Professor Stewart's Hoard of Mathematical Treasures (25 page)

But that was only to be expected. As Smith said, ‘We’ll just have to think our way out.’

Location of the sun-discs.

For once, Brunnhilde did not find this entirely reassuring. Maybe it was the earthquake and the puffs of dust thickening the air around them. Or was it the roar of approaching water? The carpet of scorpions on the floor, scuttling from cracks in the stonework? Or just the spikes round all the walls, which even now were extending towards them?

‘What do we have to do this time?’ she asked, having been in this position so often that she knew the script by heart.

‘According to the Lost Papyrus of Bentnosy, we must choose four non-overlapping connected regions, each composed of 16 slabs, so that each includes one slab with a sun-disc,’ replied Smith. ‘Then the secret exit will open and let us into the adjoining treasure chamber - the one with those caskets of
diamonds and emeralds I told you about. From there we just have to get through the underground maze that leads to the—’

‘That seems easy enough,’ said Brunnhilde, quickly sketching a solution. She caught his eye. ‘What’s the catch, Smith?’

Not like this!

‘Ah . . . Well, according to an obscure inscription on the Oxyrhincus Ostracon of Djamm-Ta’art, which is a Late Period commentary on Bentnosy’s papyrus, each of the four regions must be the same shape.’

‘Ah. That makes it harder.’ Brunnhilde smiled a hopeful smile, and tore up her sketch. ‘I suppose the answer is in Bentnosy’s papyrus?’

‘Apparently not,’ said Smith. ‘It’s not on the Ostracon, either - front or back.’

‘Oh. Well, do you think we’ll be able to work it out before that huge block of granite squashes us to the thickness of gold leaf?’

‘What block of granite?’

‘The one hanging over our heads on burning ropes.’

‘Oh, that block of granite. Strange, Bentnosy didn’t mention anything like that.’

Help Smith and Brunnhilde escape their dire predicament.

 

Well, you can if you wish - it’s a free country. Allegedly. But you won’t get the right answer.

At school we are taught an easy way to multiply fractions: just multiply the numbers on the top, and those on the bottom, like this:
But the rule for adding them is much messier: ‘Put them over the same denominator (bottom), then add the numerators (tops).’ Why can’t we add them in a similar way? Why is
wrong? And what should we do instead?

 

As soon as you say that some mathematical idea makes no sense, it turns out to be really useful and perfectly sensible. Although the rule
is not the correct way to add fractions, it is still a possible way to combine them, as the geologist John Farey, Sr, suggested in 1816 in the Philosophical Magazine. He hit on the idea of writing all fractions a/b whose denominator b is less than or equal to some specific number, in numerical order. Only fractions whose numerical values lie between 0 and 1 (inclusive) are allowed, so 0 ≤
a
≤
b
. To avoid repetitions, he also required the fraction to be in ‘lowest terms’, which means that a and b do not have a
common factor (bigger than 1). That is, a fraction like
is disallowed, because 4 and 6 both have the common factor 2. It should be replaced by
, which has the same numerical value but doesn’t involve common factors.

The resulting sequences of fractions are called Farey sequences. Here are the first few:
Farey noticed - but could not prove - that in any such sequence, the fraction immediately between a/b and c/d is the ‘forbidden sum’ (
a
+ b)/(c + d). For instance, between
and
we find
, which is (1 + 2)/(2 + 3). Augustin-Louis Cauchy supplied a proof in his Exercises de
Mathématique
, crediting Farey with the idea. Actually, it had all been published by C. Haros in 1802, but nobody had noticed.