It’s almost Christmas, and Christmas means food: turkey, dressing, candy canes, oranges, cranberries, chocolate, and, of course, pizza.
(OK, maybe pizza is not the most traditional of foods, but it’s still a popular holiday choice, so humor me.)
Pizzas normally come pre-sliced. The question is, and I’m sure you’ve asked yourself this a lot, “How do we eat this pre-sliced pizza in a way that ensures nobody gets an unfair share?”
That’s the question, as New Scientist reported on December 11, that Rick Mabry and Paul Deiermann kept asking themselves when they used to share pizza for lunch at Louisiana State University in Shreveport. They kept getting into discussions about the mathematics of slicing it up while the pizza itself congealed on their plates.
Here’s the problem that bothered them: If the waiter cuts the pizza off-centre, but all the cuts from edge to edge cross at a single point, and the angles between all the adjacent cuts are identical, will two people taking turns eating adjacent pieces get equal shares by the time they’ve worked their way around the whole pizza…and if not, who will get more?
Like I said, it’s a problem that has baffled most of us at some time or other…hasn’t it?
Well, whether it has or hasn’t, it baffled them, and after years of work, the two mathematicians have arrived at the solution that works in all cases.
It’s that “all cases” that makes it special. It’s fairly easy to see that if a pizza is sliced just once, and the cut doesn’t pass right through the centre, the piece that includes the centre is larger and hence the person who eats it gets more.
A pizza cut twice, into four parts, works the same way: whomever eats the slice that contains the centre gets the bigger portion. After that, as long as there are an even number of cuts and the diners alternate taking pieces, they end up with the same amount of pizza each.
But if there are an odd number of cuts, things get more complicated. If you cut the pizza with 3, 7, 11, 15… cuts, and no cut goes through the centre, then whomever gets the slice that includes the centre gets more pizza. But if you use 5, 9, 13, 17… cuts, then the person who gets the centre ends up with less.
There’s been a “pizza theorem” that postulates this since the late 1960s (naturally). The problem has been rigorously proving it.
That’s what Mabry and Deiermann achieved. First they came up with elegant solutions to the “three-cut problem” and the “five-cut problem.” They thought they could just proceed from there, but things got messy (sorry) as they cut the pizza more times, or, as New Scientist puts it, “the solution still included a complicated set of sums of algebraic series involving tricky powers of trigonometric functions,” summed up more succinctly as “ugly.”
So Mabry and Deiermann continued to work on it. For 11 more years. (Well, they did other things, too, but they kept revisiting it from time to time.)
The breakthrough came in 2006. Mabry was on a vacation in southern Germany, where, he says, “I had a nice hotel room, a nice cool environment, and no computer…I started thinking about it again and that’s when it all started working.” He rewrote the algebra in more elegant form, discovered there were some simple-looking sums in the middle of it, went searching to see if anyone had already worked them out, and discovered a 1999 paper that referenced a mathematical statement from 1979 that showed them what they needed to do to flesh out the proof.
If that seems like a lot of work for a trivial problem, just consider the important practical applications of it:
OK, there aren’t any. But, says Mabry, “It’s a funny thing about some mathematicians. We often don’t care if the results have applications because the results themselves are so pretty.”
Which makes this a better Christmas topic than you might have thought when I first mentioned pizza. Christmas is, among many other things, a celebration of beauty.
You may not be able to hang a mathematical proof on your Christmas tree, but that doesn’t make it any less beautiful in its own way.