A Combinatorial Problem: Finding a Group of k People Who Like Each Other
Combinatorics often presents us with brain-teasing puzzles that push our cognitive abilities to the limit. One such puzzle involves finding people who mutually like each other. This problem is a true challenge, even for the most advanced mathematicians.
Imagine a group of 30 people, where each person likes exactly k others from the group. The task before us is to determine the minimum value of k that guarantees at least one pair of people who like each other mutually.
The key result of this problem is that the minimum value of k ensuring at least two people mutually liking each other is 15. In other words, if each person likes at least 15 others, it is guaranteed that there will be at least one pair of people who like each other mutually.
Let’s consider a few illustrative examples:
- Imagine a party with 30 attendees where each person selects 15 individuals they like the most. In this scenario, it is guaranteed that there will be at least one mutual liking pair.
- If the number of liked individuals is less than 15, say 10, it is possible that no mutual liking pairs exist. However, if each person likes at least 15 others, mutual likings become inevitable.
Visualizing this problem, picture all 30 people arranged in a circle, with each person liking the next 15 people clockwise. In this setup, mutual pairs are almost guaranteed to exist.
This problem beautifully illustrates the complexity of combinatorics and shows how a small change in a condition (in this case, the value of k) can significantly alter the entire outcome. In the realm of mathematical puzzles, this is one that inspires both interest and admiration.
