What is the tightest possible way to pack identical spheres? According to Thomas Hales of the University of Michigan at Ann Arbor, the supermarket folks get it right: When they stack oranges, the first (bottom) layer consists of rows that are staggered by half an orange, and for the second layer, the oranges are placed in the hollows formed by three adjacent oranges in the first layer, and so on. In 1611, Johannes Kepler conjectured that this arrangement, known as the face-centered cubic packing, is the solution to the sphere-packing problem. Last summer, Hales (aided by Michigan graduate student Samuel Ferguson) might have succeeded in proving Kepler's conjecture. He posted his set of proofs on the Internet. The complete proof takes up more than 250 pages, and it makes heavy use of computers (the computer programs and data files take up 3 gigabytes of memory). Hale was careful to note that his results are still preliminary. While impressed with Hale's proof, John Conway of Princeton University ventured to guess that there is a very short proof of Kepler's conjecture involving totally different ideas. This sounds like a nice math topic for the Intel Science Talent Search competition. If you succeed, feel free to ask Hale or Conway for a letter of recommendation. In the mean time, don't forget to send me a copy of your proof.
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