There are n persons sitting around a circular table. They start singing a 2 minute song in pairs such that no two persons sitting together will sing together. This process is continued for 28 minutes. Find n.

Answer

To solve the problem, we need to determine the number of persons \( n \) sitting around a circular table who can sing in pairs for a total of 28 minutes, with the condition that no two persons sitting next to each other can sing together.

### Step-by-Step Solution:

1. **Understanding the Time and Pairing**:
- Each pair sings for 2 minutes.
- The total singing time is 28 minutes.
- Therefore, the total number of pairs that can sing in this time is:
\[ \text{Total pairs} = \frac{28 \text{ minutes}}{2 \text{ minutes/pair}} = 14 \text{ pairs} \]

2. **Counting Total Possible Pairs**:
- For \( n \) persons, the total number of ways to choose pairs (without any restrictions) is given by:
\[ \text{Total pairs} = \frac{n(n-1)}{2} \]
- However, since no two adjacent persons can sing together, we need to exclude those pairs.

3. **Counting Adjacent Pairs**:
- In a circular arrangement, each person has exactly 2 adjacent persons. Thus, the total number of adjacent pairs is \( n \).

4. **Setting Up the Equation**:
- The number of valid pairs that can sing together is:
\[ \text{Valid pairs} = \frac{n(n-1)}{2} - n \]
- We know from step 1 that the valid pairs must equal 14:
\[ \frac{n(n-1)}{2} - n = 14 \]

5. **Simplifying the Equation**:
- Multiply the entire equation by 2 to eliminate the fraction:
\[ n(n-1) - 2n = 28 \]
- This simplifies to:
\[ n^2 - 3n - 28 = 0 \]

6. **Factoring the Quadratic Equation**:
- We can factor the quadratic:
\[ (n - 7)(n + 4) = 0 \]
- This gives us two potential solutions:
\[ n - 7 = 0 \quad \Rightarrow \quad n = 7 \]
\[ n + 4 = 0 \quad \Rightarrow \quad n = -4 \quad (\text{not valid since } n \text{ must be positive}) \]

7. **Conclusion**:
- The only valid solution is:
\[ n = 7 \]

### Final Answer:
The number of persons \( n \) sitting around the circular table is \( 7 \).