AUSTRIAN-POLISH 1988 Problem 8

We are given $1988$ unit cubes. Using some or all of these cubes, we form three quadratic boards $A, B,C$ of dimensions $a \times a \times 1$ , $b \times b \times 1$ , and $c \times c \times 1$ respectively, where $a \le b \le c$ . Now we place board $B$ on board $C$ so that each cube of $B$ is precisely above a cube of $C$ and $B$ does not overlap $C$ . Similarly, we place $A$ on $B$ . This gives us a three-floor tower. What choice of $a, b$ and $c$ gives the maximum number of such three-floor towers?