Pizza and pie lovers: Problems of etiquette and equitable portions; (Jan. 27, 2010)
This article explains how it took 11 years for the two mathematicians, Rick Marby and Paul Deiermann, to come up with a satisfactory resolution of how to split a round (circle) pizza or a pie equitably between two eaters. Marby stated what amount to this: “If a mathematician is unable to solve a problem then, it would be stupid to take on the challenge. We decided to play stupid. Paul and I are the kinds of mathematicians who take pleasure in the beauty of the demonstration: we care less for applicability of the concept.”
There are many cases of mathematicians working on problems that “consciously” have no practical applications; they do take the time to demonstrate their concepts and theorems and produce functions: applications materialize much later for the benefit of sciences. For example, the Italian Guiseppe Peano described in 1890 a function that filled entirely a finite space (a square in that case). A century later, this function turned out to be fundamental in fractal theory (applied in biological structures, in economics, and meteorology). Personally, I think that no one invest time on a project if there was no personal practical interest; the initial interest is so personal and feels un-important to colleagues that the mathematician is shy to state his “secret” interest.
Let us try to expose the problem. First, the two eaters have to abide by the etiquette that you cannot get another slice before the other has finished his slice. The portions are picked up clockwise for easy visualization one after another. Another restriction: all slicing lines cross a single point if the server is lousy enough not to cut along the center of the perfectly circle (geometrically) pizza.
Now, if one of the lines cross the center then, there is no problem of equitable portions (in the final quantity of devouring pizza), regardless of the design on the number of lines of slicing. If the pizza is cut in two parts then you know which part is larger when the line does not cross the center. Otherwise, for any even numbers of lines the equitability is restored, assuming you are abiding by the etiquette rule.
The problem gets nasty if the number of lines is odd. If the number of lines is 3, 7, 11, or 15 then the one eating the piece including the center will be at an advantage (this advantage is relative to the quantity and not what happens after eating!). If the number of lines is 5, 9, 13, or 17 then the advantage is reversed. That should cover the equitability problem.
Now for the remaining story: In 1967, the mathematician LJ Upton resolved the problem with four lines; then he threw the challenged to resolve the problem along that concept of equitable portions. In 1968, the problem of 8 lines was solved (mathematically) and for all even numbers of lines. In 1994, Marby and Deiermann accepted the challenge to study the odd number of lines slicing the pizza and not passing by the center. For lenient practical eaters it should not matter that much if the point is close to the center: the problem is to convince mathematicians.
Paul quickly found a “gorgeous” demonstration for three lines and it got nastier later on. The problem can be solved easily by induction using the current technology of measuring areas: in this case, induction reasoning on the possibilities is accurate since we are not taking samples in complex experiment with many variables (all we have is number of odd lines as variable). Rick and Paul wanted a classical mathematical demonstration, preferably the simpler and most beautiful demonstration. They had to skim the Internet for theorems and functions and discovered that in 1979 a mathematician got fun demonstrating the complex algebraic problem of rectangular strips (don’t ask me for further details).
If you ask me to take on the psychological characters of Rick and Paul, I may venture to say that this couple is voracious but pretty cheap.
No comments:
Post a Comment