Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(π_1,\ldots,π_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive sums of $π$, i.e., $s_i:=\sum_{j=0}^{k-1}π_{i+j}$ for $i=1,\ldots,n$, where we let $π_{n+j}=π_j$. What we want to do in this paper is to know the exact value of $$\mathrm{msum}(n,k):=\min\left\{\max\{s_i : i=1,\ldots,n\} -\frac{k(n+1)}{2}: π\in S_n\right\},$$ 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 $n$.