  Let $n$ and $k$ be positive integers with $n>k$. Given a permutation
$(\pi_1,\ldots,\pi_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive
sums of $\pi$, i.e., $s_i:=\sum_{j=0}^{k-1}\pi_{i+j}$ for $i=1,\ldots,n$, where
we let $\pi_{n+j}=\pi_j$. What we want to do in this paper is to know the exact
value of $$\mathrm{msum}(n,k):=\min\left\{\max\{s_i : i=1,\ldots,n\}
-\frac{k(n+1)}{2}: \pi \in S_n\right\},$$ where $S_n$ denotes the set of all
permutations of $1,\ldots,n$. In this paper, we determine the exact values of
$\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary
of the results, we obtain $\mathrm{msum}(n,3)$, $\mathrm{msum}(n,4)$ and
$\mathrm{msum}(n,6)$ for any $n$.
