321. Spheres and a cube. Once, during transportation, it was required to pack a sphere with a diameter of 30 cm into a cubic box with a side of 32 cm. To prevent the sphere from moving during transportation, 8 identical small spheres had to be placed in the corners of the box. What is the diameter of such a small sphere?
321. Since the center of the smaller sphere of radius $r$ lies on the diagonal and is $16 \sqrt{3}-15-r$ centimeters away from the corresponding corner of the box, to solve the problem we only need to equate this value to $r \sqrt{3}$ and solve the resulting equation for $r$. As a result, we will find the desired diameter

$$
D=(16 \sqrt{3}-15)(\sqrt{3}-1)=63-31 \sqrt{3} \approx 9.308 \text{~cm}
$$

[G. Curtis, M. M., 44, 56 (January 1970).]