The Counterfeit Coin Riddle

Step-by-Step Solution

Step 1: First Weighing

1. Divide the 12 coins into three groups:

   • Group A: 4 coins
   • Group B: 4 coins
   • Group C: 4 coins (set aside)

2. Weigh Group A against Group B.

Possible Outcomes:

• If the scale balances, the counterfeit coin is in Group C (the 4 coins not weighed).
• If the scale tilts, the counterfeit coin is among the 8 coins on the scale, and we now know whether it is heavier or lighter.

---

Step 2: Second Weighing

• Take three coins from the suspected group (either Group C if the first weighing balanced, or the tilting group if not).
• Weigh them against three known good coins.

Possible Outcomes:

1. If the scale balances, the counterfeit coin is the one left out.
   • Since we already know whether the fake is heavy or light from Step 1, we now have the answer.
2. If the scale tilts, we now know the counterfeit is one of the three being weighed and whether it is heavier or lighter.

---

Step 3: Third Weighing

• Take two of the remaining suspected coins and weigh them against each other.

Possible Outcomes:

• If the scale balances, the fake coin is the one left out.
• If the scale tilts, we know exactly which one is fake and whether it is heavier or lighter.

---

Final Answer:

👉 You can always find the counterfeit coin and determine if it’s heavier or lighter in exactly 3 weighings.