  Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points
in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap
\left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every
$n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. In 1958 Steinhaus asked whether
piercing sequences can be arbitrarily long. A negative answer was provided by
Schinzel, who proved that any such sequence may have at most $74$ elements.
This was later improved to the best possible value of $17$ by Warmus, and
independently by Berlekamp and Graham.
  In this paper we study a more general variant of piercing sequences. Let
$f(n)\geq n$ be an infinite nondecreasing sequence of positive integers. A
sequence $X=(x_1,x_2,\ldots,x_{f(N)})$ is $f$-piercing if
$\{x_1,x_2,\ldots,x_{f(n)}\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right)
\neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$.
A special case of $f(n)=n+d$, with $d$ a fixed nonnegative integer, was studied
by Berlekamp and Graham. They noticed that for each $d\geq 0$, the maximum
length of any $(n+d)$-piercing sequence is finite. Expressing this maximum
length as $s(d)+d$, they obtained an exponential upper bound on the function
$s(d)$, which was later improved to $s(d)=O(d^3)$ by Graham and Levy. Recently,
Konyagin proved that $2d\leqslant s(d)< 200d$ holds for all sufficiently big
$d$.
  Using a different technique based on the Farey fractions and stick-breaking
games, we prove here that the function $s(d)$ satisfies
$\left\lfloor{}c_1d\right\rfloor{}\leqslant s(d)\leqslant c_2d+o(d)$, where
$c_1=\frac{\ln 2}{1-\ln 2}\approx2.25$ and
$c_2=\frac{1+\ln2}{1-\ln2}\approx5.52$. We also prove that there exists an
infinite $f$-piercing sequence with $f(n)= \gamma n+o(n)$ if and only if
$\gamma\geq\frac{1}{\ln 2}\approx 1.44$.
