Posted February 7, 2012 By Presh Talwalkar. Read about me , or email me .

Figuring out how to divide a regular pizza is easy. A standard pizza is usually cut into equally-sized slices, so two people can just eat the same number of slices.

But what happens when you cut the pizza in odd ways, like the following division?

Let’s play a game to test your skill at getting the most pizza.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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A pizza dividing game

You and I are to share a round pizza, perfectly shaped as a circle.

I will slice the pizza by making cuts from the middle to the crust. The slices may be of various sizes.

We agree to divide up the pizza according to the following rules of politeness:

a) We will pick pieces in alternate turns b) You get to pick the first slice c) Only slices that are adjacent to already picked slices can be chosen. That means after the first turn, there will be two available slices to pick on each turn

If I make the pizza an even number of slices, then we both end up with the same number of slices.

If I make the pizza an odd number of slices, you end up picking the first and last slice, so you end up with one more piece of pizza.

If we are both playing strategically, would you rather have me slice the pizza into an even or an odd number of slices?

Think about this carefully–finding the answer is a lot harder than it sounds! Scroll down below to read the answer.

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The solution to the game

The intuitive answer is that you are better off with an odd number of slices as you end up with one more piece of pizza. But it turns out that having more pieces is not to your benefit.

You are actually better off picking an even number of slices!

With an even number of slices, you can always guarantee that you get at least 1/2 of the pizza.

Why is that?

The proof is simple. Imagine the pizza slices are colored as red and green in alternate slices. There will be an equal number of red and green slices. By the pigeonhole principle, either the red or green slices will make up at least 1/2 of the pizza. You as the player going first can guarantee the larger set by picking the first slice of the right color. I will be forced to pick an adjacent slice of the opposite color, and in subsequent turns you can always pick the slice I just revealed of the right color, forcing me again to pick a slice of the opposite color. You’ll end up with all the slices of the right color and get at least 1/2 of the pizza.

(Also check out a similar problem: the coins in a row puzzle)

With an odd number of slices, I can actually cut the pizza so you only get 4/9 of its area.

How is that possible?

The math required to prove this is quite complicated, and I will refer you to two papers that provide detailed proofs. See How to eat 4/9 of a pizza and Solution of Peter Winkler’s Pizza Problem.

Here’s an example of a pizza in which the first person picking the slice is limited to 4/9 of the pizza (image from this paper).

The numbers represent the size of the slices and add up to 9. There are slices of size “0,” but we can also interpret that to mean really, really small slices.

Suppose Alice picks first, and Bob picks second. Here is the strategy Bob can use to limit Alice’s pizza total to 4/9:

If Alice starts with a non-zero piece, Bob picks the available piece adjacent to a thick cut. Afterwards Bob always picks the piece just revealed by Alice (so he follows Alice) unless this would mean eating from a still untouched interval. If both pieces available to Bob are from untouched intervals, he picks the piece from the interval of smaller size. One can verify (several elementary cases) that Bob always eats at least 5 with this strategy.

It’s counter-intuitive, but you’re best with an even number of slices: you end up with the same number of pieces, but you can control that you end up with more pizza.