Question

У короля было10 министров. Однажды он получил шифровку от своего шпиона с известием: "Ваше величество! Мне удалось установить, что 1. Среди ваших министров не менее одного честного 2. Среди любой пары министров выбранной наугад не более одного честного. Сколько честных министров у короля?

Answer 1

Решение: Поставив в пару обязательно честного министра и любого из оставшихся девяти министров видим по условию задачи, что каждый из девяти оставшихся нечестный.

Ответ: 1 честный министр.

Answer 2

Problem Analysis

The king received a message from his spy with the following information: 1. Among his 10 ministers, there is at least one honest minister. 2. Among any randomly chosen pair of ministers, there is at most one honest minister.

The question is: How many honest ministers does the king have?

Solution

To solve this problem, we can use a logical approach. Let's consider the possible scenarios:

1. If there is only one honest minister, then there cannot be any other honest ministers in the remaining 9 ministers. This scenario does not satisfy the second condition, as any randomly chosen pair of ministers will have at most one honest minister. Therefore, this scenario is not possible.

2. If there are two honest ministers, then the remaining 8 ministers cannot be honest. In this case, any randomly chosen pair of ministers will have at most one honest minister, satisfying the second condition. This scenario is possible.

3. If there are three or more honest ministers, then the remaining ministers cannot be honest. In this case, any randomly chosen pair of ministers will have more than one honest minister, which does not satisfy the second condition. Therefore, this scenario is not possible.

Based on the analysis above, the only possible scenario is that there are two honest ministers among the ten ministers.

Conclusion

The king has two honest ministers.