1. Place the numbers $1,2,3,4,5,6,7,8$ on the eight vertices of a cube such that the sum of any three numbers on each face is not less than 10. Find the minimum value of the sum of the four numbers on each face.
1. Let the four numbers on a certain face be $a_{1}, a_{2}, a_{3}, a_{4}$, whose sum reaches the minimum value, and $a_{1}<a_{2}<a_{3}<a_{4}$. Since the sum of three different positive integers less than 5 is at most 9, it follows that $a_{4} \geqslant 6$. Therefore,
$$
a_{1}+a_{2}+a_{3}+a_{4} \geqslant 16 \text {. }
$$

The example shown in Figure 1
demonstrates that 16 can be achieved.