You will sign up with the team to participate in a charity event, which is known to have rich bonuses. Your team's goal is to follow the rules of the event and receive as many bonuses as possible. The amount of bonus received is closely related to the number chosen by each participant. The competition has a total of 50 integer numbers from 1 to 50 to choose from, and your team needs to choose three numbers to register. The organizers allocate bonuses for each number according to the following rules:
A. Choose any number from 1-10 to receive a bonus of $1000
B. In the 11th to 50th, if you choose the 10 numbers with the lowest number of choices, you will receive a bonus of $1500
C. In the 11th to 50th, if you choose the 20 numbers with the lowest number of choices, you will receive a $500 bonus
D. In the 11th to 50th, if you choose the 10 numbers with the highest number of choices, you will receive a $100 bonus
E. If you choose any number from 41 to 50, you will lose a $100 bonus (rule BCD applies simultaneously)
Note: In numbers 1-10, if x people choose a certain number at the same time, the bonus for that number will be evenly distributed among these x people.
In numbers 11-50, first generate the 10 numbers with the fewest number of choices, then generate the 20 numbers with the fewest number of choices, and finally generate the 10 numbers with the fewest number of choices. The lower numbers will be prioritized and counted towards the higher bonus level, while the higher numbers will be counted towards the lower bonus level.
The smaller number here is included in the higher bonus layers, while the larger number is included in the lower bonus layers, which is established in this situation. For example, assuming that there are currently 11, 14, 17, 26, 28, 30, 35, 37, 42, 45, 47, 50 numbers selected, these 12 numbers have the fewest and the same number of people. So, the relatively small 10 numbers are calculated to the 10 numbers with the least number of people selected, and a bonus of 1500 dollars (higher bonus layer) is obtained. The larger numbers 47 and 50 are calculated to the next level, which is to select the 20 numbers with the least number of people, and a bonus of $500 (lower bonus layer) is obtained
There are a total of 33 teams involved, each with 3 people, and each team is very smart.
In addition, I have tried to use programming algorithms to solve this problem. I'm not sure whether it is right and I hope there can be a mathematical solution. Due to a total of three numerical ranges, there are ten options available. I assume that there are equally many teams selected for each option initially, and they randomly select numbers within a numerical range. Calculate the benefits for each team. The option chosen by the team with the lowest return decreases by one team selecting it, while increasing by one team selecting the option chosen by the team with the highest return. Conduct multiple rounds of survival of the fittest and observe which option ultimately leads to the most teams. It turns out that choosing 1 number from 1-10 and 2 numbers from 11-40 or 3 numbers from 11-40 seems a good idea.