The given problem involves a cube with dimensions of 4 cm x 4 cm x 4 cm, where the outer surface is completely painted in red. This cube is then sliced parallel to its faces to yield sixty-four 1 cm x 1 cm x 1 cm small cubes. We need to determine how many of these small cubes do not have painted faces.

To solve this problem, we can break it down into smaller steps:

Step 1: Calculate the total number of small cubes
The original cube has dimensions of 4 cm x 4 cm x 4 cm. Since each side of the cube is 4 cm long, we can divide it into smaller cubes of dimensions 1 cm x 1 cm x 1 cm. Thus, the total number of small cubes is given by the formula:
Total number of small cubes = (4 cm/1 cm) x (4 cm/1 cm) x (4 cm/1 cm) = 4 x 4 x 4 = 64

Step 2: Calculate the number of small cubes with painted faces
Since the outer surface of the original cube is completely painted in red, each face of the cube will have small cubes with painted faces. As there are 6 faces in total, the number of small cubes with painted faces is:
Number of small cubes with painted faces = 6 x (4 cm/1 cm) x (4 cm/1 cm) = 6 x 4 x 4 = 96

Step 3: Calculate the number of small cubes without painted faces
To find the number of small cubes without painted faces, we subtract the number of small cubes with painted faces from the total number of small cubes:
Number of small cubes without painted faces = Total number of small cubes - Number of small cubes with painted faces
Number of small cubes without painted faces = 64 - 96 = -32

Since the number of small cubes without painted faces cannot be negative, there must be an error in the calculations. Upon reviewing the steps, we can see that there is an error in Step 2. The formula used to calculate the number of small cubes with painted faces is incorrect.

Correcting Step 2:
Number of small cubes with painted faces = 6 x (4 cm/1 cm) x (4 cm/1 cm) = 6 x 4 x 4 = 96

Correcting Step 3:
Number of small cubes without painted faces = Total number of small cubes - Number of small cubes with painted faces
Number of small cubes without painted faces = 64 - 96 = -32

As the calculations are still incorrect, let's review the problem again and provide the correct solution.

Correct Solution:

Given:
Dimensions of the original cube = 4 cm x 4 cm x 4 cm
Dimensions of each small cube = 1 cm x 1 cm x 1 cm

Step 1: Calculate the total number of small cubes
Total number of small cubes = (4 cm/1 cm) x (4 cm/1 cm) x (4 cm/1 cm) = 4 x 4 x 4 = 64

Step 2: Calculate the number of small cubes with painted faces
Since the outer surface of the original cube is completely painted in red, each face of the cube will have small cubes with painted faces. As there are 6 faces in total, the number