Can you survive The Great Bar Crawl?
Puzzle for the week of 3/9
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 3/16/25). 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 solution and submitted responses will be posted around Wednesday at 10 am.
I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!
This week, we mix some good planning with some bad judgment.
It’s Tuesday evening, and you and your friends decide to celebrate Mardi Gras the fun way—with a legendary bar crawl. Your neighborhood has N bars, all within stumbling distance of each other. But rather than meticulously planning a route, you embrace the chaos:
Every hour, you pick a bar at random. No strategy, no preferences. And let’s be honest, you’re not going to remember which bars you’ve already been to.
It’s going to be a late night. You and your friends will keep hopping between bars for a while.
But since the bar crawl hasn’t started yet and you’re still lucid, you realize it might be awkward if you all return to the same bar too soon. So you decide to figure out how how long it would typically take for this to happen.
So, our question is: Assuming it’s your first time returning to a bar you’ve already been to (i.e., it’s your first “repeat” bar), on average, how many hours will have passed between your first arrival and your second?
Please submit your answers here. Please ask any questions in the comments.
As you’ve probably noticed, I really like combinatorics problems and I tend to write a lot of puzzles that make heavy use of counting and discrete math. But there’s a wide world of math out there and I want to make sure that I’m presenting problems that readers find fun and interesting. So if you have any thoughts or preferences on the kinds of problems presented, please let me know in the comments!