1. Finding the Counterfeit Coin. You have 5 coins. You know that one of them is counterfeit and weighs less than the others. Suppose also that you have a balance scale. Is there a strategy for finding the counterfeit coin using the balance scale only twice? If yes, what is the strategy?
2. Finding the Counterfeit Coin Revisited. Suppose you have 9 coins that look identical. Suppose, however, that eight are genuine (and identical) but one is counterfeit. The only clue to the counterfeit coin is that it weighs slightly more than the genuine coins. Suppose also that you have a balance scale. Can you find a way to determine which coin is counterfeit using the balance scale only twice?
3. Finding the Counterfeit Coin – Part 3. Suppose you have 12 coins that look identical. Suppose, however, that eleven are genuine (and identical) but one is counterfeit. The only clue to the counterfeit coin is that it has a different weight. I.e., it weighs either slightly more or slightly less than the genuine coins but you don’t know which. Suppose also that you have a balance scale. Can you find a way to determine which coin is counterfeit and tell whether it is heavier or lighter using the balance scale only three times?
1. Strategy for finding the counterfeit coin using the balance scale only twice:
To determine the counterfeit coin among the 5 coins using the balance scale, follow these steps:
Step 1:
- Divide the coins into two groups of 2 and leave one coin aside.
- Place one group of 2 coins on each side of the balance scale.
Possible outcomes:
- If the scale tips, with one side being lower, you can skip step 2. Proceed to step 3.
- If the scale balances, move to step 2.
Step 2:
- Take the remaining 2 coins not on the scale from step 1 and place one on each side of the balance scale.
Possible outcomes:
- If the scale tips, with one side being lower, the coin on that side is the counterfeit coin.
- If the scale balances, the coin you left aside in step 1 is the counterfeit coin.
Step 3 (if applicable):
- Take the group of 2 coins that caused the scale to tip in step 1.
- Place one coin on each side of the balance scale.
Possible outcomes:
- If the scale tips, the coin on the lighter side is the counterfeit coin.
- If the scale balances, the coin you left aside in step 1 is the counterfeit coin.
2. Strategy for finding the counterfeit coin among 9 coins using the balance scale only twice:
To determine the counterfeit coin among the 9 coins using the balance scale, follow these steps:
Step 1:
- Divide the coins into three groups of 3 coins each.
- Place two groups of 3 coins on each side of the balance scale.
Possible outcomes:
- If the scale tips, with one side being heavier, move to step 2.
- If the scale balances, the counterfeit coin is in the group that was not on the scale.
Step 2:
- Take the group of 3 coins that caused the scale to tip in step 1.
- Take 2 coins from that group and place one on each side of the balance scale. Leave the remaining coin aside.
Possible outcomes:
- If the scale tips, with one side being heavier, the heavier coin on that side is the counterfeit coin.
- If the scale balances, the remaining coin you left aside in step 2 is the counterfeit coin.
3. Strategy for finding the counterfeit coin among 12 coins using the balance scale only three times:
To determine the counterfeit coin among the 12 coins using the balance scale, follow these steps:
Step 1:
- Divide the coins into three groups of 4 coins each.
- Place two groups of 4 coins on each side of the balance scale.
Possible outcomes:
- If the scale tips, with one side being heavier, move to step 2.
- If the scale balances, the counterfeit coin is in the group that was not on the scale.
Step 2:
- Take the group of 4 coins that caused the scale to tip in step 1.
- Divide this group into two groups of 2 coins each, and leave 2 coins aside.
- Place one group of 2 coins on each side of the balance scale.
Possible outcomes:
- If the scale tips, with one side being heavier, move to step 3.
- If the scale balances, the remaining 2 coins you left aside in step 2 are the counterfeit coins.
Step 3 (if applicable):
- Take the group of 2 coins that caused the scale to tip in step 2.
- Place one coin on each side of the balance scale.
Possible outcomes:
- If the scale tips, the heavier coin on that side is the counterfeit coin.
- If the scale balances, the remaining coin you left aside in step 2 is the counterfeit coin.