  Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$.
The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length
$m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of
iterated derived sequences of $X$, until length 1 is reached, determines a
triangle, the \textit{Steinhaus triangle $\Delta X$ generated by the sequence
$X$}. We say that $X$ is \textit{balanced} if its Steinhaus triangle $\Delta X$
contains each element of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity.
An obvious necessary condition for $m$ to be the length of a balanced sequence
in $\mathbb{Z}/n\mathbb{Z}$ is that $n$ divides the binomial coefficient
$\binom{m+1}{2}$. It is an open problem to determine whether this condition on
$m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for
$n=2$ and generalized by John C. Molluzzo in 1976 for $n\geq3$. So far, only
the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we
answer positively Molluzzo's problem in the case $n=3^k$ for all $k\geq1$.
Moreover, for every odd integer $n\geq3$, we construct infinitely many balanced
sequences in $\mathbb{Z}/n\mathbb{Z}$. This is achieved by analysing the
Steinhaus triangles generated by arithmetic progressions. In contrast, for any
$n$ even with $n\geq4$, it is not known whether there exist infinitely many
balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. As for arithmetic progressions,
still for $n$ even, we show that they are never balanced, except for exactly 8
cases occurring at $n=2$ and $n=6$.
