Question

You have 12 visually identical coins. One is counterfeit and weighs slightly less or slightly more than the others. Using an equal-arm balance, develop a plan to determine the counterfeit coin. Assume the coins are labeled one to twelve. Your plan should describe which coins you place on the sides of the balance and what your next step is depending on the direction (if any) the balance tilts. The fact that you don't know whether it is lighter or heavier up front makes this problem much more difficult than other similar problems.

a) Compare coins 1-4 with 5-8.
b) Compare coins 1-3 with 4-6.
c) Compare coins 1-2 with 3-4.
d) Compare coins 1-6 with 7-12.

Answer 1

To determine the counterfeit coin among 12 visually identical coins, use an equal-arm balance. Compare different groups of coins on the balance, starting with coins 1-4 and coins 5-8. Depending on the balance tilting direction, continue comparing smaller groups until you identify the counterfeit coin.

Step-by-step explanation:

To determine the counterfeit coin among the 12 visually identical coins, you can use an equal-arm balance. Follow these steps:

1. Start by comparing coins 1-4 with coins 5-8 on the balance. If the balance tilts:

• If the balance tilts to the left, it means the counterfeit coin is among coins 1-4. Go to step (b).
• If the balance tilts to the right, it means the counterfeit coin is among coins 5-8. Go to step (b).
• If the balance remains balanced, it means the counterfeit coin is among coins 9-12. Go to step (d).

(b) If the balance tilted in step (a), compare coins 1-3 with coins 4-6:

• If the balance tilts to the left, it means the counterfeit coin is among coins 1-3. Go to step (c).
• If the balance tilts to the right, it means the counterfeit coin is among coins 4-6. Go to step (c).
• If the balance remains balanced, it means the counterfeit coin is either coin 7 or 8. Go to step (d).

(c) If the balance tilted in step (b), compare coins 1-2 with coins 3-4:

• If the balance tilts to the left, it means the counterfeit coin is coin 1. Go to step (d).
• If the balance tilts to the right, it means the counterfeit coin is coin 2. Go to step (d).
• If the balance remains balanced, it means the counterfeit coin is coin 3. Go to step (d).

(d) If the balance is still balanced after step (c), compare coins 1-6 with coins 7-12:

• If the balance tilts to the left, it means the counterfeit coin is either coin 1-6. Weigh one of the coins against a known genuine coin. If it is lighter, the counterfeit is among coins 1-5. If it is heavier, the counterfeit is coin 6.
• If the balance tilts to the right, it means the counterfeit coin is either coin 7-12. Weigh one of the coins against a known genuine coin. If it is lighter, the counterfeit is among coins 7-11. If it is heavier, the counterfeit is coin 12.
• If the balance remains balanced, it means the counterfeit coin is either coin 10 or 11. Weigh one of the coins against a known genuine coin. If it is lighter, the counterfeit is coin 10. If it is heavier, the counterfeit is coin 11.