帶有數字的棋盤


20

一個8x8的棋盤用從1到64的數字從左到右,逐行和從上到下依次標記。現在,將負號添加到這些數字中的32個中,以便在每一行和每一列中正好有4個正數和4個負數。現在將所有64個數字相加。

可以實現的最小總和是多少?

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33

The smallest sum that can be achieved is

Because

Reasoning


1

If the summation cannot be a negative number, the minimum sum that can be achieved is 0. If the summation can be a negative number, then, the minimum that can be achieved is -107 (if the greatest numbers in each row is negated)


3

This is one of the first on puzzling.SE that I can actually do! :)

There are many ways to get $0$. Note that each row has precise $4$ or each $+/-$, so we can translate so that it's just eight copies of $\{1, ..., 8\}$. By symmetry, we have $$ 1 + 8 - 2 - 7 = 0 \quad\text{and}\quad 3+6 - 4-5 = 0.$$ So we can easily make each row sum to $0$. Other combinations work. Simply alternate these down each row, eg \begin{matrix} 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \end{matrix} There are loads of similar combinations.