Math Professor delivers lecture on fair division

Professor Alissa Crans from Loyola Marymount University gave the McDougal Lecture in Mathematics on Tuesday, Apr. 30, at 4:30 p.m. in Steitz Hall 102. The title of her presentation was “Frosting Fairness, Finally!” 

Crans is a big fan of cake and has wondered about the fair division of cake cutting. She started the lecture with the question, “How do you cut a square cake equally into four slices?” Many possible answers were brought up. She then asked “How do you cut a square cake equally into five slices?” Cutting the cake either horizontally or vertically could divide it into five slices with equal area. 

Crans then asked a tougher question: “How do you cut a square cake into five slices so that everyone gets an equal amount of cake and frosting?” Cake indicates area, and frosting indicates perimeter. Nobody was able to provide a feasible solution immediately. This was the main puzzle Crans would solve in her lecture. 

“What do you do when you cannot solve the problem in front of you?” Crans asked. One student answered, “I solve an easier problem instead.” Since the main puzzle could not be solved, Crans continued to say that, “We should solve related but easier problems first.” She brought up a related question, asking, “How do you cut a square cake into four slices so that everyone gets an equal amount of cake and frosting?” The answer was to cut from two diagonal lines. 

Based on the solution for the last question, one could cut the cake equally into five slices by simply shaving off a little bit at each slice. Crans said, “If we cut straight lines from the center, then any way of equally dividing the perimeter equally divides the slices.” For mathematicians, it is not enough to just answer a single question; they want to systematically generalize all similar questions. 

Crans moved to the next question, which was, “What are some other shapes of cake for which we can obtain slices with equal cake and frosting using straight cuts from a single point?” A circle is an obvious answer. So far, Crans had led the audience to do mathematics research. Mathematics research is question-driven and once you solve one question, you go on and solve other related problems. Ultimately, a method is created to solve all generalized questions. This is so called, “if it is true globally, it is true infinitesimally.”

Crans explain the statement: “Area swept out to be proportional to the arc length.” The conclusion was that the shapes that are formed using either arcs of a single circle or tangents to that circle would be viable for the aforementioned method of cutting. In other words, any polygons with an inscribed circle can be applied to this method. A reception followed the lecture, where a triangular cake was cut into slices so that everyone got the same amount of cake and frosting. 

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