Professor Stewart's Hoard of Mathematical Treasures (43 page)

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.

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.

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:

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?

 

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
x
⁶
= (
x
²
)
³
= (
x
³
)
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.

 

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 underneath.

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?

 

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 . . .