Appeasing the Cherry Blossom Horde

Puzzle for the week of 3/30

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 4/6/25). 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 weekend marks the peak bloom for D.C.’s cherry blossoms 🌸, and as always, there's a ton of foot traffic around the Tidal Basin—a circular-ish body of water just off the Potomac.

Now, imagine that in an alternate universe, the D.C. government decided (questionably) to build a floating walking path straight across the middle of the basin. The idea was to help with pedestrian traffic flow — even if it meant ruining some of the scenic views 😅.

While the path was being constructed, the parks department needed to fence it off to keep curious visitors from using it too soon. So they decided to throw up a temporary barrier somewhere across the basin. The only rule? It had to cut across the floating path to block it. Beyond that, they didn’t overthink it—they just picked two random spots along the shoreline and stretched a rope between them.

Now that construction is done, you — a humble parks worker — are on your way to remove the barrier and finally open the path to the public. But halfway there, you realize…you forgot the map! The map would’ve told you which of the two ends of the floating path the barrier is closer to, so you could have picked the shorter walk.

You’re not sure if it’s worth turning back, so instead you ask:
What’s the expected ratio of the shorter segment of the path to the longer one, given that the barrier randomly intersects the path?

In other words, if the barrier cuts the floating path into two segments, what is the expected value of the ratio:

\(\frac{\text{length of shorter segment}}{\text{length of longer segment}}\)

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