Twelve balls are identical in all ways except one has a different weight. Three weighings on a balance scale will not only identify the odd ball, but also tell whether it is heavier or lighter. How many balls must be put on each side of the scale in the first weighing, the second weighing, and the third weighing?
First weighing: four against four
Second weighing: two against two
Third weighing: one against one
Let’s name the balls 1-12.
First we weigh {1,2,3,4} on the left and {5,6,7,8} on the right. There are three scenarios which can arise from this.
If they balance, then we know 9, 10, 11 or 12 is odd. Weigh {8, 9} and {10, 11} (Note: 8 is not odd)
If they balance, we know 12 is the odd one. Just weigh it with any other ball and figure out if it is lighter or heavier.
If {8, 9} is heavier, then either 9 is heavy or 10 is light or 11 is light. Weigh {10} and {11}. If they balance, 9 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).
If {8, 9} is lighter, then either 9 is light or 10 is heavy or 11 is heavy. Weigh {10} and {11}. If they balance, 9 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).
If {1,2,3,4} is heavier, we know either one of {1,2,3,4} heavier or one of {5,6,7,8} is lighter but it is guaranteed that {9,10,11,12} are not odd. Weigh {1,2,5} and {3,6,9} (Note: 9 is not odd).
If they balance, then either 4 is heavy or 7 is light or 8 is light. Following the last step from the previous case, we weigh {7} and {8}. If they balance, 4 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).
If {1,2,5} is heavier, then either 1 is heavy or 2 is heavy or 6 is light. Weigh {1} and {2}. If they balance, 6 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).
If {3,6,9} is heavier, then either 3 is heavy or 5 is light. Weigh {5} and {9}. They won’t balance. If {5} is lighter, 5 is odd (lighter). If they balance, 3 is odd (heavier).
If {5,6,7,8} is heavier, it is the same situation as if {1,2,3,4} was heavier. Just perform the same steps using 5,6,7 and 8. Weigh {5,6,1} and {7,2,9} (Note: 9 is not odd).
If they balance, then either 8 is heavy or 3 is light or 4 is light. We weigh {3} and {4}. If they balance, 8 is odd (heavier). If they don’t balance then whichever one is lighter is odd (lighter).
If {5,6,1} is heavier, then either 5 is heavy or 6 is heavy or 2 is light. Weigh {5} and {6}. If they balance, 2 is odd (lighter). If they don’t balance then whichever one is heavier is odd (heavier).
If {7,2,9} is heavier, then either 7 is heavy or 1 is light. Weigh {1} and {9}. If they balance, 7 is odd (heavier). If they don’t balance then 1 is odd (lighter).
Note: There are other possible solutions to this problem as well.