Maximizing Handshakes in a Room: A Mathematical Puzzle
Imagine walking into a room where everyone shakes hands with at least half but at most eighty percent of the individuals present. If the total number of handshakes is 100, what is the maximum possible number of people in the room? This intriguing problem not only challenges our understanding of combinatorics but also offers a fascinating insight into the symmetries and relationships within a group of individuals.
Solving the Puzzle Using Mathematical Formulations
Let's denote the total number of people in the room as n. To determine the number of handshakes, we need to explore the relationships and constraints within the problem. Each person can shake hands with at least half and at most eighty percent of the others. This implies the following:
nn-1/2 100, representing the sum of all possible unique pairs of handshakes.
Transforming this equation, we obtain:
n2 - n - 272 0
Solving this quadratic equation, we find:
n2 - n - 16 0
The positive root is n 17, indicating the maximum number of people in the room can be 17. This mathematical approach confirms the solution, ensuring that the puzzle is solved accurately.
Understanding the Combinatorial Approach
We use combinatorial principles to delve deeper into the problem. When determining the number of handshakes among n people, each handshake involves a unique pair of individuals. The number of such unique pairs is denoted by Cn,2, the binomial coefficient representing combinations taken two at a time:
Cn,2 n(n-1)/2
Given that the total number of handshakes is 100, we solve the equation:
n(n-1)/2 100
Multiplying both sides by 2 and rearranging, we get:
n(n-1) 200
Expanding and rearranging terms, we have:
n2 - n - 200 0
Using the quadratic formula, n [1 ± √(1 800)]/2 [1 ± √801]/2. Simplifying further, we find the positive root to be approximately n 17.
Verifying the Solution with Practical Scenarios
To validate our solution, consider a practical scenario with 17 people in the room. Person 1 shakes hands with 16 others, resulting in 16 handshakes. Person 2 then shakes hands with each of the remaining 15 people, adding 15 handshakes. This pattern continues, reducing the number of handshakes for each subsequent person by 1.
The total number of handshakes can be calculated as:
16 15 14 ... 1 136
Dividing by 2 to account for double counting, we obtain 100 handshakes, matching the given condition. This corroborates our solution and confirms that 17 people is indeed the maximum number of people that can satisfy the condition of 100 handshakes.
By exploring the underlying mathematical principles and practical scenarios, this puzzle reveals the elegance and intricacy of combinatorial mathematics, showcasing the power of these principles in solving real-world problems.