Misplaced Priorities - Part 1
Puzzle for the week of 1/5/25
Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 1/12/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem!
The answers of all those that volunteered their solutions will be posted around Wednesday at 10 am.
In many cases, I expect the readers will be better puzzlers than me so I make no guarantees the solutions are correct. I also make no promises about having worked out the solutions to the puzzles ahead of time so it may be the case that they’re very challenging. Part of the fun is finding out!
I, like many people, was a less-than-enthusiastic customer of the airline industry this holiday season. Perhaps others can relate, but while I’m boarding I can help but wondering if there’s a more efficient way of getting people on a plane. One of the first thoughts we might consider is lining people up from outer-back to inner-front. In other words, we would line everybody up single file where the first person is seated in the back row window, the second person is seated in the second to last row window, and so on. After all of the window seats, the next person will be seated in the back row middle spot (right next to the window spot). and so on. Let’s ignore the left vs. right side of the plane for now and assume we’re filling one side at a time)
Of course, such a scheme first would require that people queue up properly!
Let’s assume for a moment that the first 10 people are in line but they are a completely random order. What’s the probability that everyone is in the correct position except two people (which are definitely in the incorrect position)? What about exactly 3 people in incorrect positions? Exactly 4? Now let’s generalize. What’s the probability that out of N randomly ordered people, there are exactly k people in an incorrect position?1
Good luck!
Please submit your thoughts, progress, or answers here.
A little google searching or asking ChatGPT should easily point you to a formula making the problem very simple. I think this problem is more fun, challenging, and satisfying if you try to figure it out yourself and derive the formula.