Problem 505

Suppose that the base of the prism is $b \mathrm{~cm}$ by $w \mathrm{~cm}$ and the height of the prism is $h \mathrm{~cm}$. Since Aaron has 144 cubes with edge length 1 cm, then the volume of the prism is $144 \mathrm{~cm}^{3}$, and so $bwh = 144$. Since the perimeter of the base is 20 cm, then $2b + 2w = 20$ or $b + w = 10$. Since $b$ and $w$ are positive integers, then we can make a chart of the possible combinations of $b$ and $w$ and the resulting values of $h = \frac{144}{bw}$, noting that since $b$ and $w$ are symmetric, then we can assume that $b \leq w$:

$$\begin{array}{c|c|c} b & w & h \\ \hline 1 & 9 & 16 \\ 2 & 8 & 9 \\ 3 & 7 & \frac{48}{7} \\ 4 & 6 & 6 \\ 5 & 5 & \frac{144}{25} \end{array}$$

Since $h$ must itself be a positive integer, then the possible values of $h$ are 16, 9, and 6. The sum of the possible heights is $16 \mathrm{~cm} + 9 \mathrm{~cm} + 6 \mathrm{~cm} = 31 \mathrm{~cm}$.