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 [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
1. **Define the problem and variables:**
   We are given the set $\{1, 2, 3, 4, 5, 6, 7\}$ and need to place each number in a circle such that the sum of any three aligned numbers is the same. Let this common sum be denoted by $s$.

2. **Set up the equations:**
   Consider the three vertical columns:
   \[
   (A + D + E), (A + C + F), (A + B + G)
   \]
   The sum of these three columns is:
   \[
   (A + D + E) + (A + C + F) + (A + B + G) = 3s
   \]

3. **Rearrange the terms:**
   Notice that the sum can also be written as:
   \[
   3A + (D + C + B) + (E + F + G) = 3s
   \]

4. **Equate the two expressions:**
   Since both expressions represent the same total sum, we have:
   \[
   3A + (D + C + B) + (E + F + G) = 3s
   \]
   and
   \[
   (A + D + E) + (A + C + F) + (A + B + G) = 3s
   \]

5. **Simplify the equation:**
   By comparing the two expressions, we get:
   \[
   3A + 2s = 3s
   \]
   Subtract $2s$ from both sides:
   \[
   3A = s
   \]
   Therefore,
   \[
   A = \frac{s}{3}
   \]

6. **Determine the value of $A$:**
   Since $A$ must be an integer and $s$ is the sum of three integers from the set $\{1, 2, 3, 4, 5, 6, 7\}$, $s$ must be divisible by 3. The only integer in the set $\{1, 2, 3, 4, 5, 6, 7\}$ that can be the average of three aligned numbers (i.e., $A = \frac{s}{3}$) is 4.

7. **Conclusion:**
   Since $A = 4$, the number 4 cannot be placed in the circle $E$ because it must be placed in the circle $A$.

The final answer is $\boxed{4}$