Solution:

There are at most three different halls in the castle. Because rooms $n$ and $n+10$ are on the same hall, any two rooms with the same units digit must be on the same hall.

Now, repeatedly using the rule that rooms $n$ and $3 n+1$ are on the same hall, we find that room $1$ is on the same hall as rooms $4$, $13$, $40$, and $121$, so all rooms with units digit $0,1,3$, or $4$ are on the same hall. There need not be any other rooms on this hall, since whenever $n$ has units digit in this set, so does $3 n+1$, and so does $(n-1) / 3$, if it is a whole number.

Similarly, room $2$ is on the same hall as rooms $7$ and $22$, so all rooms with units digit $2$ or $7$ are on the same hall, and there need not be any other rooms on this hall. Finally, rooms $5$, $16$, $49$, and $148$ are on the same hall, so all rooms with units digit $5,6,8$ or $9$ are on the same hall. Thus, we see that there are at most three different halls in the castle.