Problem 105

Special Subset Sums: Testing

Let $S (A)$ represent the sum of elements in set $A$ of size $n$ . We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$ , the following properties are true:

$S (B) \neq S (C)$ ; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S (B) > S (C)$ .

For example, ${81, 88, 75, 42, 87, 84, 86, 65}$ is not a special sum set because $65 + 87 + 88 = 75 + 81 + 84$ , whereas ${157, 150, 164, 119, 79, 159, 161, 139, 158}$ satisfies both rules for all possible subset pair combinations and $S (A) = 1286$ .

Using sets.txt (right click and “Save Link/Target As…”), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, $A_{1}, A_{2}, \dots, A_{k}$ , and find the value of $S (A_{1}) + S (A_{2}) + \dots + S (A_{k})$ .

NOTE: This problem is related to Problem 103 and Problem 106.

特殊的子集和：检验

记 $S (A)$ 是大小为 $n$ 的集合 $A$ 中所有元素的和。若从 $A$ 中任取两个非空且不相交的子集 $B$ 和 $C$ 始终满足下列条件，则称 $A$ 是一个特殊和集：

$S (B) \neq S (C)$ ；也就是说，任意子集的和都不相同。
如果 $B$ 中的元素比 $C$ 多，则 $S (B) > S (C)$ 。

例如， ${81, 88, 75, 42, 87, 84, 86, 65}$ 不是一个特殊和集，因为 $65 + 87 + 88 = 75 + 81 + 84$ ；而 ${157, 150, 164, 119, 79, 159, 161, 139, 158}$ 满足上述规则，相应的 $S (A) = 1286$ 。

在文本文件sets.txt中有一百组包含七至十二个元素的集合（文档中的前两个集合就是上述样例），找出其中所有的特殊和集 $A_{1}, A_{2}, \dots, A_{k}$ ，并求 $S (A_{1}) + S (A_{2}) + \dots + S (A_{k})$ 的值。

注意：此题和第103题及第106题有关。