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?

Question

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?

Steve · Answer

Starting with the biggest,

18+7=25
17+8=25
16+9=25

Now all the numbers 7-10 and 15-18 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.

MathMate · Answer

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.