Use your brains -3

In this cube we can put the white cubes 'inside' the bigger cube which is contains 2x2x2 = 8 cube in the center of the bigger cube, so minimum case we have 20-8 = 12 white cube that is visible in the bigger cube

A 4x4x4 cube is made up of four 4x4x1s layers , two inside squares sandwiched between two outside squares.

In the two outside 4x4x1s layers, all cubes are visible once, the edges one more time, and the corners a third.

int the two inside 4x4x1 layers, the cubes on the edges are visible once, and the corners twice, and the insides, not at all.

Thus the layers look like this (numbers indicate visible faces):

Thus the breakdown of cubes by visibility is:

0 x 8

1 x 24

2 x 24

3 x 8

Now we just have to place the white cubes such that they have the least faces visible. We have 20 white faces, and putting the first 8 into the inside section with 0 visible faces leaves 12 cubes left. Locations with only one visible face can completely absorb the 12 with the minimal visibility, and so you just have 1 visible face per each of the twelve remaining, and so the answer is twelve.