You have 12 identical-looking balls, but one of them is either heavier or lighter than the rest. You have a balance scale that you can use only three times.

How can you determine which ball is the different one and whether it is heavier or lighter?

Answer:

Here’s how you can determine the odd ball and whether it is heavier or lighter in three weighings:

1. First Weighing:
   • Divide the 12 balls into three groups of 4 balls each: Group A, Group B, and Group C.
   • Weigh Group A against Group B.

   Scenario 1: If the scales balance, the odd ball is in Group C. Scenario 2: If the scales do not balance, the odd ball is in either Group A or Group B.

2. Second Weighing:
   • If Scenario 1: Take 3 balls from Group C and weigh them against 3 balls from Group A or B (known normal balls).
     • If they balance, the odd ball is the remaining one in Group C.
     • If they do not balance, the odd ball is among the 3 balls being weighed, and you know whether it’s heavier or lighter.
   • If Scenario 2: Take 2 balls from the heavier side and 2 balls from the lighter side from the first weighing, and swap them with each other (heavier group to lighter side and lighter group to heavier side). Weigh them.
     • If they balance, the odd ball is among the 2 balls not weighed this time from the unbalanced groups.
     • If they do not balance, you can determine if the odd ball is heavier or lighter based on the movement of the scale.
3. Third Weighing:
   • If you narrowed it down to 3 balls, take 2 of them and weigh them against each other.
     • If they balance, the odd ball is the remaining one.
     • If they do not balance, you can determine which one is the odd ball and whether it is heavier or lighter based on the known imbalance.

By following these steps, you can identify the odd ball and determine if it is heavier or lighter within three weighings.