An organizer of a 64-person meeting wants to start by having every person individually greet each other person for 30 seconds. Indicate whether the proposed solution achieves the goal without using excessive time. Before answering, think of a possible solution approach for this seemingly simple problem. Form two lines of 32 each, with attendees matched up. Every 30 seconds, have one line shift left one position (with the person on the left end wrapping to right).
Once the person that started on the left is back on the left, then have each line split into two matched lines, and repeat until each line has just 1 person.
The Correct Answer and Explanation is :
The proposed solution seems like a clever approach to minimizing the time and ensuring that every person interacts with every other person. Let’s break it down:
Initial Setup
- The 64 attendees are divided into two lines of 32.
- Each person greets the person they are directly opposite in the other line, which takes 30 seconds per greeting.
Shift Process
- After each round of greetings, one line shifts left by one position. This means that the person at the left end of the line wraps around to the right end of the line.
- This continues for 31 rounds, and in each round, each person greets a different person.
Refining the Process
Once the rounds of shifting have occurred, we then divide each line into two smaller lines of 16 people. The process of shifting and greeting continues, reducing the size of the groups at each step.
Analysis of the Approach
Let’s calculate the total number of greetings:
- In the initial 32-person lines, each person greets the 31 others on the opposite side, so 31 rounds are needed for everyone to greet everyone.
- Once the lines split, there are 16 people in each line, and each person needs to greet 15 others, taking 15 rounds.
- This process continues as the lines are further split.
At each step, the number of people decreases, reducing the number of rounds required. The time for each greeting is 30 seconds, so the total time can be calculated by summing the time for each round and splitting process.
Conclusion
This approach is efficient because it maximizes the number of people interacting each round while minimizing the total number of rounds required. It’s an optimal solution that reduces time significantly compared to having each person greet everyone individually.
Now, let me generate an image to visually represent the process.
The diagram visualizes the proposed approach for organizing the greetings among the 64 people. As you can see, people initially greet those across from them, and after each round, one of the lines shifts to facilitate new greetings. The lines eventually split and continue the process in smaller groups. This method ensures that all attendees meet each other while minimizing the total time.
This is an efficient solution to the problem. It avoids the excessive time required if everyone were to greet every other person individually.
