Question

. 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 гарантированнополучить крону? б) Какую наибольшую сумму может гарантировать себе Петсон?в) Какую наибольшую сумму может гарантировать себе Финдус? (даю 50 баллов но ответьте нормально пж)

Answer 1

Ответ:

а) Гарантированно крону может получить Петсон.

Ему достаточно в первый раз взять 2 монеты, а после брать так чтобы в сумме с Финдусом у них вышло 3 монет(Например:Петсон взял первым две монеты, после Финдус берёт 1 монету, тогда Петсон должен взять 2 монеты или если Финдус взял 2 монеты, то Петсон должен взять 1 монету и т.д.. В конце остаютсю 3 монеты(последняя это крона) и должен брать Финдус, ну и так и сяк монету получит Петсон(Финдус берёт 2 монеты, тогда Петсон забирает крону, если Финдус берёт 1 монету, то Петсон забирает 2 оставшиеся монетки))

б-в) Гарантированное максимальное количество монет

Петсон как и Финдус может получить только 25 монет. Т.к. им для получения ещё большего кол-ва монет им может помешать только опонент. Например Петсон будет брать только по максимальному количеству монет(по два). И Петсону получить больше 25монет может помешать только Финдус, если будет брать то же по 2 монеты за каждый ход.

Аналогично и с Финдусом.он получит гарантированно не меньше 25 монет.

# лучший ответ пожак. хотя бы за количество слов

Answer 2

Problem Analysis

In this problem, there are 49 coins worth 1 "ere" each and one coin worth 1 "krona" (100 "ere"). Findus and Petson take turns picking coins from left to right, with Petson starting. They can take either 1 or 2 coins at a time. The goal is to determine who can guarantee getting the krona and what is the maximum amount of money each person can guarantee for themselves.

Solution

To solve this problem, we can analyze the possible scenarios and strategies for Findus and Petson.

a) Who can guarantee getting the krona?

To guarantee getting the krona, the player who starts (Petson) needs to ensure that they leave an odd number of coins for the other player after their turn. This way, the other player will be forced to take the last coin, which is the krona.

Since Petson starts and can take either 1 or 2 coins at a time, he can ensure that he leaves an odd number of coins for Findus after his turn. Therefore, Petson can guarantee getting the krona.

b) What is the maximum amount of money Petson can guarantee for himself?

To maximize the amount of money Petson can guarantee for himself, he needs to leave the minimum amount of coins for Findus after his turn. This means Petson should always take 2 coins if there are at least 2 coins remaining, and take 1 coin if there is only 1 coin remaining.

Let's analyze the scenarios:

1. If there are 49 coins remaining, Petson takes 2 coins, leaving 47 coins for Findus. 2. If there are 47 coins remaining, Petson takes 2 coins, leaving 45 coins for Findus. 3. If there are 45 coins remaining, Petson takes 2 coins, leaving 43 coins for Findus. 4. If there are 43 coins remaining, Petson takes 2 coins, leaving 41 coins for Findus. 5. If there are 41 coins remaining, Petson takes 2 coins, leaving 39 coins for Findus. 6. If there are 39 coins remaining, Petson takes 2 coins, leaving 37 coins for Findus. 7. If there are 37 coins remaining, Petson takes 2 coins, leaving 35 coins for Findus. 8. If there are 35 coins remaining, Petson takes 2 coins, leaving 33 coins for Findus. 9. If there are 33 coins remaining, Petson takes 2 coins, leaving 31 coins for Findus. 10. If there are 31 coins remaining, Petson takes 2 coins, leaving 29 coins for Findus. 11. If there are 29 coins remaining, Petson takes 2 coins, leaving 27 coins for Findus. 12. If there are 27 coins remaining, Petson takes 2 coins, leaving 25 coins for Findus. 13. If there are 25 coins remaining, Petson takes 2 coins, leaving 23 coins for Findus. 14. If there are 23 coins remaining, Petson takes 2 coins, leaving 21 coins for Findus. 15. If there are 21 coins remaining, Petson takes 2 coins, leaving 19 coins for Findus. 16. If there are 19 coins remaining, Petson takes 2 coins, leaving 17 coins for Findus. 17. If there are 17 coins remaining, Petson takes 2 coins, leaving 15 coins for Findus. 18. If there are 15 coins remaining, Petson takes 2 coins, leaving 13 coins for Findus. 19. If there are 13 coins remaining, Petson takes 2 coins, leaving 11 coins for Findus. 20. If there are 11 coins remaining, Petson takes 2 coins, leaving 9 coins for Findus. 21. If there are 9 coins remaining, Petson takes 2 coins, leaving 7 coins for Findus. 22. If there are 7 coins remaining, Petson takes 2 coins, leaving 5 coins for Findus. 23. If there are 5 coins remaining, Petson takes 2 coins, leaving 3 coins for Findus. 24. If there are 3 coins remaining, Petson takes 2 coins, leaving 1 coin for Findus.

