Fair Sharing of Triangular Pizzas

摘要

An email from Grégoire Nicollier pointed me on the one hand to a beautiful problem of elementary geometry, and on the other hand to a beautiful solution [5]. The problem was about sharing a pizza fairly: Imagine a pizza shaped like an equilateral triangle. If the pizza is shared fairly, then everybody should get the same amount of "area" and the same amount of "crust," and this was achieved by making the cuts in the way described in the problem’s solution. Of course, in reality there are no such pizzas, and it was not clear how to produce the solution’s indicated cuts in reality. The problem of fair pizza division has received considerable attention from mathematicians (see, e.g., Carter and Wagon [1], Humenberger [3], and Mabry and Deiermann [4]), but the theory and reasoning behind the solution are more important than the illusion of modeling real life situations.