IBM Ponder This #327 · July 2025

This puzzle was suggested by Latchezar Christov - thanks!

A wire with length $L$ is stretched between two poles. Swallows come, one by one, and land on the wire. All swallows have a width equal to 1 and cannot overlap on the wire, and each new swallow selects where to land at random with uniform probabilities (taken over all the perimissible options). This repeats until no swallow can land anymore.

In this moment, there are $S$ swallows on the wire. The number $S$ is a discrete random variable whose expectancy $N=\mathbf{E}[S]$ depends on $L$. We will call "flock density" (FD) the ratio $\mathbf{FD}(L)=\frac{N(L)}{L}$. For example

FD(1.1)=0.909091
FD(2.2)=0.606061
FD(3.3)=0.665421
FD(10)=0.722358

etc.

Your goal: Find the lengths $L_1$ and $L_2$ ($L_1,L_2>2$) for which the flock density is accordingly $\mathbf{FD}(L_1)=\frac{36}{51}$ and $\mathbf{FD}(L_2)=\frac{38}{51}$. Provide eight decimal digits of accuracy.

A bonus "*" will be given for finding the median of the distribution of the distance between two consecutive landing points for $L\to \mathrm{\infty }$, where a "landing point" of a swallow is in the middle of the space it occupies. Provide four decimal digits after the decimal point.

Solution

Best opened after a real attempt

To be added.