jmc

To make the sum the greatest, put the three largest numbers $(13,14$ and $15)$ in the corners. Then, balance the sides by putting the least integer $(10)$ between the greatest sum $(14$ and $15)$. Then put the next least integer $(11)$ between the next greatest sum ($13+15$). Fill in the last integer $(12)$ and you can see that the sum of any three numbers on a side is (for example) $14+10+15=39$ $\boxed{39}$.