# Thread: Pairing up to form perfect squares

1. ## Pairing up to form perfect squares

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 must follow this rule, the student assigned which number must be paired with the student assigned the number 1? (of course, and why?)

EDIT: the original question was presented as a multiple choice.
The possible answers might help in determining the correct answer, but "why" is still needed.

(A) 16, (B) 15, (C) 9, (D) 8, (E) 3

2. I wasn't sure whether this fairly lengthy chain of reasoning is what you're looking for. Anyway, in trying to determine the pairings, 18-7, 17-8, and 16-9 are forced (being that 18+17 falls short of 36). (I soon realized that 15 through 10 could be paired with 1 through 6, respectively, but that doesn't prove that there are no other solutions, hence the following.) This forces 2 to be paired with 14 (because 7 is taken), which in turn forces 11 to be paired with 5 (because 14 is taken), which in turn forces 4 to be paired with 12 (because 5 is taken), which in turn forces 13 to be paired with 3 (because 12 is taken), which in turn forces 6 to be paired with 10 (because 3 is taken), which leaves 1 and 15.

3. PERFECT! Nice going.

4. A teacher assigns each of her 18 students a different integer from 1 through 18
Coming up with a different logic and answer.

4 is a perfect square. Combinations of 4 are: 0+4, 1+3, 2+2. Since 0 is not a possible assigned number and there is only one 2, the only possible combination for 4 is 1+3

Maud

5. Perhaps I read too much into the question. From David's post, I am now assuming that not all the perfect squares need to have a combination rather that all of the combined numbers must equal any perfect square.

got it!