整数子集表示

若子集有 $k$ 个数，每个数可“+用 / -用 / 不用”三种状态，所以最多得到 $3^k$ 个不同和值。

要覆盖 $1\sim 40$（连同对应负数与 0，至少需要 81 个值：$-40\sim 40$），故

$$3^k\ge 81\Rightarrow k\ge 4.$$

4 个数是否可行？取

$$\{1,3,9,27\}.$$

任意系数取 $-1,0,1$，可得到平衡三进制表示，覆盖区间

$$-(1+3+9+27)\sim (1+3+9+27)= -40\sim 40.$$

因此最小规模为 4，最小子集可取

$$\boxed{\{1,3,9,27\}}.$$

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英文解析

If a subset has $k$ numbers, and each number can be in one of three states ("used plus", "used minus", or "not used"), then at most $3^k$ distinct sums can be obtained.

To cover the range $1\sim 40$ (including the corresponding negative numbers and $0$, requiring at least $81$values: $-40\sim 40$), we have

$$3^k\ge 81\Rightarrow k\ge 4.$$

Is $k=4$ feasible? Taking

$$\{1,3,9,27\}.$$

By choosing any coefficients from $\{-1,0,1\}$, we can obtain a balanced ternary representation, covering the interval

$$-(1+3+9+27)\sim (1+3+9+27)= -40\sim 40.$$

Thus, the minimum size is $4$, and a possible minimum subset is

$$\boxed{\{1,3,9,27\}}.$$