# 帶有數字的棋盤

The smallest sum that can be achieved is

Because

Reasoning

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)

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.