Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
Lets call the sum of any three aligned numbers $s$. Then the sum of the three vertical columns is $(A + D + E ) + (A + C + F) + (A + B + G) = 3s$, but this is also equal to $3A + (D + C + B) + (E + F + G) = 3A + 2s$, and hence $3A = s$, or $A = \frac{s}{3}$. Hence $A$ must be the average of any three aligned numbers. It is then easy to see that the only thing $A$ can be is $4$, and hence $\boxed{4}$ cannot be placed in the circle $E$.