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Problem 103


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:

  1. $S(B) \neq S(C)$; that is, sums of subsets cannot be equal.
  2. 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.

$n = 1: \{1\}$
$n = 2: \{1, 2\}$
$n = 3: \{2, 3, 4\}$
$n = 4: \{3, 5, 6, 7\}$
$n = 5: \{6, 9, 11, 12, 13\}$

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。若任取$A$的任意两个非空且不相交的子集$B$和$C$都满足下列条件,我们称$A$是一个特殊的和集:

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

对于给定的$n$,我们称使得$S(A)$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。

$n = 1: \{1\}$
$n = 2: \{1, 2\}$
$n = 3: \{2, 3, 4\}$
$n = 4: \{3, 5, 6, 7\}$
$n = 5: \{6, 9, 11, 12, 13\}$

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$B = \{b, a_1+b, a_2+b, … ,a_n+b\}$的形式,其中$b$是集合$A$“正中间”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,我们的方法仅仅只能找出近似最优特殊和集。对于$n = 6$,最优特殊和集是$A = \{11, 18, 19, 20, 22, 25\}$,相应的$S(A) = 115$,对应的集合数字串是:$111819202225$。

若集合$A$是$n = 7$时的最优特殊和集,求其对应的集合数字串。

注意:此题和第105题第106题有关。