Professor Stewart's Hoard of Mathematical Treasures (16 page)

Putting the pieces together, we have
which is the required Egyptian representation.

The greedy algorithm doesn’t always give the
simplest
Egyptian representation. For instance, when applied to
it produces
and fails to spot a simpler answer:

The Erdős-Straus conjecture states that every fraction of the form 4/n has a unit fraction representation with three terms:
It is true for all
n
< 10
¹⁴
. Exceptions, if they exist, must be very thin on the ground, but no proof or disproof exists.

There are also some interesting variations on the greedy algorithm that you can try. I suggest using fractions with small numerators and denominators to avoid monsters like the one we’ve just seen. First, try it with the extra condition that every fraction involved must be one over an even number. Surprisingly, the greedy algorithm still works - it has been proved that every fraction less than one is a sum of unit fractions with distinct even denominators.

Now try it with odd denominators. Computer experiments suggest that it also works in this case. For instance,
But now nobody has a proof. For all we know there might be some peculiar fraction for which the odd-denominator greedy algorithm goes on for ever.

Now that’s really greedy.

We have only scratched the surface of the mathematics of Egyptian fractions. For more, see:
en.wikipedia.org/wiki/Egyptian_fraction

William Feller.

William Feller was a probability theorist at Princeton University. One day he and his wife wanted to move a large table from one room of their house to another, but, try as they might, they couldn’t get it through the door. They pushed and pulled and tipped the table on its side and generally tried everything they could, but it just wouldn’t go.

Eventually, Feller went back to his desk and worked out a mathematical proof that the table would never be able to pass through the door.

While he was doing this, his wife got the table through the door.

Form five rectangles by choosing their sides from the list 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, with each number being chosen exactly once. Then assemble the rectangles, without overlaps, to form an 11×11 square.

 

Answer on page 293

When Newton saw an apple fall, he found
A mode of proving that the earth turn’d round
In a most natural whirl, called gravitation;
And this is the sole mortal who could grapple
Since Adam, with a fall or with an apple.

Isaac Newton

George Gordon Byron.

‘Shiver me timbers and brace the mainsplice!’ declared Roger Redbeard, the pirate captain. ‘What have we here, me hearties? Methinks it be a treasure map, aaargh, for plainly I sees an X.’

Here be treasure me hearties, aaargh!

‘I knows that island,’ said the bosun. ‘It be where we marooned that lily-livered swine Admiral Ponsonby-Ffynche and his crew, when we seized the Vainglorious. Dead Man’s Key, it were. Not a drop o’ water on the island, may their bones whiten in the merciless sun.’

‘Set sail for Dead Man’s Key!’ ordered Redbeard. As the crew hoisted the mains’l, he looked round to make sure no one was watching, and turned the map over. On the back, in letters of blood, were instructions to locate the loot:

The rest was torn away.

Roger cursed a foul pirate curse, for he was a foul pirate and knew how to curse the sort of curse that foul pirates curse. ‘I swear,’ he swore, ‘I’ll dig up the entire island if I have to, aaargh!’ For he knew that pirates never placed their X’s on maps in the
correct location, as that would make it too easy for others to discover their hoard.