To identify the counterfeit coin among the 8 identical-looking coins using a balance scale only twice, we can follow a three-step algorithm. Let’s denote the coins by A, B, C, D, E, F, G, H.
**Step 1: First weighing**
Divide the 8 coins into 3 groups: Group 1 with 3 coins A, B, C, Group 2 with 3 coins D, E, F, and Group 3 with 2 coins G, H.
For the first weighing, balance coins from Group 1 against coins from Group 2 on the scale. Now, there are three possible outcomes:
1. If the scale balances: The counterfeit coin must be in Group 3.
2. If Group 1 is heavier: The counterfeit coin must be in Group 1 and is heavier.
3. If Group 1 is lighter: The counterfeit coin must be in Group 1 and is lighter.
**Step 2: Redistribute the coins and prepare for the second weighing**
Depending on the first weighing’s outcome, we proceed differently:
1. If counterfeit coin is in Group 3: Put coin G on the left pan and coin H on the right pan.
2. If counterfeit coin is in Group 1 and is heavier: Put coins A, D, and E on the left pan; and coins B, F, and G on the right pan.
3. If counterfeit coin is in Group 1 and is lighter: Put coins A, D, and E on the right pan; and coins B, F, and G on the left pan.
**Step 3: Second weighing**
Again, there are three possible outcomes in each case:
1. If counterfeit coin is in Group 3:
- If G and H balance: This contradicts the assumption that the counterfeit coin is in Group 3 as no coin should have its weight different from the rest.
- If G is heavier or lighter: This indicates that G is the counterfeit coin, and we also learn whether it’s heavier or lighter.
- If H is heavier or lighter: This indicates that H is the counterfeit coin, and we also learn whether it’s heavier or lighter.
2. If counterfeit coin is in Group 1 and is heavier:
- If A, D, E and B, F, G balance: This contradicts the assumption that the counterfeit coin is in Group 1 as no coin is heavier.
- If A, D, E is heavier: The counterfeit coin must be A and is heavier.
- If B, F, G is heavier: The counterfeit coin must be B and is heavier.
3. If counterfeit coin is in Group 1 and is lighter:
- If A, D, E and B, F, G balance: This contradicts the assumption that the counterfeit coin is in Group 1 as no coin is lighter.
- If A, D, E is lighter: The counterfeit coin must be A and is lighter.
- If B, F, G is lighter: The counterfeit coin must be B and is lighter.
As you can see, using only 2 weightings, we’re able to identify the counterfeit coin and determine whether it’s heavier or lighter than the rest.