What If Robots Cut Your Pizza?

Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here: math, logic and probability. These problems, puzzles and riddles come from lots of top-notch puzzle folks around the world — including you! You’ll find this week’s puzzle below.

Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Bill Dugger 👏 of Sunnyvale, California, our big winner. You can find a solution to the previous Riddler at the bottom of this post.

Now here’s this week’s Riddler, a pizza pie puzzle that comes to us from Zach Wissner-Gross, a physics Ph.D. and entrepreneur from Brookline, Massachusetts.

At RoboPizza™, pies are cut by robots. When making each cut, a robot will randomly (and independently) pick two points on a pizza’s circumference, and then cut along the chord connecting them. If you order a pizza and specify that you want the robot to make exactly three cuts, what is the expected number of pieces your pie will have?

Need a hint? You can try asking me nicely. Want to submit a new puzzle or problem? Email me.

And here’s the answer to last week’s Riddler, another food puzzle concerning a picky eater going to great lengths to avoid a sandwich’s crust. If you eat only the portion of the square sandwich closer to its center than to its edges, you will eat $$(4\sqrt{2}-5)/3$$, or about 21.9 percent, of the sandwich. Very wasteful, your neuroses.

Why? I’ve adapted the excellent, thorough and very mathy explanation from reader Daniel Franke here.

If the sandwich is the unit square, i.e., the area bounded by $$|x|<1, |y|<1$$, then the edible portion is characterized by those points which are closer to the center than to the edges, or $$\sqrt{x^2+y^2}<\min(1-|x|,1-|y|).$$ Plotting that, we see that we should expect an answer of something a little less than 1/4.

The edible section, in blue, looks like this:

Let’s cut out just the top diagonal quadrant of the graph, where $$0<|x|<y$$; the rest will immediately follow from symmetry. In this case, the equation simplifies to $$\sqrt{x^2+y^2}<1-y.$$ Solving that for $$y$$, we obtain $$y < -\frac{x^2}{2} + \frac{1}{2},$$ showing that the curves which form the edges of that figure are parabolas. The top parabola terminates at the boundaries of the quadrant where $$y=|x|$$. Substituting and solving, this places its endpoints at $$(\sqrt{2}-1,\sqrt{2}-1)$$ and $$(1-\sqrt{2},\sqrt{2}-1)$$. Integrating, we obtain the area enclosed between the parabola and the $$x$$-axis: $$\int_{1-\sqrt{2}}^{\sqrt{2}-1}\left(-\frac{x^2}{2} + \frac{1}{2}\right) dx = \frac{2}{3}\left(2-\sqrt{2}\right).$$ However, some of that area falls outside our quadrant, specifically two right triangles each with a width and height of $$\sqrt{2}-1$$. Subtracting these out gives: $$\frac{2}{3}\left(2-\sqrt{2}\right) – \left(\sqrt{2}-1\right)^2 = \mathbf{\frac{4\sqrt{2}-5}{3}},$$ or approximately 0.219. Since the portion of the bread occupying the top quadrant has an area of 1 (the square has a total area of 4, so each quadrant has an area of 1), this is our final answer.

Who knew eating a sandwich was so complicated? Here’s a nice interactive of the geometry from a solving team at the company Desmos, and another nice explanation from Richard Morey. And here’s the lovely pencil-and-paper work from Bill, this week’s winner:

I offered extra credit to those exploring other sandwich shapes. Here are the illustrated portions of the sandwiches you can eat, as the number of sides on your regular polygonal bread increases:

And from Laurent Lessard, this is the fraction of those sandwiches you can safely eat:

As the number of sides increases to infinity, the bread more and more closely resembles a circle, and the portion you can eat approaches 1/4, or 25 percent. If you’re a picky eater, seek circular bread. Maybe create a line of crusty pita breads. Or invest in crumpets. Crumpets are tasty.

Elsewhere in the puzzling world:

Happy summer, have a sunny weekend, and enjoy your pizza and sandwiches — perhaps with some wine.

Oliver Roeder was a senior writer for FiveThirtyEight. He holds a Ph.D. in economics from the University of Texas at Austin, where he studied game theory and political competition.