In the end, Petson can guarantee getting the krona and can guarantee a maximum of 24 coins for himself.

c) What is the maximum amount of money Findus can guarantee for himself?

Since Petson starts and can guarantee getting the krona, Findus will always have fewer coins to choose from. Therefore, Findus cannot guarantee getting the krona.

To maximize the amount of money Findus can guarantee for himself, he needs to leave the minimum amount of coins for Petson after his turn. This means Findus should always take 2 coins if there are at least 2 coins remaining, and take 1 coin if there is only 1 coin remaining.

Let's analyze the scenarios:

1. If there are 48 coins remaining, Findus takes 2 coins, leaving 46 coins for Petson. 2. If there are 46 coins remaining, Findus takes 2 coins, leaving 44 coins for Petson. 3. If there are 44 coins remaining, Findus takes 2 coins, leaving 42 coins for Petson. 4. If there are 42 coins remaining, Findus takes 2 coins, leaving 40 coins for Petson. 5. If there are 40 coins remaining, Findus takes 2 coins, leaving 38 coins for Petson. 6. If there are 38 coins remaining, Findus takes 2 coins, leaving 36 coins for Petson. 7. If there are 36 coins remaining, Findus takes 2 coins, leaving 34 coins for Petson. 8. If there are 34 coins remaining, Findus takes 2 coins, leaving 32 coins for Petson. 9. If there are 32 coins remaining, Findus takes 2 coins, leaving 30 coins for Petson. 10. If there are 30 coins remaining, Findus takes 2 coins, leaving 28 coins for Petson. 11. If there are 28 coins remaining, Findus takes 2 coins, leaving 26 coins for Petson. 12. If there are 26 coins remaining, Findus takes 2 coins, leaving 24 coins for Petson. 13. If there are 24 coins remaining, Findus takes 2 coins, leaving 22 coins for Petson. 14. If there are 22 coins remaining, Findus takes 2 coins, leaving 20 coins for Petson. 15. If there are 20 coins remaining, Findus takes 2 coins, leaving 18 coins for Petson. 16. If there are 18 coins remaining, Findus takes 2 coins, leaving 16 coins for Petson. 17. If there are 16 coins remaining, Findus takes 2 coins, leaving 14 coins for Petson. 18. If there are 14 coins remaining, Findus takes 2 coins, leaving 12 coins for Petson. 19. If there are 12 coins remaining, Findus takes 2 coins, leaving 10 coins for Petson. 20. If there are 10 coins remaining, Findus takes 2 coins, leaving 8 coins for Petson. 21. If there are 8 coins remaining, Findus takes 2 coins, leaving 6 coins for Petson. 22. If there are 6 coins remaining, Findus takes 2 coins, leaving 4 coins for Petson. 23. If there are 4 coins remaining, Findus takes 2 coins, leaving 2 coins for Petson.

In the end, Findus cannot guarantee getting the krona, but he can guarantee a maximum of 23 coins for himself.

Summary

To summarize: a) Petson can guarantee getting the krona. b) Petson can guarantee a maximum of 24 coins for himself. c) Findus cannot guarantee getting the krona, but he can guarantee a maximum of 23 coins for himself.

I hope this explanation helps! Let me know if you have any further questions.