Blue too busy, I try...

I think it is from 1+3+...+2n-1=n^2.
There are n*n points,2 sides are 2n-1 points,(1,n),(2,n),...(n,n),(n,n-1)...(n,1).If we take isolate point (i,n),that means one person one room,(i+1,n),(i+2,n),..(n,n),(n,n-1),...(n.j),means they all in same room j.We have 1 to one correspondence.so...