2007 AMC 12B 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\qquad \mathrm {(B)} 13\qquad \mathrm {(C)} 16\qquad \mathrm {(D)} 18\qquad \mathrm {(E)} 21

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.