Question

ДАМ 20 БАЛЛОВ В ряд слева направо лежат 49 монет по 1 эре и 50-я монета — 1 крона (100 эрэ). Финдус и Петсон берут монеты слева, по 1 или 2 монеты за раз, начинает Петсон. а) Кто может гарантированно получить крону? б) Какую наибольшую сумму может гарантировать себе Петсон? в) Какую наибольшую сумму может гарантировать себе Финдус?

Answer 1

Ответ:

А) финдус б) 25эр в) 124 эры

Answer 2

Problem Analysis

In this problem, there are 49 coins worth 1 "ere" each, and the 50th coin is worth 1 "krona" (100 "ere"). Findus and Petson take turns taking 1 or 2 coins from the left side of the row. Petson starts the game. We need to determine who can guarantee getting the "krona" and calculate the maximum amount of money that Petson and Findus can guarantee for themselves.

Solution

To solve this problem, we can analyze the game step by step and determine the optimal strategy for each player.

1. Who can guarantee getting the "krona"? - Since Petson starts the game, he has the advantage of being able to take the first coin(s). - Petson can guarantee getting the "krona" by following the optimal strategy. - Therefore, Petson can guarantee getting the "krona".

2. What is the optimal strategy for Petson to guarantee the "krona"? - Petson's goal is to ensure that the number of coins remaining after his turn is always divisible by 3. - If the number of coins remaining is already divisible by 3, Petson should take 1 coin to maintain this property. - If the number of coins remaining is not divisible by 3, Petson should take 2 coins to make it divisible by 3. - By following this strategy, Petson can guarantee getting the "krona".

3. What is the maximum amount of money that Petson can guarantee for himself? - Petson can guarantee getting the "krona", which is worth 100 "ere". - Therefore, the maximum amount of money that Petson can guarantee for himself is 100 "ere".

4. What is the maximum amount of money that Findus can guarantee for himself? - Findus plays after Petson and can only take 1 or 2 coins. - Findus cannot guarantee getting the "krona" because Petson can always follow the optimal strategy to ensure that the number of coins remaining is divisible by 3. - Therefore, the maximum amount of money that Findus can guarantee for himself is less than the "krona".

Conclusion

In conclusion: - Petson can guarantee getting the "krona". - The maximum amount of money that Petson can guarantee for himself is 100 "ere". - Findus cannot guarantee getting the "krona". - The maximum amount of money that Findus can guarantee for himself is less than the "krona".