TITLE: Steinhaus's Easter Egg Problem
QUESTION [19 upvotes]: The following is the text of Steinhaus's so-called Easter egg problem. According to this article of Roman Duda, this was recorded in the New Scottish Book around Easter 1955 and "Steinhaus offered an Easter egg (decorated by him personally) as a prize."
This question has two parts:

Provide an English translation of the Polish text 
Report on the status of the problem

REPLY [8 votes]: This is Włodzimierz Holsztyński's translation (that he deleted) with my finish:
A circle $K$ has circumference $1$. $O$ belongs to $K$, and it is a beginning of the time axis which is wounded around $K$. On this axis the time intervals of consecutive lengths $1,\frac 12,\frac 13,\dots,\frac 1n,\ldots$ are positioned in a way that $n$-th interval $I_n$ is given by $\ I_n = \langle\sum_{k=1}^{n-1}\frac 1k,\sum_{k=1}^n\frac 1k\rangle$. The time-points $t_n=n\alpha$, where $\alpha$ is irrational (e.g. $\alpha=\frac{\sqrt 5-1}2$), are considered to be signals if and only if $t_n\in I_n$ on $K$. 
CLAIM to be PROVED [Włodzimierz Holsztyński stopped here]: Show that the signals constitute a Poisson process of rate parameter $1$, i.e. the relative measure of those $x$'s for which the interval $\langle x, x+\ell\rangle$ contains exactly $k$ signals is $\frac{\ell^k}{k!}e^{-\ell}$.