Professor Stewart's Hoard of Mathematical Treasures (21 page)

Mathophila agreed with that, but felt it was a silly puzzle. She really liked the first version better.

Can you solve the original version? You can turn cards through a right angle if you wish.

 

This is a traditional puzzle that goes back to the Renaissance Italian mathematician Tartaglia in the 1500s, but its solution has systematic features that were not noticed until 1939, discussed in the Answer. There are many similar puzzles.

You have three jugs, which respectively hold 3 litres, 5 litres, and 8 litres of water. The 8-litre jug is full, the other two are empty. Your task is to divide the water into two parts, each of 4 litres, by pouring water from one jug into another. You are not allowed to estimate quantities by eye, so you can only stop pouring when one of the jugs involved becomes either full or empty.

Divide the water into two equal parts.

Answer on page 299

If you draw a closed curve in the plane which doesn’t cross itself, then it seems pretty obvious that it must divide the plane into two regions: one inside the curve, the other outside it. But mathematical curves can be very wiggly, and it turns out that
this obvious statement is difficult to prove. Camille Jordan gave an attempted proof, more than 80 pages long, in a textbook published in several volumes between 1882 and 1887, but it turned out to be incomplete. Oswald Veblen found the first correct proof of this ‘Jordan curve theorem’ in 1905. In 2005, a team of mathematicians developed a proof suitable for computer verification - and verified it. The proof was 6,500 lines long.

A closed curve, with the inside shaded.

A subtler topological feature of such a closed curve is that the regions inside and outside the curve are topologically equivalent to the regions inside and outside an ordinary circle. This too may seem obvious, but, remarkably, the corresponding statement in three dimensions, which seems equally obvious, is actually false. That is: there is a surface in space, topologically equivalent to an ordinary sphere, whose inside is topologically equivalent to the inside of an ordinary sphere, but whose outside is not topologically equivalent to the outside of an ordinary sphere! Such a surface was discovered by James Waddell Alexander in 1924, and is called Alexander’s horned sphere. It is like a sphere that has sprouted a pair of horns, which divide repeatedly and intertwine.

Alexander’s horned sphere.

The daredevil adventurer and treasure-hunter Colorado Smith, who is not like a real archaeologist at all, ducked a passing shower of blazing war-arrows to check the crude sketch-map scribbled in his father’s battered notebook.

‘The holy sanctuary of Pheedme-Pheedme the goddess of eating and sleeping,’ he read, ‘is formed from 64 identical square cushions stuffed with ostrich-down, arranged in an 8×8 array. The five sacred avatars of Pheedme-Pheedme, represented in overstuffed fabric, are to be placed on the cushions so that they “watch over” every other cushion: that is, every other cushion must be in line with one occupied by an avatar. This line can be horizontal, vertical or diagonal, relative to the array, where “diagonal” means “sloping at 45° ”.’

‘Look out!’ shrieked his sidekick Brunnhilde, taking cover beneath a large stone altar.

‘I wouldn’t do that if I were you,’ said Smith, and hauled her out a split second before the supporting slabs exploded in puffs of dust and the 10-tonne altar stone crashed to the ground. ‘Now, Dad’s notebook says something about the principles of—uh - Mat?’

‘Ma’at was the Egyptian concept of justice and rightful place,’ Brunnhilde pointed out. ‘But this temple is Burmalayan.’

‘True. Can’t be ma’at . . . No, it’s definitely the Principle of Mat. Apparently the goddess reclines on a mat, surrounded by her sacred avatars. We have to leave a space for the holy reclining mat, which is square. Hmm . . . maybe this would do.’

Is this how to lay out the five sacred avatars and Pheedme-Pheedme’s reclining mat?

‘That seems suspiciously easy,’ said Brunnhilde. ‘What else do we have to do?’

Smith quietly removed a deadly kamikaze-scorpion from her hair, hoping she wouldn’t notice. ‘Uh, we have to arrange the avatars to leave the largest possible space for a square mat. Bearing in mind that they must watch over every cushion. I doubt we can do better than my picture.’

‘Those ancient priests were sneaky, though,’ said Brunnhilde. She tried not to listen to the approaching bloodcurdling cries, and racked her brains. If they could solve the riddle of the sacred mat, they could proceed to the enigma of the potted dormice, and then only 17 more puzzles would lie between them and the Hoard of Treasures. ‘Does the mat have to be arranged with its sides parallel to those of the cushions? Could it be tilted?’

‘I don’t see anything in the 999 pages of The Book of the Ninth Life to prohibit that,’ said Smith. ‘The only restriction is that the mat can’t overlap any cushion bearing one of the sacred avatars. The edges of the mat and the cushion can touch, but there mustn’t be a genuine overlap.’

How can the biggest mat be fitted in without breaking the sacred rules?

 

If n is a whole number, then the sum of its divisors, including n itself, is the divisor-sum σ(n). So, for example,
The divisor-sum is the key to a very ancient recreation, the search for perfect numbers. A number is abundant if it is smaller than the sum of its ‘proper’ divisors - those that exclude the number itself. It is deficient if it is greater than that sum, and perfect if it is equal to it. In terms of the divisor-sum, these conditions become
Here we see 2n, rather than n, because σ(n) includes the divisor n as well as all the others. This is done so that the nice formula σ(mn) = σ(m)σ(n) holds when m and n have no common factor greater than 1.

Lots of numbers are deficient; for example 10 has proper divisors 1, 2, 5, which sum to 8. Abundant numbers are rarer: 12 has proper divisors 1, 2, 3, 4, 6, with sum 16. Perfect numbers are very rare; the first few are:
followed by 496 and 8,128. Euclid discovered a pattern in these perfect numbers: he proved that whenever 2
^p
- 1 is a prime, the number 2
^p
-1
(2
^p
- 1) is perfect. Much later, Euler proved that every even perfect number must be of this form. Primes of the form 2
^p
- 1 are called Mersenne primes (Cabinet, page 151).

No one knows whether any odd perfect numbers exist;
however, Carl Pomerance has given a non-rigorous but plausible argument that they don’t. There is a solid proof that if an odd perfect number does exist then it must be at least 10
³⁰⁰
, and have at least 75 prime factors. Its largest prime factor must be greater than 10
⁸
.

A related, equally ancient pastime is to find pairs of amicable numbers - each equal to the sum of the proper divisors of the other. That is,
so σ(n) = σ(m) = m + n. For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, adding to 284; the proper divisors of 284 are 1, 2, 4, 71, 142, adding to 220. The next few amicable pairs are (1184, 1210), (2620, 2924), (5020, 5564) and (6232, 6368).

In all known examples, the numbers in an amicable pair are either both even or both odd. Every known pair shares at least one common factor; it is not known whether a pair of amicable numbers with no common factor can exist. If there is such pair, then their product is at least 10
⁶⁷
.

An integer is multiply perfect if it divides the sum of its divisors exactly; the multiplicity is the quotient. Here it makes no difference whether we include the number itself, or not, except that the multiplicity goes down by 1 if we don’t. But it’s normal to include it. If we do, then ordinary perfect numbers have multiplicity 2, triperfect numbers have multiplicity 3, and so on. The smallest triperfect number is 120, as Robert Recorde knew in 1557: the sum of its divisors is
Here are a few other multiply perfect numbers. Many more are known. (The dots between the numbers mean ‘multiply’.)