There's Real Math to Cutting Fair Slices of Pizza

December 23, 2009

Sharing a pizza should involve very little conflict, except maybe over who gets the last piece, but for Rick Mabry and Paul Deiermann, sharing a pizza plunges them into the mathematics of how to slice it up properly.

"We went to lunch together at least once a week," says Mabry, recalling the early 1990s when they were both at Louisiana State University, Shreveport. "One of us would bring a notebook, and we'd draw pictures while our food was getting cold."

Their main concern was cutting the pizza off-center, something a harried waiter could do and still retain the edge-to-edge cuts crossing at a single point and with the same angle between adjacent cuts. For two people to share a pizza, it must be sliced by n straight, concurrent, equiangular cuts. However, if the pizza-cutter misses the mark—the point (p) of concurrency is not at the center—then the pizza won't be shared equally when each takes the same number (n) of slices

The solutions, of course, rely on mathematics, as seen in several articles in MAA journals over the years and in a recent article in New Scientist.

Since the 1960s, problem solvers have shown that when n is even and greater than 2, one approach is to choose alternate slices about the point (p) of concurrency. This scheme also results in equal shares for any n, if the center (o) lies on one of the cuts. However, if n is odd, and if o does not lie on a cut, then, as has been known since the 1990s, this method of alternating slices does not result in equal sharing. The question then is: Who does get the most pizza?

Here are the solutions: for n=3,7,11,15,…, whoever gets the center gets the most pizza; while for n=5,9,13,17,…, whoever gets the center gets the least.

Mabry and Deiermann outlined the pizza problems and solutions in "Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results (pdf)," by Rick Mabry (Louisiana State University, in Shreveport) and Paul Deiermann (Southeast Missouri State University, Cape Girardeau). Their article appeared in the American Mathematical Monthly (May 2009).

In short, the authors proved their pizza conjecture by demonstrating an equivalence to a trigonometric inequality, which they proved with the aid of a theorem that counts lattice paths. Moreover, their main theorem is general enough to obtain results for equiangular slicing of other dishes.

"It's a funny thing about some mathematicians," Mabry said in the New Scientist article. "We often don't care if the results have applications because the results are themselves so pretty."

More important is the fact that some solutions to abstract mathematical problems do have applications in unexpected arenas. Another example is the "space-filling curve," a 19th-century mathematical curiosity that has resurfaced as a model for the shape of the human genome.

Source: New Scientist (December 11, 2009).

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