8 ball puzzle

Been thinking about this puzzle for the last couple of days and finally solved it today, after some discussion with BT, Chris and De'en during lunch, it allowed me to pursue my own train of thoughts. Anyway, the puzzle goes like this, you have 8 balls of which one weighs differently from the others. Using a mass balance and only 3 weighings, figure out which ball weighs differently and whether it is heavier or lighter.









Solution:
Case 1(1st weighing)






=> 1, 2, 3, 4, 5, 6 = k
=> (7 or 8) not k

Therefore weigh 7 and 8 against k to find which ball weighs differently.

Case 2 (1st weighing)





=> (1 or 2 or 3) < k
=> or (4 or 5 or 6) > k
=> 7, 8 = k

Case 2a (2nd weighing)






=> (5 or 6) > k
=> 1, 2, 3, 4, 7, 8 = k
Therefore, weigh 5 against k, if 5 = k, => 6 > k

Case 2b (2nd weighing)





=> 3 < 4
=> 1, 2, 5, 6, 7, 8 = k
Therefore, weigh 3 against k, if 3 = k => 4 > k.

Case 2c (2nd weighing)

=> (1 or 2) < k
=> 3, 4, 5, 6, 7, 8 = k
Therefore weigh 1 against k, if 1 = k, 2 < k.

I missed out a couple of steps but the idea is essentially there.
SOLVED!!!