Now the smallest possible weight is $1+2=3$ while the largest possible weight is $5+6=11$. As there are exactly nine edges, we deduce that for the weights for each edge to be different they must take the values $3, 4, 5, 6, 7, 8, 9, 10$ and $11$. However, the total of these is $63$, an odd number, so the task is impossible.