Solving the Coin Puzzle: Identifying One Heavier or Lighter Coin Among 12 Using Three Weighings

Imagine you have 12 similar coins, but one of them is either slightly lighter or heavier than the others. Your goal is to find this unique coin and determine whether it is lighter or heavier using only an equal-arm balance and making just three weighings. This puzzle is not only a fun challenge but also a great exercise in logical thinking and problem-solving.

Step-by-Step Solution

(1) **Divide the Coins**

Step 1: Divide the 12 coins into three groups of 4 coins each. Group A: Coins 1, 2, 3, 4 Group B: Coins 5, 6, 7, 8 Group C: Coins 9, 10, 11, 12

Begin by dividing the 12 coins into three groups of four. This initial setup helps to narrow down the possibilities with each subsequent weighing.

Step 2: First Weighing

Weigh Group A against Group B. If the scales balance:
Outcome 1: All coins in Groups A and B are normal, and the odd coin is in Group C (Coins 9, 10, 11, 12). If one side is heavier:
Outcome 2: The odd coin is in the heavier group or the lighter group.

For the first weighing, place Group A on one side and Group B on the other. If they balance, the odd coin must be in Group C. If one group is heavier or lighter, proceed accordingly.

Step 3: Second Weighing

**Analyze the Outcomes:** **If Outcome 1 (A B):** Weigh Coin 9 against Coin 10. If they balance, the odd coin is between Coins 11 and 12. Weigh Coin 11 against any of the normal coins to determine if it is heavier or lighter. If one side is heavier or lighter, determine whether Coin 9 or 10 is the odd coin and if it is heavier or lighter. **If Outcome 2 (A ≠ B):** Assume Group A is heavier (the logic is similar if Group B is heavier). Take 3 coins from the heavier side (A) and 3 coins from the lighter side (B) for the second weighing. Weigh 1, 2, 5 against 3, 6, 7. If they balance, the odd coin is between 4 and 8, and if one side is heavier or lighter, determine the odd coin within these.

If the first weighing results in a balance, proceed to the next step with the remaining group (Coins 9, 10, 11, 12). If one side is heavier or lighter, take the respective coins and weigh them in the second step.

Step 4: Third Weighing

Final Identification: Based on the results of the second weighing, you will have a group of 3 coins that contain the odd one. Weigh any 2 of the remaining coins against each other. If they balance, the coin that wasn't weighed is the odd one, and you can determine if it is heavier or lighter based on previous weighings. If they don't balance, you can determine which one is heavier or lighter.

With the narrowing down to 3 coins, the third weighing will definitively identify the odd coin and whether it is heavier or lighter based on the results of the previous weighings.

Summary

This method allows you to effectively identify the odd coin and determine whether it is heavier or lighter using only three weighings by systematically narrowing down the possibilities based on the outcomes of each weighing.