AMC 12 2007b Problem 18

Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. What is $a+b+c$?

$\mathrm {(A)} 10 \mathrm {(B)} 13 \mathrm {(C)} 16 \mathrm {(D)} 18 \mathrm {(E)} 21$