Mathematical Puzzles
Expert
10 min
You have 12 identical-looking coins. Exactly one coin is counterfeit and may be either heavier or lighter than the others. Using a balance scale only three times, determine which coin is counterfeit and whether it is heavier or lighter.
Hint
Divide the coins into groups of four and use the first weighing to establish whether the unusual coin is in a balanced group or an unbalanced group.
Answer
The counterfeit coin can always be identified in three weighings using a carefully designed 4-4-4 strategy.
Explanation
This is the classic 12-coin balance-scale deduction problem. The first weighing establishes whether the counterfeit is among the eight weighed coins and whether it is heavier or lighter, or whether it is among the four unweighed coins. The remaining two weighings are then designed to distinguish the remaining possibilities.
Step-by-Step Solution
Weigh 1: Coins 1-4 against 5-8.
Depending on the result, assign candidate coins and possible heavy/light states. Weigh 2 using a mixture of known-good coins and candidate coins. Weigh 3 isolates the final candidate and determines its weight status.
Depending on the result, assign candidate coins and possible heavy/light states. Weigh 2 using a mixture of known-good coins and candidate coins. Weigh 3 isolates the final candidate and determines its weight status.