Consider a party with 60 people. Assume each person knows more than m other people. How large must m be to guarantee that 3 people know each other?
To determine the value of m required to guarantee that 3 people know each other in a party of 60, we can use the concept of the pigeonhole principle.
The pigeonhole principle states that if you have n+1 objects to put into n containers, there must be at least one container with more than one object.
In the context of this problem, each person in the party knows more than m other people. If we consider the number of people each person knows as the "containers," and the total number of people in the party as the "objects," we need to find the minimum value of m that guarantees at least one container with three or more objects.
Let's consider the worst-case scenario where each person knows exactly m people. The maximum number of people that a person can know is m. In that case, we can set up an equation:
60 ≤ m + 2
The "+2" accounts for the minimum of 3 people (two other people and oneself) required to form a group of individuals who know each other.
Simplifying the equation:
m ≥ 58
Thus, m must be greater than or equal to 58 to guarantee that at least 3 people know each other in a party of 60 individuals.