Problem 103
Special subset sums: optimum
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 $S(B)>S(C)$.
If $S(A)$ is minimised for a given $n$, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.
$$\begin{aligned}
n = 1:& \{1\} \\
n = 2:& \{1, 2\} \\
n = 3:& \{2, 3, 4\} \\
n = 4:& \{3, 5, 6, 7\} \\
n = 5:& \{6, 9, 11, 12, 13\}
\end{aligned}$$
It seems that for a given optimum set, $A = \{a_1, a_2, … , a_n\}$, the next optimum set is of the form $B = \{b, a_1+b, a_2+b, … ,a_n+b\}$, where $b$ is the “middle” element on the previous row.
By applying this “rule” we would expect the optimum set for $n = 6$ to be $A = \{11, 17, 20, 22, 23, 24\}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = \{11, 18, 19, 20, 22, 25\}$, with $S(A) = 115$ and corresponding set string: $111819202225$.
Given that $A$ is an optimum special sum set for $n = 7$, find its set string.
NOTE: This problem is related to Problem 105 and Problem 106.
特殊的子集和：最优解
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$S(B)>S(C)$。
对于给定的$n$，称使得$S(A)$最小的集合$A$为最优特殊和集。前$5$个最优特殊和集如下所示。
$$\begin{aligned}
n = 1:& \{1\} \\
n = 2:& \{1, 2\} \\
n = 3:& \{2, 3, 4\} \\
n = 4:& \{3, 5, 6, 7\} \\
n = 5:& \{6, 9, 11, 12, 13\}
\end{aligned}$$
似乎对于一个给定的最优特殊和集$A = \{a_1, a_2, … , a_n\}$，下一个最优特殊和集将是$B = \{b, a_1+b, a_2+b, … ,a_n+b\}$，其中$b$是位于集合$A$“中间”位置的元素。
应用这条“规则”，我们猜测对于$n = 6$的最优特殊和集将是$A = \{11, 17, 20, 22, 23, 24\}$，相应的$S(A) = 117$。然而事实并非如此，上述“规则”仅仅只能给出近似最优解。对于$n = 6$，最优特殊和集是$A = \{11, 18, 19, 20, 22, 25\}$，其相应的$S(A) = 115$，并可以表示为数字字符串：$111819202225$。
求$n = 7$时的最优特殊和集$A$，并给出其数字字符串表示。