jmc

Once all the numbers are placed in the figure, the sum of all the numbers is $2+3+\cdots +9$. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum is $(2+9)/2\cdot 8=44$. Since each of the four smallest squares have the same sum, they each have a sum of $44/4=\boxed{11}$.