After my fumbled puzzle from 5/24, solvers Izumihara Ryoma and Eric Widdison kicked up a conversation helping clarify where I erred in my solution (which I swear I will correct one of these days 🫠). As part of that discussion, they also discussed how they approached the problem and some other interesting questions that would emerge if we assumed different distributions of trees. As part of this discussion, Izumihara came up with a follow up question to share with everyone. So not only did he save my ass with his solution write-up for that problem, but he also gifted us another puzzle. Cheers to Izumihara!

[I will assume you are familiar with the Dividing Mr. Once-ler’s Estate puzzle. See the link if you need a refresher.]

From Izumihara:

Obviously, the number of fence posts will depend on the particular distribution of trees in the Truffula forest. With this in mind:

1. Over all probability distributions in the first quadrant, what are the infimum and supremum of the expected number of fence poles?

2. Can you find a distribution (or a sequence of distributions) whose expected pole count approaches the infimum or supremum?

3. What changes if we restrict to radially symmetric distributions?

Good luck solving! Please submit your answers here. Please ask any questions in the comments.