I'm not making an assumption that the lists don't have elements in common on separate rows.
First create a table of integers.
mysql> create table n (n int primary key);
mysql> insert into n values (1),(2),(3),(4),(5),(6);
You can join this to your rooms table using the FIND_IN_SET() function. Note that this cannot be optimized. It will execute N full table scans. But it does create an interim set of rows.
mysql> select * from n inner join rooms on find_in_set(n.n, rooms.csv) order by rooms.room, n.n;
+---+----+-------------+-------+
| n | id | room | csv |
+---+----+-------------+-------+
| 2 | 3 | brown_room | 2,4,6 |
| 3 | 4 | brown_room | 3 |
| 4 | 3 | brown_room | 2,4,6 |
| 6 | 3 | brown_room | 2,4,6 |
| 1 | 1 | gold_room | 1,2,3 |
| 2 | 1 | gold_room | 1,2,3 |
| 3 | 1 | gold_room | 1,2,3 |
| 4 | 5 | gold_room | 4,5,6 |
| 5 | 5 | gold_room | 4,5,6 |
| 6 | 5 | gold_room | 4,5,6 |
| 1 | 2 | silver_room | 1,2,3 |
| 2 | 2 | silver_room | 1,2,3 |
| 3 | 2 | silver_room | 1,2,3 |
+---+----+-------------+-------+
Use GROUP BY to reduce these rows to one row per room. Use GROUP_CONCAT() to put the integers together into a comma-separated list.
mysql> select room, group_concat(distinct n.n order by n.n) as csv
from n inner join rooms on find_in_set(n.n, rooms.csv) group by rooms.room
+-------------+-------------+
| room | csv |
+-------------+-------------+
| brown_room | 2,3,4,6 |
| gold_room | 1,2,3,4,5,6 |
| silver_room | 1,2,3 |
+-------------+-------------+
I think this is a lot of work, and impossible to optimize. I don't recommend it.
The problem is that you are storing comma-separated lists of numbers, and then you want to query it as if the elements in the list are discrete values. This is a problem for SQL.
It would be much better if you did not store your numbers in a comma-separated list. Store multiple rows per room, with one number per row. You can run a wider variety of queries if you do this, and it will be more flexible.
For example, the query you asked about, to produce a result with numbers in a comma-separated list is more simple, and you don't need the extra n
table:
select room, group_concat(n order by n) as csv from rooms group by room
See also my answer to Is storing a delimited list in a database column really that bad?