Small) Dinner Parties
Mrs. X has seven acquaintances, A, B, C, D, E, F and G, that have never met one
another (none of the seven acquaintances has ever been introduced to the other
six people (for example, A has never been introduced to B, C, D, E, F and G),
each of the seven has only met with Mrs. X).
Mrs. X would like each of the seven acquaintances to meet each of the other six
people. To this purpose, she wishes to organize a sequence of dinner parties.
Unfortunately, she lives in a very small apartment and, for each party, she can
only accommodate three people (other than herself) at a dinner table. For
example, the first party may include Mrs. X together with A, B and C.
What is the minimum number of such parties that will allow each of the seven
acquaintances to meet at least once with each of all the other six acquaintances?

The problem you have posed is essentially about finding the minimum number of dinner parties where each of Mrs. X's seven acquaintances can meet every other acquaintance at least once, while only three of them can attend each dinner.

This problem is related to combinatorics, specifically to a variant of the famous **complete graph** problem in graph theory. Let’s break it down:

### Step 1: Representing the Problem
We can represent this problem as a **graph**. Each of the seven acquaintances can be thought of as a vertex in a graph. We need every vertex to be connected to every other vertex at least once (this would make it a **complete graph** \(K_7\)).

In graph terms:
- There are 7 people, so the graph will have 7 vertices.
- Every person needs to meet every other person, which means we need all possible edges between pairs of vertices.
- There are \(\binom{7}{2} = 21\) edges (since \(\binom{7}{2}\) represents the number of ways to choose 2 people from a group of 7, i.e., the number of pairs who need to meet).

### Step 2: Organizing Dinner Parties
At each dinner party, Mrs. X invites three people. In graph terms, this means we are forming a triangle (a set of 3 vertices fully connected to each other). Each triangle accounts for 3 edges (pairs of acquaintances that meet during the party).

### Step 3: Minimum Number of Parties
To solve this, we need to cover all 21 edges with as few triangles as possible. Each dinner party allows us to cover 3 edges, so in theory, the minimum number of dinner parties needed is:

\[
\frac{21 \text{ edges}}{3 \text{ edges per party}} = 7 \text{ parties}.
\]

Thus, the minimum number of dinner parties required is **7**.

### Step 4: Can 7 Parties Really Work?
Now, we need to check if it is possible to design 7 parties in such a way that all 21 pairs of acquaintances meet at least once. One possible arrangement is:

- **Party 1**: A, B, C
- **Party 2**: A, D, E
- **Party 3**: A, F, G
- **Party 4**: B, D, F
- **Party 5**: B, E, G
- **Party 6**: C, D, G
- **Party 7**: C, E, F

If we list out the pairs of people who meet in each party, we can verify that every pair meets at least once:
- **Party 1**: AB, AC, BC
- **Party 2**: AD, AE, DE
- **Party 3**: AF, AG, FG
- **Party 4**: BD, BF, DF
- **Party 5**: BE, BG, EG
- **Party 6**: CD, CG, DG
- **Party 7**: CE, CF, EF

This covers all possible pairs of acquaintances, ensuring that each pair meets exactly once.

### Conclusion:
The minimum number of dinner parties required for each of the seven acquaintances to meet every other acquaintance at least once is **7**.

Would you like to see the detailed step-by-step verification of each pair meeting, or do you have any further questions?

Here are five related questions to expand your understanding:
1. How does this problem change if Mrs. X could invite four people instead of three?
2. What if there were eight acquaintances instead of seven?
3. How many edges would be in a complete graph with 10 people?
4. Can you explain the concept of a complete graph in simple terms?
5. What would be the result if Mrs. X could only invite two people at a time?

**Tip**: When solving combinatorial problems like this, breaking the problem into smaller steps and verifying the solution with a logical arrangement can help ensure accuracy.

Mrs. X has seven acquaintances, A, B, C, D, E, F, and G, who have never met each other. Mrs. X wants to organize a series of dinner parties, with each party consisting of herself and three acquaintances. The goal is for each acquaintance to meet every other one at least once. What is the minimum number of dinner parties required?

Solve Mrs. X's dinner party problem, where 7 acquaintances must meet every other one at least once. Discover how graph theory and combinatorics provide the solution in just 7 parties, using the complete graph concept and edge covering. This problem is suitable for students in Grades 10-12.

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