Find the Odd Ball


Denote the left pan of the scale by LP and the right pan by RP, and follow this recipe:
  1. Place 4 balls in each pan of the scale. If the scale balances, mark each of these 8 balls with the letter N to denote that they are normal. Then go to Step 2. If the scale does not balance, mark each of the 4 unweighed balls with an N, since they are normal, and go to Step 5.
  2. The odd ball is among the 4 unweighed balls. Place two of them in LP, and one of them plus an N ball in RP. If RP drops, got to Step 3. If LP drops, go to Step 4.
  3. This means either that one of the balls in LP is light, or the unmarked ball in RP is heavy. Weigh the potentially light balls against each other. If balance is achieved, the unmarked ball previously in RP is the odd ball and is heavy. If, instead, the scale does not balance the potentially light balls, the pan that rises isolates the odd ball, establishing that it is truly light.
  4. This means either that one of the balls in LP is heavy, or the unmarked ball in RP is light. Weigh the potentially heavy balls against each other. If balance is achieved, the unmarked ball previously in RP is the odd ball and is light. If, instead, the scale does not balance the potentially heavy balls, the pan that lowers isolates the odd ball, establishing that it is truly heavy.
  5. Here the results of the first weighing made one pan drop and one rise. Mark those balls in the rised pan with the letter L, as being potentially light, and those balls in the lowered pan with an H, as being potentially heavy. Now place two H and one L ball in LP, and place an H, an L, and a N ball in RP. If balance is achieved, go to Step 8. If the left pan rises, go to Step 6. If the left pan drops, go to Step 7.
  6. Here either the L ball in LP is light, or the H ball in RP is heavy. Now weigh the L ball against an N ball. If balance is achieved, then the H ball is the odd ball and truly heavy. Lack of balance isolates the L ball as being odd, and truly light.
  7. Here one of the balls in LP is heavy, or else the L ball in RP is light. Weigh the heavy balls against each other. Balance implies the L ball as odd, and truly light. Scale imbalance, on the other hand, nails down the truly heavy ball.
  8. In this case, the three H and two L balls just weighed are normal, and the odd ball must be among the remaining H and two L balls that did not participate in the second weighing. Thus weigh the two L balls against each other. If the scale balances, the H ball is the odd one, and truly heavy. If the scale fails to balance, the rising pan isolates the L ball that is truly light.
This completes the consideration of all possible outcomes, demonstrating that three weighings are sufficient to isolate the Odd Ball, and establish whether it is lighter or heavier than the rest.