The coin false between doce

1. Each weighing has three possible results: left low, right low or balance.
2. With two weighings there is a maximum of 9 patterns, but there are 24 possible cases.
3. A suitable first weighing is to compare the coins 1, 2, 3, 4 against 5, 6, 7, 8.

Answer: the minimum is 3 weighings. First let's see why 2 weighings cannot be enough. Each weighing has three possible results, so two weighings only allow us to distinguish: $$
3^2=9

$$ patterns of results. But there are 24 cases to distinguish: any of the 12 coins can be fake, and they can also be heavier or lighter. Therefore, 2 weighings are not enough. With 3 weights there are: $$

3^3=27

$$ possible patterns, so in principle there is enough information. Now we give a strategy that achieves this. Number the coins from 1 to 12. **First weighing** Weight: $$

1,2,3,4\quad\t\t\t\t\t\t\t\t\text{contra}\quad 5,6,7,8.
$$ There are two types of case. Case 1: the first weighing balances So coins 1 to 8 are authentic. The false one is between 9, 10, 11 and 12. Second weighing: weigh 9, 10, 11 against 1, 2, 3. - If they balance, the false one is 12. Third weighing: weigh 12 against 1 to know if it is heavier or lighter.

• If 9, 10, 11 weigh more, the fake one is between 9, 10 and 11, and is heavier. Third weighing: weighs 9 against 10. If one weighs more, that is the false one; If they balance, the false is 11.
• If 9, 10, 11 weigh less, the fake is between 9, 10 and 11, and is lighter. Third weighing: weighs 9 against 10. If one weighs less, that is the false one; If they balance, the false is 11. Case 2: the first weighing does not balance Suppose, for the sake of ideas, that 1, 2, 3, 4 outweigh 5, 6, 7, 8. Then the false one is among these eight possibilities: - 1, 2, 3, 4 heavy;
• 5, 6, 7, 8 light. Coins 9, 10, 11 and 12 are authentic. Second weighing: $$

1,2,5\quad\t\t\t\t\t\t\t\t\text{contra}\quad 3,6,9.
$$ There are now three possible outcomes. - If they balance, there are only 4 heavy, 7 light or 8 light. Third weighing: weighs 7 against 8. If one weighs less, that is the false one; If they balance, the false one is 4 and is heavier.

• If the left side weighs more, the only compatible possibilities are 1 heavy, 2 heavy or 6 light. Third weighing: weigh 1 against 2. If one weighs more, that is the false one and is heavier; If they balance, the false one is 6 and is lighter.
• If the right side weighs more, the only compatible possibilities are 3 heavy or 5 light. Third weighing: it weighs 3 against a genuine coin, for example 9. If 3 weighs more than 9, the fake is 3 and is heavier; If they balance, the false one is 5 and is lighter. If the opposite had occurred in the first weighing—if 5, 6, 7, 8 had weighed more than 1, 2, 3, 4—the same reasoning is used, exchanging the sides and changing “heavy” to “light” where appropriate. Thus, 3 weighings are always enough, and 2 cannot be enough.