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Monday, August 10, 1998 Published at 18:03 GMT 19:03 UK


Staggering solution to sphere stacking

Greengrocers have been getting it right for centuries

BBC correspondent Emma Simpson explains the problem
An American professor claims he has solved a problem which has baffled scientists for centuries - how to stack oranges in a box.

Thomas Hales posted his solution to Kepler's stacking problem on the Internet on Sunday.

Professor Hales has proved what no scientist has been able to for 300 years - in a solution which has been practiced by greengrocers through the centuries.

[ image: The way to do it]
The way to do it
The problem dates from the beginning of the 17th century when Sir Walter Raleigh asked the English mathematician Thomas Harriot to study how to stack cannonballs.

Harriot subsequently wrote to German astronomer Johannes Kepler, who discovered the laws that describe the orbits of the planets.

In 1611, Kepler suggested the most efficient way to pack spheres in the smallest space was an arrangement known as face-centred cubic packing, which is how greengrocers stack their oranges.

The bottom layer is arranged in lines which are alternately staggered by half an orange.

The stack is built up by placing the oranges in the dimples of the layer below.

Kepler could never prove there was a more efficient way and ever since generations of mathematicians have since been trying to find conclusive proof about whether he was correct.

[ image: Oranges have kept mathematicians puzzled for years]
Oranges have kept mathematicians puzzled for years
The solution by Professor Hales, of the University of Michigan, that Kepler's suggestion is the most efficient could be the biggest breakthrough in mathematics since Andrew Wiles proof of Fermat's Last Theorem in 1994.

It contains 250 pages of logic and relies on computer programs which require more than three gigabytes of storage space.

Now Professor Hales is taking a break from the research which has taken him several years to complete along with research student Samuel P Ferguson.

Dr Simon Singh said the problem is one every mathematician would have liked to have solved.

He said: "There's an infinite number of arrangements.

"The question is how you can ever prove there isn't some other arrangement you haven't encountered.

"That's what people have been doing for the last 400 years - trying to find out whether or not there is a better arrangement than Kepler's suggestion."

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