Identifying the False Coin Among 12 with Three Weighings: An Effective Strategy
Have you ever encountered a puzzle that challenges your logical reasoning and problem-solving skills? One such classic puzzle involves identifying a single false coin among 12 coins, where the only known information is that the false coin weighs differently. With only three weighings on a simple balance scale, can you find which coin is the false one and determine whether it is heavier or lighter? This article will guide you through the logical steps to solve this puzzle.
Overview of the Puzzle
The task at hand is to identify a false coin among 12 coins using a balance scale with three weighings. The challenge lies in not only finding the false coin but also determining if it is heavier or lighter than the genuine coins. This puzzle requires careful planning and strategic weighing to achieve the goal within the given constraints.
Step 1: First Weighing
Divide the 12 coins into three groups of four coins each.
Group A: Coins 1, 2, 3, 4
Group B: Coins 5, 6, 7, 8
Group C: Coins 9, 10, 11, 12
Weigh Group A against Group B.
Case 1: Groups Balance
If the balance is even, the false coin is in Group C (Coins 9, 10, 11, 12).
Case 2: Group A is Heavier
If Group A is heavier, the false coin is either in Group A or Group B, and it is heavier.
Case 3: Group B is Heavier
If Group B is heavier, the false coin is either in Group B or Group A, and it is lighter.
Step 2: Second Weighing
Based on the result of the first weighing, narrow down the search to a group of four coins.
Case 1: False Coin in Group C
Divide Group C into two pairs: (9, 10) and (11, 12).
Subcase 1.1: Weigh (9, 10) against (11, 12)
If they balance, the false coin is either 9 or 10, and you can determine if it is heavier or lighter in the next weighing. If one side is heavier, you know the false coin is heavier and which pair it belongs to.Case 2: False Coin in Group A or B, and it is Heavier
Divide the heavier group (either A or B) into two pairs: (1, 2) and (5, 6).
Subcase 2.1: Weigh (1, 2) against (5, 6)
If they balance, the false coin is either 3 or 4, and you can determine if it is heavier or lighter in the next weighing. If one side is heavier, you know the false coin is heavier and which pair it belongs to.Case 3: False Coin in Group A or B, and it is Lighter
Divide the lighter group (either A or B) into two pairs: (3, 4) and (7, 8).
Subcase 3.1: Weigh (3, 4) against (7, 8)
If they balance, the false coin is either 9 or 10, and you can determine if it is lighter or heavier in the next weighing. If one side is lighter, you know the false coin is lighter and which pair it belongs to.Step 3: Third Weighing
Based on the outcome of the second weighing, further narrow down to two coins and weigh one against a known good coin to determine the false coin and its weight.
Case 1: False Coin in (9, 10) or (3, 4) or (7, 8)
Weigh one of the suspected coins against a known good coin.
If they balance, the other coin is false and you can determine if it is heavier or lighter. If they don’t balance, you know which coin is the false one and whether it is heavier or lighter.Case 2: False Coin in (3 or 4) or (1 or 2)
Weigh one of the suspected coins against a known good coin.
If they balance, the other coin is false and you can determine if it is heavier or lighter. If they don’t balance, you know which coin is the false one and whether it is heavier or lighter.Summary of the Approach
This method strategically divides the coins and interprets the outcomes of each weighing to determine which coin is false and whether it is heavier or lighter within just three weighings. It effectively uses the balance scale to eliminate possibilities and narrow down to the false coin.
By employing logical reasoning and a systematic approach, you can solve this classic puzzle efficiently. The same principles can be applied to various similar problems that require careful planning and strategic decision-making.