Dividing Mr. Once-ler's Estate

Puzzle for the week of 5/04

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 5/11/25). 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!

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You and your three siblings are at the reading of the will for your late father—a wealthy outdoorsman—that owned a forest of very rare and valuable trees known as Truffula trees. The Truffula forest has a single clear center, and from that center outward, the density of trees drops by 50% for every kilometer you travel outward.

In his will, your father decides to split the forest evenly between you and your three siblings by creating 4 equal quadrants emanating from the center of the forest

Each sibling has very different intentions for their forest so you all agree to the creation of a fence that will section off each sibling’s quadrant. For your quadrant, you also want to make to keep the Truffula trees protected so you decide to create a fence around the outside of your quadrant (something like a circumference). To minimize fencing costs, you plan to stretch the fence snugly around just the trees at the forest’s outer edge, keeping all other trees safely within the boundary (see the red line). To actually build this fence, you'll need special posts every time the fence changes direction around these outermost trees. You also find that your section of the forest has 1000 trees.

The question for this week is, on average, how many fence posts will you need?

To clarify, you only need to worry about fencing the circumference part of your region i.e. you can ignore any fencing needed along the quadrant boundaries.

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