The intermediate value theorem shows us how to find an even center on an irregular shape

By Manon Bischoff edited by Daisy Yuhas

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I don't like sharing my food. But sometimes I have to cut a sandwich or pizza in half. If the other person is equally hungry, they'll be scrupulous about the division being fair. Still there are tricky cases. What happens, for example, if we split a pizza with toppings that have not been evenly distributed but instead are clustered in clumps? Mathematicians have given these scenarios some thought.

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How Do You Cut a Pizza in Half Fairly?

For simplicity’s sake, let’s start by imagining a perfectly circular pizza, evenly topped with cheese and tomato sauce and holding randomly distributed slices of pepperoni. Now you can make a straight slice, cutting the dough, tomato sauce and cheese in half. To do this, you pass a pizza cutter through the center. But suppose 30 percent of the pepperoni is on the left half and 70 percent on the right. This division would be unfair: one person would have more toppings than the other.

If you rotate the pizza clockwise but keep the cutting edge along the center, you can vary the proportion of pepperoni on both halves. After a rotation of 180 degrees, for instance, the situation will reverse: on the left half (from the point of view of the person cutting), there is 70 percent of the topping; on the right, there is merely 30 percent. Thinking about this case, one important observation is that the proportion of pepperoni can only change continuously during the rotation—that is, the proportion on the left side must increase steadily from 30 to 70 percent. But that also means there is a point along this rotation at which both sides have exactly 50 percent of the topping!

This observation is a consequence of what mathematicians call the intermediate value theorem. If a continuous function (that is, a mathematical function with no holes or breaks) takes values f(a) < s and f(b) > s, then, at some point between a and b, there is a point x such that f(x) = s. To give some examples, if the temperature is 20 degrees Celsius at 8 A.M. and 30 degrees C at 3 P.M., then there was a time between 8 A.M. and 3 P.M. when it was exactly 25 degrees. The same principle applies to our pizza.

A vertical cut in the center of a pepperoni pizza creates two halves with seven slices of pepperoni on the left and 10 on the right. Rotating the pie clockwise by 180 degrees and keeping the same cutting line results in the reverse: the left half has 10 slices, and the right half has seven. Therefore, there must be a cut between them that divides the pizza fairly.

A real pizza—particularly a handmade one—is not always perfectly circular. Still, it can be divided fairly. Irregular shapes lack symmetry and so there is no clear center point for cutting through an irregular pie. But it’s still possible to find a center of mass.

You can identify a line to cut dough, cheese and tomato sauce in half and then rotate—adjusting the placement of the line based on the handmade pie’s shape—until, after 180 degrees rotation, you are back in the starting position, with the pepperoni portions reversed for the left and right halves. Thus, one can again use the intermediate value theorem to prove the conjecture: yes, every pepperoni pizza can be fairly divided into two halves.

To all who are headed to a pizza party: you’re welcome! And next week in Part Two, we’ll take a more complicated case of splitting a meal fairly.