To solve the problem of identifying the defective coin among 15 coins, where we have the additional information that one specific coin is not defective, we can use a systematic approach. This method involves a balance scale and logical reasoning to narrow down the possibilities effectively.
Understanding the Setup
Initially, we have 15 coins, and we know that one of them is defective, but we don't know if it is heavier or lighter than the others. The twist in this scenario is that the owner has identified one coin as non-defective, which we can set aside for our investigation.
Step-by-Step Approach
Here’s how we can proceed to find the defective coin:
- Step 1: Set Aside the Non-Defective Coin
Let’s call the non-defective coin "Coin A." We now have 14 coins left to examine.
- Step 2: Divide the Remaining Coins
Split the 14 coins into three groups: two groups of 5 coins each and one group of 4 coins. Label them as Group 1 (5 coins), Group 2 (5 coins), and Group 3 (4 coins).
- Step 3: First Weighing
Weigh Group 1 against Group 2. There are three possible outcomes:
- If they balance, then the defective coin is in Group 3.
- If Group 1 is heavier, the defective coin is either heavier in Group 1 or lighter in Group 2.
- If Group 2 is heavier, the defective coin is either heavier in Group 2 or lighter in Group 1.
- Step 4: Narrowing Down
Depending on the outcome of the first weighing, you will have narrowed it down to either 4 or 5 coins. If it’s 4 coins, proceed to weigh any two of them against each other. If they balance, the defective coin is one of the two not weighed. If they don’t balance, you can determine which one is defective based on the direction of the imbalance.
- Step 5: Final Weighing
If you are left with 5 coins after the first weighing, take 3 coins and weigh them against 3 coins from the non-defective group (Coin A). If they balance, the defective coin is one of the remaining 2 coins. If they don’t balance, you can determine whether the defective coin is heavier or lighter based on the scale's tilt.
Example Scenario
Let’s say after the first weighing, Group 1 (5 coins) is heavier than Group 2 (5 coins). This indicates that either one of the coins in Group 1 is heavier or one of the coins in Group 2 is lighter. You can then take two coins from Group 1 and one from Group 2 to weigh against two known good coins (Coin A and one from Group 3). This process will help you pinpoint the defective coin.
Conclusion
By systematically eliminating possibilities through weighing and logical deduction, you can effectively identify the defective coin among the 15, even with the initial uncertainty of its weight. This method not only showcases problem-solving skills but also emphasizes the importance of strategic thinking in tackling complex challenges.
