Question

Имеется 2009 одинаковых по виду монет, среди которых либо 2 фальшивые, либо фальшивых нет. Как за 2 взвешивания на чашечных весах без гирь узнать, есть фальшивые или их нет, если фальшивые монеты имеют массу, отличную от настоящих монет.

Answer 1

Отложить 1 монету на стол так чтобы осталось 2008 монет; разложить их поровну на чашках весов, если чаши будут равны, то среди монет нет фальшивой, если различны, то 1 фальшивая монета лежит на столе

Answer 2

Problem Analysis

You have 2009 identical-looking coins, among which either 2 are fake or there are no fake coins. The fake coins have a different weight than the genuine coins. You need to determine if there are any fake coins using only 2 weighings on a balance scale without weights.

Solution

To solve this problem, we can divide the 2009 coins into three groups: A, B, and C.

1. Divide the coins into two equal groups, A and B, each containing 1004 coins. 2. Weigh group A against group B on the balance scale: - If the two groups weigh the same, it means there are no fake coins in these groups. In this case, we move to step 3. - If one group is heavier than the other, it means there are fake coins in that group. In this case, we move to step 4.

3. Divide group A into two equal subgroups, A1 and A2, each containing 502 coins. 4. Weigh group A1 against group A2 on the balance scale: - If the two groups weigh the same, it means there are no fake coins in group A. In this case, we move to step 5. - If one group is heavier than the other, it means there are fake coins in that group. In this case, we move to step 6.

5. Divide group B into two equal subgroups, B1 and B2, each containing 502 coins. 6. Weigh group B1 against group B2 on the balance scale: - If the two groups weigh the same, it means there are no fake coins in group B. In this case, we move to step 7. - If one group is heavier than the other, it means there are fake coins in that group. In this case, we move to step 8.

7. Weigh any two coins from group C on the balance scale: - If the two coins weigh the same, it means there are no fake coins in group C. - If one coin is heavier than the other, it means that coin is fake.

8. Weigh any two coins from the heavier group (either A1, A2, B1, or B2) on the balance scale: - If the two coins weigh the same, it means there are no fake coins in that group. - If one coin is heavier than the other, it means that coin is fake.

By following these steps, we can determine if there are any fake coins and identify which coins are fake if they exist.

Example

Let's go through an example to illustrate the solution. Suppose we have 2009 coins, among which 2 are fake.

1. Divide the coins into two groups, A and B, each containing 1004 coins. 2. Weigh group A against group B on the balance scale: - If group A is heavier than group B, it means there are fake coins in group A. - If group A and group B weigh the same, it means there are no fake coins in these groups.

Let's assume that group A is heavier than group B.

3. Divide group A into two subgroups, A1 and A2, each containing 502 coins. 4. Weigh group A1 against group A2 on the balance scale: - If group A1 is heavier than group A2, it means there are fake coins in group A1. - If group A1 and group A2 weigh the same, it means there are no fake coins in group A.

Let's assume that group A1 is heavier than group A2.

5. Divide group B into two subgroups, B1 and B2, each containing 502 coins. 6. Weigh group B1 against group B2 on the balance scale: - If group B1 is heavier than group B2, it means there are fake coins in group B1. - If group B1 and group B2 weigh the same, it means there are no fake coins in group B.

Let's assume that group B1 and group B2 weigh the same.

7. Weigh any two coins from group C on the balance scale: - If the two coins weigh the same, it means there are no fake coins in group C. - If one coin is heavier than the other, it means that coin is fake.

Let's assume that the two coins from group C weigh the same.

8. Weigh any two coins from the heavier group (in this case, group A1) on the balance scale: - If the two coins weigh the same, it means there are no fake coins in group A1. - If one coin is heavier than the other, it means that coin is fake.

In this example, we have determined that there are 2 fake coins, and they are in group A1.

Conclusion

By following the described weighing process, we can determine if there are any fake coins among the 2009 identical-looking coins and identify which coins are fake if they exist.