1. ## Set Sums

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

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

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

Does there exist six such sets where $\displaystyle \sum_{a \in A_1\cap A_2 \cap A_3} a \neq \sum_{b \in B_1 \cap B_2 \cap B_3} b$?

2. ## Re: Set Sums

I got it.

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

The set sums are all 14.

This gives

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

The set sums are all 7.

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