sum sets are equal

Question

I have two sequence of positive integers say $a_1, a_2, a_3, a_4, a_5, a_6$ and $b_1, b_2, b_3, b_4, b_5, b_6$ (they may be repeated) such that the multisets $\{a_1, a_1+a_2, \cdots , a_1+a_2+\cdots +a_6, a_6, a_6+a_5, \cdots ,a_6+a_5+\cdots +a_1\}=\{b_1, b_1+b_2, \cdots , b_1+b_2+\cdots +b_6, b_6, b_6+b_5, \cdots ,b_6+b_5+\cdots +b_1\}$.

This means that the multiset containing all consecutive partial sums involving $a_1$ and $a_6$ is equal to the multiset containing all consecutive partial sums involving $b_1$ and $b_6$.

From here can we say that either $a_i=b_i$ for all $i$ or $a_i=b_{7-i}$ for all $i$ ? I have tried with some particular examples and it worked.

Answer 1

It's false: $$a_i = 1, 2, 1, 1, 3, 2 \quad b_i = 1, 1, 3, 1, 1, 3 $$ Set of partial sums for $a_i$ in order is $$ 1, 3, 4, 5, 8, 10, 2, 5, 6, 7, 9, 10 $$ Set of partial sums for $b_i$ in order is $$ 1, 2, 5, 6, 7, 10, 3, 4, 5, 8, 9, 10 $$

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Edit: This is not true even for 5 elements...

$$a_i = 1, 2, 1, 1, 2 \quad b_i = 1, 1, 1, 1, 3 $$ Set of partial sums for $a_i$ in order is $$ 1, 3, 4, 5, 7, 2, 3, 4, 6, 7 $$ Set of partial sums for $b_i$ in order is $$ 1, 2, 3, 4, 7, 3, 4, 5, 6, 7 $$

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Edit2: This is not true even for 4 elements...

$$a_i = 1, 2, 1, 2 \quad b_i = 1, 1, 1, 3 $$ Set of partial sums for $a_i$ in order is $$ 1, 3, 4, 6, 2, 3, 5, 6 $$ Set of partial sums for $b_i$ in order is $$ 1, 2, 3, 6, 3, 4, 5, 6 $$

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Edit3: This is not true even for 3 elements...

$$a_i = 1, 2, 2 \quad b_i = 1, 1, 3 $$ Set of partial sums for $a_i$ in order is $$ 1, 3, 5, 2, 4, 5 $$ Set of partial sums for $b_i$ in order is $$ 1, 2, 5, 3, 4, 5 $$