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. 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 👏 John Swartzentruber 👏 of Swarthmore, Pa., 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, which is something a little different. With encryption prominent in the news this week, I figured you might want to get your hands dirty decoding some top-secret messages. So I crafted some for you.
Here are four different coded messages. Each has a different encryption scheme and they progress, I think, from easiest to toughest to crack. Submit the decrypted messages as your answers.
- A zsnw kmuuwkkxmddq kgdnwv lzw XanwLzajlqWayzl Javvdwj!
- xckik acvlbeg oz mmqn xnlautw. gzag, mwcht, kbjzh… ulw cpeq edr mom dhqx lksxlioil?
- hy vg nw rh ev pr is or tf?
- 😎😊, 😓😇😀😓’😒 😈😓. 😍😎😖 😆😄😓 😁😀😂😊 😓😎 😖😎😑😊. 😇😄😘, 😀😓 😋😄😀😒😓 😈😓’😒 😅😑😈😃😀😘.
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And here’s the solution to last week’s Riddler, concerning a nightmarish airplane boarding process.
Given how the other passengers are choosing their seats, you have a 50 percent chance of finding your assigned seat empty. The solution may seem counterintuitive — at first glance it seems like there will be a growing wave of wrong-seat-sitters, mayhem will ensue, fights will break out, and the odds that you get your assigned seat will plummet. But really, there are only two seat assignments you care about: yours and the one meant for the world’s worst passenger, the guy who boards first. These are the only two seats that can be the final unoccupied seat. This can be proven by contradiction: If the seat belonging to the Nth passenger (where N isn’t you or the first passenger) is the final open seat, then it was also open when the Nth passenger boarded, and she would’ve taken it then, so it can’t be the final open seat. Q.E.D. The two potential final seats — your and the first guy’s — are otherwise identical, so when you board, there’s a 50-50 chance yours is the one open at the end.
Another tack is a sort-of induction argument, starting with a smaller plane and seeing what happens to the solution when it gets larger. Imagine the plane has just two seats. The world’s worst person will clearly take your seat half the time, leaving you with your assigned seat half the time. Now imagine the plane has three seats. The world’s worst person will take your seat a third of the time. Another third of the time he’ll take the other passenger’s seat, in which case that other passenger will take your seat half the time. Your seat is taken ⅓+(⅓*½)=½ the time. Again, you get your assigned seat half the time. And so on and so forth. The answer is 50 percent regardless of how large the plane is.
You could also turn to code to approximate the answer, like in this great simulator from reader John Pettitt.
The Oscars are Sunday, but we’ve got an envelope of our own to open right now. The 💌 Frank Gorshin Memorial Coolest Riddler Extension Award 💌 goes to [drumroll] Ryan Davis. Ryan altered the plane to mimic a Boeing 747-400, and baked in seat-rating information from TripAdvisor. In his setup, passengers whose seats are taken prefer to steal a high-quality seat. Here are his results. The poshest sections tend to have the greatest number of passengers not in their assigned seats.
Honorable mentions: Zach Wissner-Gross, a surefire first-ballot Riddler Hall of Famer, extended the problem so that bumped passengers sit in the closest open seat. The corners are more appealing if you want to sit in your assigned seat:
And Gavin Byrnes emailed me with the following spooky-cool extension:
In my scenario, there are two types of passengers on a plane: Big Todds and Random Roberts. When a Random Robert boards the plane, he will sit in any random unoccupied seat. When a Big Todd boards the plane, he will sit in his assigned seat, even if there is already a Random Robert in it. If there is a Random Robert in a Big Todd seat, he will be crushed to death and become a Vengeful Ghost. I’m interested in what the math of this ends up being.
According to Gavin’s analysis, if there are 50 Big Todds, there will end up being about 15 Vengeful Ghosts. Yikes. Thank God it’s Friday.