You have 12 coins, and they all look identical. However, one of these coins is either heavier or lighter than the others. You have a balance scale but can use it only 3 times to determine which coin is different and whether it is heavier or lighter.

Objective:
Identify the unique coin (either heavier or lighter) using the balance scale no more than 3 times.

Solution

This puzzle requires a strategic approach to minimize the number of weighings while ensuring that you can identify the odd coin and whether it’s heavier or lighter.

Step-by-Step Solution for The Coin Weighing Puzzle:

Step 1: Divide the Coins into Three Groups

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

Step 2: First Weighing

• Weigh Group 1 (coins 1, 2, 3, 4) against Group 2 (coins 5, 6, 7, 8).
  • Scenario A: If the scales balance, the odd coin is in Group 3 (coins 9, 10, 11, 12).
  • Scenario B: If the scales do not balance, the odd coin is in either Group 1 or Group 2.

Step 3: Second Weighing (If the Scales Balanced in Step 2)

• Take 3 coins from Group 3 (coins 9, 10, 11) and weigh them against 3 coins from either Group 1 or Group 2 (which you know contains only normal coins).
  • Scenario A1: If the scales balance, the odd coin is the one not weighed (coin 12), and you can determine whether it’s heavier or lighter by comparing it against a known normal coin.
  • Scenario A2: If the scales do not balance, you now know the odd coin is one of the 3 coins being weighed, and you can determine which one and whether it’s heavier or lighter by comparing any two of these coins.

Step 4: Second Weighing (If the Scales Did Not Balance in Step 2)

• Identify the heavier and lighter sides from the first weighing. Take 3 coins from the group that was heavier and 3 coins from the group that was lighter. Weigh them against each other.
  • Scenario B1: If the scales balance, the odd coin is in the remaining unweighed coin from the two groups, and you can determine whether it’s heavier or lighter by comparing it against a known normal coin.
  • Scenario B2: If the scales do not balance, you now know the odd coin is one of the 3 coins being weighed, and you can determine which one and whether it’s heavier or lighter by comparing any two of these coins.

Step 5: Final Weighing

• If necessary, use the third weighing to compare the remaining possible odd coins and identify the odd one and whether it’s heavier or lighter.

Final Answer

By carefully following these steps, you can identify the odd coin and whether it’s heavier or lighter using just 3 weighings on the balance scale.