Set up volume equivalence

The volume of the large sphere is equal to the sum of the volumes of 64 small spheres. Let $R$ be the radius of the large sphere and $r$ be the radius of a small sphere.

$\frac{4}{3} \pi R^3 = 64 \times \frac{4}{3} \pi r^3$

Simplify the equation

Cancel $\frac{4}{3} \pi$ from both sides to find the relationship between radii.

$R^3 = 64r^3$

Solve for the small radius $r$

Take the cube root of both sides, substituting $R = 12$ cm.

$r = \frac{R}{\sqrt[3]{64}} = \frac{12}{4} = 3 \text{ cm}$

Calculate the diameter

The question asks for the diameter of the small sphere, which is $2r$.

$d = 2 \times 3 = 6 \text{ cm}$