How to equally share a pizza?

I remember when I was in college. The brothers in my fraternity and I developed quite a taste for stromboli from the pizzeria around the corner from our house. Generally I would order a large and eat it all by myself. (I had the digestive tract of a Denebian sandworm in those days and because I was a teaching assistant, I could generally spring for the full cost of $6.)

But I remember one day when, for some reason, I split the cost with J.C. (who was my “big brother”). J.C. opened the box and cut the strom in half. Then he invited me to choose which half I wanted. As anyone who grew up in a house with brothers knows, this is the fair way to split food between two people. Why I’d not seen it until then I had no idea. It was a beautiful and elegant solution to the problem. I was more impressed with the elegance than I was with my stromboli. And that’s saying something.

So today, when I stumbled across a paper describing how a pair of mathematicians worked out a rigorous methodology for sharing a pizza fairly, I just had to read on.

It turns out that sharing a pizza cut into even or odd numbers of slices is a bit more complicated. It’s simple if one of the cuts of the pizza goes through the center of the pizza – two people can easily then have the exact same share.

But if the slice misses the center – or if the pizza isn’t exactly circular, then things get complicated. And the solutions to the problem bifurcate between cases with an even number of slices and ones with an odd number:

“The first proposes that if you cut a pizza through the chosen point with an even number of cuts more than 2, the pizza will be divided evenly between two diners who each take alternate slices. This side of the problem was first explored in 1967 by one L. J. Upton in Mathematics Magazine (vol 40, p 163). Upton didn’t bother with two cuts: he asked readers to prove that in the case of four cuts (making eight slices) the diners can share the pizza equally. Next came the general solution for an even number of cuts greater than 4, which first turned up as an answer to Upton’s challenge in 1968, with elementary algebraic calculations of the exact area of the different slices revealing that, again, the pizza is always divided equally between the two diners (Mathematics Magazine, vol 41, p 46).

With an odd number of cuts, things start to get more complicated. Here the pizza theorem says that if you cut the pizza with 3, 7, 11, 15… cuts, and no cut goes through the centre, then the person who gets the slice that includes the centre of the pizza eats more in total. If you use 5, 9, 13, 17… cuts, the person who gets the centre ends up with less (see diagram).

Rigorously proving this to be true, however, has been a tough nut to crack. So difficult, in fact, that Mabry and Deiermann have only just finalised a proof that covers all possible cases.”

From here.

It took Mssr’s Mabry and Deiermann 11 years to find the proof they began to seek back when they were waiting around for the pizza they were having for lunch to cool.

They initially tried to prove it with computer aid in what seems to have been a proof by exhaustion. But recently, putting the computer aside, they found a more elegant algebraic expression of the problem. And writing the problem in that form gave them a more fundamental understanding of the problem. Which led them, via a literature search to a set of papers written beginning in the ’70s that ultimately gave them to the tools they needed to solve the problem.

What strikes me about this odyssey is that it was ultimately solved by the researches putting the question into an elegant form that allowed them to gain a new perspective on what they were seeking to find.

My Master’s advisor in Physics used to quip “Physics is easy – it’s just a matter of asking the right question.”

I have a sense that what is true for the Queen of the Academy (Physics) is true too for the King (Theology).

I wonder if so many of the problems that are vexing us theologically these days are vexing because we’ve yet to formulate them in a way that will give us a hint about how to answer them.

Perhaps that’s the way out of the hermeneutics problems we keep stumbling across in modern Christianity.

More on this as I have time…

Author: Nick Knisely

Episcopal bishop, dad, astronomer, erstwhile dancer...

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