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Will Someone Be Sitting In Your Seat On The Plane?

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, the readers. And readers, I need more riddles! Email me with submissions.

You’ll find this week’s puzzle below.

Mull it over on your commute, dissect it on your lunch break, and argue about it with your friends and lovers. When you’re ready, submit your answer using the form at the bottom. I’ll reveal the solution next week, and a correct submission (chosen at random) will earn a shoutout in this column. Important small print: To be eligible for the shoutout, I need to receive your correct answer before 11:59 p.m. EST on Sunday — have a great weekend!

Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Guy D. Moore 👏 of Darmstadt, Germany, 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 take on a classic that captures the essence of one of my greatest phobias. Just reading it fills me with panic and dread. Enjoy!

There’s an airplane with 100 seats, and there are 100 ticketed passengers each with an assigned seat. They line up to board in some random order. However, the first person to board is the worst person alive, and just sits in a random seat, without even looking at his boarding pass. Each subsequent passenger sits in his or her own assigned seat if it’s empty, but sits in a random open seat if the assigned seat is occupied. What is the probability that you, the hundredth passenger to board, finds your seat unoccupied?

Extra credit: It’s time to offer up a 🏆 Coolest Riddler Extension Award 🏆. Spice up this travel hellscape for your fellow readers. Change the size of the plane, the ticketing scheme, the boarding rules, or something far more creative than I can think of. Submit a description and analysis in the form below, or shoot me a link to your work on Twitter. We’ll publish a winner next week.

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

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And here’s the solution to last week’s Riddler, concerning a dog on the hunt, a duck on the lam, and a circular pond. As I warned last week, this one is really tough. Only 3.1 percent of submissions were correct, a new record low. But I heartily applaud the effort. As every famous old person has been reported to have said, “Our greatest glory is not in never falling, but in rising every time we fall.”

The most commonly submitted answer, by far, was π, the mathematical constant roughly equal to 3.14. That answer is wrong. It’s correct if the duck follows a very simple strategy: make a beeline toward the shore opposite where the dog starts. In that case, the duck would have to cover a distance equal to the pond’s radius to escape, but the dog would have to cover a distance equal to half the pond’s circumference to catch it, which is π times farther.

But give the duck some credit! The duck can be far cleverer than that. The duck can escape a dog that is much faster — in fact, the duck can escape a dog that is about 4.6033 times faster than it!

The basic idea is that the duck should swim out from the center of the pond a bit, and start swimming in circles around the pond’s center. Specifically, it should swim out a distance 1/4.6033 of the radius, and begin circling the center. The dog’s only viable strategy is to follow the duck as closely as it can. Since the duck is swimming around in a smaller circle than the dog is running, the duck has the angular advantage, and the dog will fall behind. The math involved is first figuring out the duck’s optimal small circle, and then figuring out when and in what direction the duck should ultimately bolt for the shore. The duck’s optimal strategy will look roughly like this, cooked up by Friend of the Riddler™ Zach Wissner-Gross:

I’m going to spare you all the gnarly math details to arrive at that 4.6 number. For those who take pleasure in the gnarly details, see this thorough explanation at DataGenetics, or this one at Mathematical Recreations. (The first swaps a monster for the dog and a row boater for the duck, but otherwise the problems are the same.) Or follow along with a discussion of the solution from the great folks over at BoardGameGeek.

You can also play a version of this duck-escape game yourself (you take on the role of the duck, recast as a ladybug) here.

Before we go, here’s an embed of a tweet of a video taken of computer screen displaying a visualization of dots representing a duck and a dog, from FotR™ Jon Wiesman:

And here is last week’s Riddler, come to life! (Hat tip to reader Patrick Hanna.)

God bless the Internet. Have a super cool weekend!

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.