23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes?

To solve the problem, we need to determine the minimum number of handshakes that can occur at a party attended by 23 people, where each person shakes hands with at least two other people.

### Step-by-Step Analysis

1. **Graph Representation:**
   - We can represent this problem as a graph where each person is a vertex, and each handshake is an edge connecting two vertices.
   - The problem requires that each vertex (person) has a degree of at least 2, meaning each person shakes hands with at least two others.

2. **Minimum Number of Edges:**
   - The graph must be connected (since everyone shakes hands with at least two others), and the minimum degree of each vertex is 2.
   - To minimize the number of edges, consider constructing a graph where every vertex has exactly 2 edges.

3. **Constructing the Graph:**
   - A simple way to achieve this is to arrange the 23 people in a cycle. In a cycle, each person shakes hands with exactly two other people: one to their left and one to their right.
   - In a cycle with \( n \) vertices (people), there are exactly \( n \) edges (handshakes).

4. **Calculate the Handshakes:**
   - For 23 people arranged in a cycle, there would be exactly 23 edges.
   - Therefore, the minimum number of handshakes is 23.

### Conclusion
The minimum possible number of handshakes is **23**.

Would you like further details on how this was derived, or do you have any other questions?

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Here are some related questions for further exploration:

1. How would the number of handshakes change if each person shakes hands with exactly 3 people?
2. What is the maximum number of handshakes that can occur among 23 people?
3. How would the minimum number of handshakes change if the number of attendees increased to 25?
4. What is the general formula for the minimum number of handshakes in a party with \( n \) people, where each shakes hands with at least two others?
5. How would the scenario change if the condition was at least three handshakes per person?
6. How would the solution change if the group was allowed to split into smaller groups?
7. If 23 people form a complete graph, how many handshakes occur?
8. How would the graph theory concept of Eulerian paths apply to this handshake problem?

**Tip:** In graph theory, a cycle graph is a simple, closed graph where each vertex is connected to exactly two other vertices, making it an efficient structure for minimizing connections while ensuring every vertex has the required degree.

Explore the minimum number of handshakes problem with 23 people attending a party, where each person shakes hands with at least two others. Learn how to solve this using graph theory concepts, ensuring every vertex has a degree of at least 2. Suitable for Grades 7-9.

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