#### Identifying the odd ball from a group of 12 balls in 3 weighings with a beam balance if you do not whether the odd ball is heavier or lighter

Mahesh Godavarti

1

Let the balls be numbered 1 through 12. In the description below the first weighing is in black, the second weighing is in red and the third weighing is in blue.

Divide the balls into three equal groups.

- Step 1: Weigh balls 1, 2, 3, 4 against balls 5, 6, 7, 8. If the two sides are balanced go to step 2 else go to Step 4.
- Step 2: We know the odd ball is among balls 9, 10, 11, 12. We know balls 1 through 8 are normal. Weigh balls 1, 2, 3 against 9, 10, 11. If the two sides are balanced, go to Step 3 else go to Step 7.
- Step 3. We know ball 12 is the odd ball. Weigh ball 12 against any of the other balls to determine whether it is heavier or lighter (DONE)
- Step 4: Assume without any loss of generality that balls 1, 2, 3, 4 or on the lighter side of the beam balance and balls 5, 6, 7, 8 are on the heavier side. Then we know that the odd ball is EITHER lighter AND one of 1, 2, 3, 4 OR is heavier AND one of 5, 6, 7, 8. Weigh balls 3, 5, 6 against balls 4, 7, 8. If the two sides are equal go to step 5 else go to step 6.
- Step 5: We know the odd ball is one of 1 and 2 and is lighter. Weigh 1 and 2 against each other to determine the odd ball (DONE).
- Step 6. Without any loss of generality, let the side with balls 3, 5, 6 be the lighter side and the side with balls 4, 7, 8 be the heavier one. Then we know that the odd ball is either 3 and it's lighter or one of 7 or 8 and is heavier. Weigh 7 and 8 against each other to determine the odd ball. (DONE).
- Step 7. The side with balls 9, 10, 11 will either be heavier or lighter. We know the odd ball is one of 9, 10, or 11 and is either heavier or lighter. Weigh 9 and 10 against each other to determine which one of 9, 10, or 11 is the odd ball (DONE).