Coin Weighing 3

Let’s label the coins 1 through 12 and divide them into three groups of 4: A = {1, 2, 3, 4}, B = {5, 6, 7, 8}, and C = {9, 10, 11, 12}.

For the first weighing, place group A on the left pan and group B on the right pan.

Case 1: A and B balance. All 8 coins in groups A and B are genuine. The counterfeit is in group C. For the second weighing, take two coins from C — say 9 and 10 — and weigh them against two genuine coins.

• Case 1A: {9, 10} and the genuine coins balance. The counterfeit is either 11 or 12. We don’t yet know if it’s heavier or lighter. Weigh 11 against a genuine coin. If they balance, 12 is the counterfeit — weigh it against a genuine coin to determine if it’s heavier or lighter. If 11 doesn’t balance, it’s the counterfeit, and the direction tells us whether it’s heavier or lighter.

• Case 1B: {9, 10} doesn’t balance with the genuine coins. The counterfeit is 9 or 10, and we now know whether the counterfeit is heavier or lighter from the direction of the imbalance. Weigh 9 against a genuine coin. If they balance, 10 is the counterfeit. If they don’t balance, 9 is the counterfeit.

Case 2: A and B don’t balance. Say A is heavier than B. The counterfeit is either a heavy coin in A or a light coin in B. All coins in C are genuine.

For the second weighing, we rearrange the coins strategically. Take coins {1, 2, 5} and weigh them against {3, 6, 9}, where coin 9 is from the genuine group C.

• Case 2A: {1, 2, 5} and {3, 6, 9} balance. The counterfeit is one of {4, 7, 8}. From the first weighing, we know coin 4 could be heavy and coins 7 or 8 could be light. Weigh 7 against 8. If they balance, 4 is the heavy counterfeit. If 7 is lighter, it’s the counterfeit. If 8 is lighter, it’s the counterfeit.

• Case 2B: {1, 2, 5} is heavier than {3, 6, 9}. The counterfeit must be one of {1, 2}, both potentially heavy, or coin 6, potentially light. Since {1, 2, 5} is heavier, coins 1 or 2 being heavy would explain it, but coin 6 being light would also make the right side lighter. We weigh 1 against 2. If they balance, 6 is the light counterfeit. If 1 is heavier, it’s the counterfeit. If 2 is heavier, it’s the counterfeit.

• Case 2C: {1, 2, 5} is lighter than {3, 6, 9}. The counterfeit must be coin 3 or 4 (both potentially heavy) or coin 5 (potentially light). Coin 5 being light would make the left side lighter, which matches what we observe. Coins 3 or 4 being heavy would make the right side heavier, which also matches. Weigh 3 against 4. If they balance, 5 is the light counterfeit. If 3 is heavier, it’s the counterfeit. If 4 is heavier, it’s the counterfeit.

This method guarantees we identify the counterfeit and determine its weight in 3 weighings.