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Math Help - Set Sums

  1. #1
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    Set Sums

    Let A_1,A_2,A_3,B_1,B_2,B_3 all be finite subsets of \mathbb{N} with the following properties:

    1. For n=1,2,3, \sum_{a \in A_n} a = \sum_{b \in B_n} b.

    2. For 1\le m < n \le 3, \sum_{a \in A_m \cap A_n} a = \sum_{b \in B_m \cap B_n} b.

    Does there exist six such sets where \sum_{a \in A_1\cap A_2 \cap A_3} a \neq \sum_{b \in B_1 \cap B_2 \cap B_3} b?
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  2. #2
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    Re: Set Sums

    I got it.

    \begin{matrix}A_1 = \{3,4,7\}, & A_2 = \{1,6,7\}, & A_3 = \{2,5,7\} \\ B_1 = \{1,6,7\}, & B_2 = \{2,5,7\}, & B_3 = \{1,2,5,6\}\end{matrix}

    The set sums are all 14.

    This gives

    \begin{matrix} A_1\cap A_2 = \{7\}, & A_1\cap A_3 = \{7\}, & A_2 \cap A_3 = \{7\} \\ B_1 \cap B_2 = \{7\}, & B_1 \cap B_3 = \{1,6\}, & B_2 \cap B_3 = \{2,5\}\end{matrix}

    The set sums are all 7.

    But, A_1\cap A_2 \cap A_3 = \{7\} while B_1 \cap B_2 \cap B_3 = \emptyset, which obviously have different sums.
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