  Digital Century Revisited

  Innumeratus wrote the nine non-zero digits down in order, with gaps, like this:

  1 2 3 4 5 6 7 8 9

  ‘I want you to . . . ’ he began.

  ‘ . . . make 100 by inserting standard arithmetical symbols,’ said Mathophila. ‘That’s easy, it was in Professor Stewart’s Cabinet of Mathematical Curiosities, which you gave me for Christmas, but it goes back a lot further than that.’ And she wrote:

  123 - 45 - 67 + 89 = 100

  ‘No, that’s cheating,’ said Innumeratus. ‘I left gaps! You can’t consider 1 2 3 to be one hundred and twenty-three, and ... ’

  ‘Oh. No concatenation of symbols allowed, then.’

  ‘Yeah. No caterwaulification . . . whatever.’

  She thought for a moment, and wrote down

  (1 + 2 - 3 - 4)×(5 - 6 - 7 - 8 - 9)

  ‘Sorry, no brackets,’ said Innumeratus.

  Mathophila shrugged, and wrote

  1 + 2×3 + 4×5 - 6 + 7 + 8×9

  ‘You don’t mind me using the rule that multiplication precedes addition, so I don’t need to put brackets round individual multiplications, do you?’

  ‘No, that’s OK. But ... uh . . . look, sorry, but no subtraction symbols either.’

  There was a silence. ‘I’m not sure that’s possible,’ said Mathophila.

  ‘Wanna bet?’ asked Innumeratus smugly.

  What should Mathophila do?

  Answer on page 331

  An Infinity of Primes

  Euclid proved that there is no largest prime. Here’s a quick way to see this: if p is prime then p! + 1 is not divisible by any of the numbers 2, 3, . . . , p, since any such division leaves remainder 1. So all its prime factors are bigger than p. Here, p! = p × (p - 1)× (p - 2)×. . .×3×2×1.

  Euclid’s proof was slightly different. He stated it geometrically, and in modern terms he used a typical example to show that if you have any finite list of primes, then you can get a bigger one by multiplying them all together, adding 1, and then taking any prime factor of the result.

  This suggests an interesting sequence of primes, all guaranteed to be different:

  For example,

  p3 = the smallest prime factor of 2×3 +1 = 7, namely 7

  p4 = the smallest prime factor of 2×3×7 +1 = 43, namely 43

  p3 = the smallest prime factor of 2×3×7×43 +1 = 1807, namely 13

  (because 1807 = 13×139), and so on.

  The first few terms are

  2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139,

  2801, 11, 17, 5471, 52662739, 23003, 30693651606209,

  37, 1741, 1313797957

  and the sequence is highly irregular. Occasionally the product p1 × p2 × ‧‧‧ ×pn + 1 is prime, and the size goes up enormously, but when it’s not prime, the smallest factor is often very small indeed. This behaviour is pretty much what you might expect, wild though it may be.

  Despite (or perhaps because of) this tendency to swing madly between huge numbers and tiny ones, the first 13 terms include the first seven primes: 2, 3, 5, 7, 11, 13, 17. Which raises an interesting - and probably difficult - question: does every prime occur somewhere in this sequence?

  I have no idea how to answer that, though if I had to guess I’d say it’s true.

  A Century in Fractions

  The famous English puzzlist Henry Ernest Dudeney remarked that the fraction

  is equal to 100, and uses every digit 1-9 exactly once. He found ten other ways
to achieve this, one of which has only one digit before the fractional part. What was this solution?

  Answer on page 332

  Ah, That Explains It . . .

  • Knowledge is power

  • Time is money

  But, by definition,

  • Power = work/time

  So,

  • Time = work/power

  which implies that

  • Money = work/knowledge

  Therefore:

  • For a fixed amount of work, the more you know, the less money you get.

  Life, Recursion and Everything

  Readers of Douglas Adams’s The Hitch Hiker’s Guide to the Galaxy will recall the prominent role of the number 42 - the answer to the Great Question of Life, the Universe and Everything. The question turned out to be ‘what is six times nine?’, which was vaguely disappointing. Anyway, Adams chose 42 because a quick poll of his friends suggested that this was the most boring number they could think of.

  It’s true that interesting properties of 42 don’t exactly trip off the tongue, but we know (Cabinet, page 105) that all numbers are interesting. However, the proof is non-constructive. So I was pleased to find out about a natural occurrence of 42 as an interesting number. It arises in a sequence of numbers introduced by F. Göbel. Suppose we define