Update: There's Real Math to Slicing Pizza

January 26, 2010

The following “Math in the News” item was originally published on Dec. 23, 2009. Rick Mabry recently contacted us about some errors in our original news item. The MAA has added his corrections and republished the entire article.

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 it is sliced.

"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 a pizza having off-center cuts. Ideally, for two people to share a pizza equally, it would 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 might not be shared equally when each takes the same number (n) of slices.

There are mathematical answers to the quandary, as noted in several articles in MAA journals over the years and reported 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. Amazingly, such an alternation of slices results in equal shares for the two parties. It is easy to see, by symmetry, that this scheme also results in equal shares for any n if the center (O) lies on one of the cuts.

Here are the conjectured answers for an odd number of sharers: 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.

This was originally conjectured in the 1990's by Larry Carter, John Duncan, and Stan Wagon. 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 the 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 provide 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."

Nonetheless, solutions to abstract mathematical problems may often have applications in unexpected arenas.

Source: New Scientist (December 11, 2009).

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