Meathead Math - Word of Mouth

Puzzle for the week of 2/23/25

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 3/2/25). 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!

You’re the owner of the Meathead Math gymnasium and are starting a new series of yoga classes. At first you think you’ll need to do some advertising but then you wonder if word of mouth will be sufficient. In particular, you estimate that every member of the gym has some probability p of being friends with any other member of the gym. For example, if you’re gym member i, there’s a probability p that you’re friends with gym member j, a probability p that you’re friends with gym member k, etc. You estimate that once a gym member learns about the new yoga class, they will tell each of their other gym member friends about it.

If you select one gym member at random to tell about the promotion, what value of p is required for there to be at least a 50% chance of at least half of the gym hearing about the promotion assuming every gym member will tell their friends when they hear about it? You may assume the number of members at the gym N is large.

Please submit your answers here. Please ask any questions in the comments.