麻豆传媒

Try to solve two of Professor Stewart鈥檚 puzzles here

WHEN I was 14 years old, I started a notebook of every interesting thing I could find out about mathematics that wasn鈥檛 taught at school. My notebook grew to a set of six, which I still have, and then spilled into a filing cabinet. Its contents are a miscellany of mathematical games, puzzles, stories and factoids. I recently compiled this miscellany into the book .

It was fun to go through all those old oddities, not least because it reminded me about, what in my view, is the strangest conjecture in the whole of mathematics 鈥� the sausage conjecture.

A conjecture is a statement that mathematicians think could be true, but which no one has yet proved or disproved. It is a problem waiting to be solved, where we have reason to think we know what answer to expect.

The sausage conjecture appears to deal with a simple problem, yet a proof has proved elusive. It asks how efficiently circles or spheres can be wrapped. If you arrange a number of identical circles in the same plane and tie a length of string tightly around them, which arrangement minimises the area inside the string? Santa and his elves would encounter exactly this problem if they were looking to free up some space on the sleigh.

Mathematicians could offer them some help because they have thought long and hard about how to pack things into the smallest possible space.

Not that they have such problems completely wrapped up, though. It took nearly 400 years to prove what every grocer knows: that the most compact way to stack oranges in a box is to pile them up in hexagonal layers so that each orange touches 12 others. And when mathematicians look at these problems in more than three dimensions, as they inevitably do, there isn鈥檛 always a proof. It is the same story when it comes to minimising the size of parcels, and this is where the sausage conjecture comes in.

Suppose Santa鈥檚 elves tie a ribbon around, say, six circular mince pies in a flat pack, so that the package fits into the smallest possible area. If they experimented, they would find that the best arrangement of the pies is in a line, so that the string round the outside forms a sausage shape. However, if the elves were tying a ribbon round seven identical mince pies, a sausage would no longer give the smallest area. They would do better arranging the mince pies in a hexagon, so that the one in the middle touches the other six (see diagram).

More prosaically, if you have six circles or fewer, it is best to lay them out in a straight line. But for seven circles or more, you can reduce the area inside the string by avoiding sausages and making the arrangement rounder. This statement is a genuine theorem, meaning that it has a mathematical proof. The conjecture arises when we ask similar questions in higher dimensions.

Suppose, for example, that Santa鈥檚 elves have to wrap up identical spherical Christmas puddings, minimising the volume inside the wrapping. It is a three-dimensional version of the mince-pie problem.

Mathematicians have proved that for 56 or fewer spheres, the arrangement that achieves that goal is again a sausage: string the spheres out in a long line and wrap them inside a cylinder with rounded ends (see diagram).

With 57 spheres, however, a sausage ceases to be the optimal arrangement. There is a trade-off between keeping the arrangement thin and keeping its volume small, and the break point is between 56 and 57 spheres. For 57 or more spheres, a more compact form minimises the volume of the package.

Again, this is a theorem with a proof. But mathematicians have developed a reflex action of generalising a problem to any number of dimensions. So what happens when you go beyond three dimensions and into hyperspace? What is the most efficient way to wrap four-dimensional hyperspheres?

For a start, you have to use three-dimensional wrapping paper and make the total 4D hypervolume of the package as small as possible. It turns out the answer is also a sausage 鈥� up to a point. If you have 50,000 or fewer 4D hyperspheres, then the way to minimise the hypervolume of the package is to lay them out in a straight line and make a very long sausage. With more than 100,000, something more compact is better, though we don鈥檛 know precisely what is best; we just know that some unspecified arrangement is better than a sausage.

No one knows the exact break point where compact arrangements take over from sausages, but it is definitely somewhere between 50,000 and 100,000 hyperspheres. The surprise here is that sausages hang on as the best solution for as long as they do. This happens because the trade-off between the hypervolume of the package and the 鈥渟ausageyness鈥� of the arrangement is loaded in favour of long, thin configurations 鈥� until they get really, really long. But you have to calculate areas, volumes and so on to see why.

Given all this, you might suppose that even higher dimensions follow the same pattern, with the sausage shape working best up to a certain point before another arrangement takes over. However, in 1975, the Hungarian mathematician L谩szl贸 Fejes T贸th concluded that everything almost certainly changes at five dimensions and that this very natural guess is completely wrong. He came up with the sausage conjecture, which says that for five dimensions and up, sausages are best no matter how many objects you are wrapping.

鈥淚n five dimensions or more, sausage-shaped parcels are best no matter how many objects you wrap鈥�

He could have been wrong, but most mathematicians now think he was probably correct. One big advance was made in 1998 by mathematicians Ulrich Betke, Martin Henk and J枚rg Wills. They proved that the sausage conjecture is true for 42 or more dimensions. Which leaves only 5, 6, 7,鈥� 41 for the rest of us to think about.

And there are further generalisations. So far the sausage conjecture only works for Euclidean space, in which space is flat, parallel lines never cross and the angles of a triangle add up to 180 degrees 鈥� the kind of geometry we learn at school. To generalise further, we need to take into account curved spaces too. If Santa told his elves to pack 600 non-Euclidean teddy bears, they would be in big, big trouble.