## The Pizza Race ProblemJanuary 9, 2013 One way to get children interested in mathematics is to offer them a treat, such as a pie or cake, and have them divide it into fair portions by themselves. If there are just two children, the task can be reduced from a mathematical to a political task. One child cuts, and the other child chooses which half he or she prefers. When there are three children involved, an impromptu lesson in trigonometry emerges in which 120 angular degrees and the definition of the center are the objects of concern. What happens when an object has already been cut into irregular portions, and the diners need to successively take portions? How much of a quantity advantage could one diner have over the others in this sort of game? Also, what happens when a portion of the object has been previously removed; for example, one slice taken from a pizza.
(1) They choose pieces in an alternating fashion;On each turn, except the first and the last, the diner has a choice of two available pieces. Alice, in selecting first, would take the largest slice, and she would seem to always have an advantage. This is true for an even number of pieces. In that case, Alice, if she chooses wisely, will always get more than half the pizza. What's interesting is that Alice can actually come away with less than half in the odd piece case. Although Alice gets both the first and last piece in an odd piece pizza, Winkler conjectured she might at a minimum get only 4/9 of the pizza. This conjecture was proven by Kolja Knauer, Piotr Micek and Torsten Ueckerdt.[2]
2π, or whatever you want. This makes a computer simulation very easy, since in each case you can just spit out a bunch of random numbers and not worry about their sum.
As is my custom, I've written a simple computer simulation for this pizza problem (source code, here). It generates random pizzas, allows Bob and Alice to make random selections, and it keeps track of the largest portion for Alice. As it turns out, Alice will, on average, do very well in sharing an odd piece pizza, even when she chooses her pieces at random.
Part of this might be because Alice always selects the largest piece as her first piece; and, for some random pies, this piece can be quite a bit larger than others. The following histogram shows the results of 10,000 trials for an eleven piece pizza in which Alice chooses the largest piece first, and then randomly thereafter.
0.400 vs 0.444...).[1]
## References:- Keyue Gao, "Pizza Race Problem," arXiv Preprint Server, December 10, 2012
- Kolja Knauer, Piotr Micek and Torsten Ueckerdt, "How to eat 4/9 of a pizza," arXiv Preprint Server, January 24, 2011.
Linked Keywords: Children; mathematics; pie; cake; politics; political; trigonometry; angular degree; center; pizza; family; knife; fork; Wikimedia Commons; arXiv; New York University; mathematician; Peter M. Winkler; Laszlo Lovasz; birthday; Bob and Alice; cryptography; algorithm; geometry; geometrical; circular buffer; ring; random number; computer simulation; source code; pizza.c; randomness; random; histogram; etiquette; Gnumeric; constraint; Keyue Gao; Pizza Race Problem. |
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