A teacher assigns each of her 18 students a different integer

A teacher assigns each of her 18 students a different integer 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?

  1. 16
  2. 15
  3. 9
  4. 8
  5. 3

Answer

The correct answer is B.

The sum of any two different numbers between 1 and 18 must be between 3 and 35. The only perfect squares between 3 and 35 are 4, 9, 16, and 25. Therefore, the sum of each of the nine pairs must be 4, 9, 16, or 25. The possible pairs for 1 are 3, 8, or 15.

Because 16, 17, and 18 are each greater than or equal to 16, we must pair them with a number so that the sum is 25. Therefore, 16 must be paired with 9, 17 with 8, and 18 with 7. After pairing these, we are left with 1 - 6 and 10 - 15.

Now, tow cases are left - 1 can paired with 3 or with 15. If 1 is paired with 3, then 15 would have to pair with 10, 2 would have to pair with 14, 11 with 5, and 12 with 4. We are then left with 13 and 6 which cannot be paired. For that reason 1 cannot be paired with 3. Thus, 1 must be paired with 15.