Two positions of a same dice are as shown below. Identify the number at the bottom when the top is $1$.
[Note: Consider the two states sharing one common number, say $4$, with the clockwise loops identifying $4-2-1$ and $4-3-6$.]

Let's find the opposite faces using the single-common-face rule:
1. Identify the common number in both positions, which is $4$.
2. Write down the numbers clockwise starting from the common number $4$:
* Position 1: $4\to 2\to 1$
* Position 2: $4\to 3\to 6$

3. By comparing the corresponding positions in the sequences:
* $2$ is opposite to $3$
* $1$ is opposite to $6$
* $4$ is opposite to the remaining number ($5$)

Therefore, when the top number is $1$, the number at the bottom face will be $6$.