algebra
posted by Linda .
My science class is pretty small. There are just 18 students in the class. My teacher, Mr. Burnett, has an unusual system for picking lab partners. He has given each student a number from 1 to 18, and on lab days, he pulls two numbers out of a bag to match people up. During our last lab I noticed that the sum of each pairing was a perfect square. How were the partners paired with each other?

Starting with the biggest,
18+7=25
17+8=25
16+9=25
Now all the numbers 710 and 1518 are taken. That leaves 1,2,3,4,5,6,10,11,12,13,14,15
Taking the easy way out,
1+15=16
2+14=16
3+13=16
4+12=16
5+11=16
6+10=16
There may be other arrangements, but this one falls out the easiest. 
This is a neat problem!
The hardest ones to match are those from 16 to 18, since we can only match to a total of 25. The rest can match to a total of 16 and the job is done.