Divide the $n{}$ positive number into two set $S_1,S_2$ such that any number in $S_1$ is even while any number in $S_2$ is odd. Assume that $S_1=\{a_1,a_2,\cdots ,a_k\},S_2=\{b_1,b_2,\cdots ,b_s\}$. Because $\frac{a_i+b_j}{2}$ is not an integer.So $a_ib_j$ is a square number. $a_1b_1,a_1b_2$ are both square number.So $b_1b_2$ is square number too.We're done. Note:if $S_1$ or $S_2$ is empty.Then the blackboard is full of integer.