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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 $k$. 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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 $k$. 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