That’s pretty impressive if you actually sat down and worked it out. In any case for clarity’s sake I’m copying and pasting the answer from a riddle website:
For the first weighing let us put on the left pan balls 1,2,3,4 and on the right pan balls 5,6,7,8. There are two possibilities. Either they balance, or they don’t. If they balance, then the different ball is in the group 9,10,11,12. So for our second weighing we would put 1,2 in the left pan and 9,10 on the right. If these balance then the different ball is either 11 or 12. Weigh ball 1 against 11. If they balance, the different ball is number 12. If they do not balance, then 11 is the different ball. If 1,2 vs 9,10 do not balance, then the different ball is either 9 or 10. Again, weigh 1 against 9. If they balance, the different ball is number 10, otherwise it is number 9. That was the easy part. What if the first weighing 1,2,3,4 vs 5,6,7,8 does not balance? Then any one of these balls could be the different ball. Now, in order to proceed, we must keep track of which side is heavy for each of the following weighings. Suppose that 5,6,7,8 is the heavy side. We now weigh 1,5,6 against 2,7,8. If they balance, then the different ball is either 3 or 4. Weigh 4 against 9, a known good ball. If they balance then the different ball is 3, otherwise it is 4. Now, if 1,5,6 vs 2,7,8 does not balance, and 2,7,8 is the heavy side, then either 7 or 8 is a different, heavy ball, or 1 is a different, light ball. For the third weighing, weigh 7 against 8. Whichever side is heavy is the different ball. If they balance, then 1 is the different ball. Should the weighing of 1,5, 6 vs 2,7,8 show 1,5,6 to be the heavy side, then either 5 or 6 is a different heavy ball or 2 is a light different ball. Weigh 5 against 6. The heavier one is the different ball. If they balance, then 2 is a different light ball.
Interestingly enough, the answer that chesedname gave would provide us with the knowledge of whether the odd ball is heavier or lighter in every case.