Let $ a_1$, $ a_2$, $ a_3$, $ a_4$ the numbers on a side (face). Let $ b_1=a_1+a_2+a_3$, $ b_2=a_1+a_2+a_4$, $ b_3=a_1+a_3+a_4$, $ b_4=a_2+a_3+a_4$ Then we have that all $ a_i$ are distinct, so all $ b_i$ are distinct and $ C=b_1+b_2+b_3+b_4=3(a_1+a_2+a_3+a_4)$ We have to find the minimum of $ C$. W.L.O.G. $ b_4>b_3>b_2>b_1 \ge 10$ Then $ C\ge 46$ But $ C$ is a multiple of $ 3$, then $ C\ge 48$, and I have an example for $ C=48$ $ (0;0;0)=1$; $ (0;0;1)=7$; $ (0;1;0)=6$; $ (1;0;0)=4$; $ (0;1;1)=3$; $ (1;1;0)=5$; $ (1;0;1)=8$; $ (1;1;1)=2$ The minimum possible is $ 16$

really nice solution