A teacher assigns each of her 18 students a different interger from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?

Options

A. 16
B. 15
C. 9
D. 8

Answer

To solve the problem, we need to find out which number must be paired with the student assigned the number 1, given that the sum of the paired numbers must be a perfect square.

### Step-by-Step Solution:

1. **Identify Perfect Squares**:
The maximum number assigned to students is 18. The largest possible sum of two different integers from 1 to 18 is 18 + 17 = 35. The perfect squares less than or equal to 35 are:
- 1² = 1
- 2² = 4
- 3² = 9
- 4² = 16
- 5² = 25
- 6² = 36 (not included since it's greater than 35)

Thus, the perfect squares we consider are: 4, 9, 16, and 25.

2. **Determine Possible Pairings for 1**:
We need to find numbers that can be paired with 1 such that their sum is a perfect square:
- For 4: 1 + x = 4 → x = 3
- For 9: 1 + x = 9 → x = 8
- For 16: 1 + x = 16 → x = 15

Therefore, the possible pairings for the student assigned the number 1 are with the students assigned the numbers 3, 8, or 15.

3. **Check Pairings for 8 and 15**:
- If 1 is paired with 3, then 8 can still be paired with 17 (since 8 + 17 = 25).
- If 1 is paired with 8, then 15 can be paired with 10 (since 15 + 10 = 25).
- If 1 is paired with 15, then 8 can be paired with 17.

4. **Analyze the Pairing Scenarios**:
- **If 1 is paired with 3**:
- 8 can pair with 17 (8 + 17 = 25).
- 15 can pair with 10 (15 + 10 = 25).
- **If 1 is paired with 8**:
- 15 can pair with 10 (15 + 10 = 25).
- But then 8 cannot pair with 17 since 8 is already taken.
- **If 1 is paired with 15**:
- 8 can pair with 17 (8 + 17 = 25).
- This pairing works without conflict.

5. **Conclusion**:
After analyzing all possible pairings, the only consistent pairing that works without conflicts is when the student assigned the number 1 is paired with the student assigned the number 15.

### Final Answer:
The student assigned the number **15** must be paired with the student assigned the number **1**.