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:
$S(B) \ne S(C)$; that is, sums of subsets cannot be equal.
If $B$ contains more elements than $C$ then $S(B) \gt S(C)$.
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 $S(A_1) + S(A_2) + \cdots + S(A_k)$.
NOTE: This problem is related to Problem 103 and Problem 106.

Here’s a python code we can use to validate if a set follow the rules provided:

“`python
def is_special(A):
from itertools import combinations
n = len(A)
for i in range(1, n//2 + 1):
for comb in combinations(A, i):
s = sum(comb)
if any(s == sum(other) for other in combinations(A, i)):
return False
if any(s < sum(other) for other in combinations(A, n - i)): return False return True ``` In the code above, we iterate over all combinations of A with length i using the itertools.combinations function. If the sum of any combination equals the sum of any other combination with the same length, or if the sum of a combination is less than the sum of a longer combination, we return False. Otherwise, we return True. To use this function, you would iterate over all sets in your file, compute their sums if they are special as follows: ```python with open("sets.txt") as file: lines = file.readlines() S = sum(sum(map(int, line.split(','))) for line in lines if is_special(map(int, line.split(',')))) ``` Remember to replace "sets.txt" with the actual path to your file. This code reads the file line by line, splits each line by commas (assuming that's how the numbers are separated), and parses the elements to integers. It then checks if the set is special using the function previously defined, and if the set is special, it computes its sum. Please note that the given Python solution is straightforward but not the most efficient.

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