I recently wrote a short essay for Chris Pritchard's book "The Changing Face of Geometry". For people who want to find out more about the subject, there is an article below, followed by a series of websites on the Kepler Stacking Conjecture.

The Mathematics of Packing Oranges
The Daily Telegraph

This week Thomas Hales at the University of Michigan he announced a solution to a 400-year-old mathematical riddle - which is the best way to stack oranges? Hales’s work includes 250 pages of logic and, more controversially, relies on 3 gigabytes of computer files, which poses serious problems for the referees who must now begin the daunting task of scrutinising the calculation.

The so-called sphere packing problem was born in 1611, when the German astronomer Johannes Kepler asked himself which is the most efficient way to pack spheres leaving as few gaps as possible. Having studied the way that sailors stack cannonballs, and the way that particles of water stack together to form snowflakes, Kepler eventually settled on an arrangment known as the face-centred cubic, which also happens to be the way that greengrocers stack oranges.

Using this arrangement, oranges occupy 74.04% of the total space. Kepler could not find a more efficient way to stack spheres, but he could not be sure that no such arrangement exists. With an infinite number of possible arrangements, the challenge has been to categorically prove whether or not Kepler’s suggested arrangement is the best one.

Professor Hales’s approach to the problem is based on a single equation with over 150 variables. The variables can be changed to describe every conceivable arrangment, thereby allowing the equation to calculate the packing efficiency for each one. Traditionally mathematicians would simply alter the variables to maximise the packing efficiency for the equation, and then see which arrangement is associated with the variables, however the equation is immensely complicated, which puts the maximisation process beyond paper and pencil calculations, and even challenges the limits of computers.

Over the last decade Hales, helped by his research student Samuel P. Ferguson, has been studying the maximisation process, inventing shortcuts which bring it within the realm of computability.. At last, having thrown enough computer power at the problem and effectively testing all possible arrangements, Hales has concluded that no arrangement beats the face-centred cubic for efficiency. In other words Kepler and greengrocers have been right all along.

Hales is currently taking a well deserved holiday, which will allow other mathematicians to examine his work in detail. His proof will not be officially accepted until it has been refereed and published. In 1990 Wu-Yi Hsiang of the University of California at Berkeley announced a solution to the stacking problem, but subsequently his work has been shown to be flawed. Similarly, in 1993 Andrew Wiles announced a proof of Fermat’s Last Theorem and later that year an error was found in his work too, although in this case the mistake was eventually fixed.

Checking Hales’s work will be made harder because of its reliance on computer programmes, which will have to be checked line by line, in case there has been an error introduced by software programmers. In addition, there is the possibility that there is an error in the hardware running the programmes.

In 1976 Wolfgang Appel and Kenneth Appel used computers to answer the so-called four-colour problem, which had remained unsolved since 1852. This was one of the first significant problems to succumb to the power of computing, and sparked concerns about how such proofs should be checked. Ever since there has been a debate among mathematicians as to whether such proofs are in the spirit of the subject.

With respect to Hales’s proof, there is general optimism that it will in due course turn out to be valid. According to Professor John Conway, co-author of the standard text on sphere packing,

  “For the last decade Hales’s work on sphere packing
  has been painstaking and credible. If he says he’s
  done it, then he’s quite probably right.”

Thomas Hale's site