The arithmetic mean of the six numbers $9$, $12$, $15$, $18$, $20$, and $22$ is increased by $1$ if the number $15$ is replaced by which of the following numbers?

1. Total number of data items ($n$) = $6$.
2. Let the initial sum of the numbers be $S$. If the arithmetic mean increases by $1$, the new sum ($S_{\text{new}}$) must satisfy:

$$\frac{S_{\text{new}}}{6}=\frac{S}{6}+1⟹S_{\text{new}}=S+6$$

3. This implies that the total sum of the numbers must increase by exactly $6$.
4. If $15$ is replaced by a new number $x$, the change in sum is given by $x-15=6$.

$$x=15+6=21$$

Therefore, $15$ must be replaced by $21$.