Finding Impossible Averages in the Sequence 10,11,12,13,14: No-Calculator Method

To solve this problem, let us examine the range of the numbers given:

The smallest number in the set is $10$, and the largest is $14$. Hence, any average of these numbers must fall within this range, i.e., between 10 and 14.

Considering the choices:

• $12$ is within the range $10$ to $14$.
• $9$ is outside of this range.
• $13$ is within the range $10$ to $14$.
• $11$ is within the range $10$ to $14$.

Thus, it is clear that the number that cannot be the average of $(10, 11, 12, 13, 14)$ is $9$, as it does not lie within the required range.