All faces of a 4-inch cube have been painted. if the cube is cut into 1-inch smaller cubes, how many of them have no paint on all their faces?

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Only the 2x2x2 inner cubes: 8

To find the number of smaller cubes with no paint on any of their faces, we need to break down the problem step by step:

Step 1: Calculate the total number of smaller cubes.
Since each edge of the larger cube is 4 inches, it will be divided into 4 smaller cubes. Therefore, the total number of smaller cubes in each edge would be 4.

Step 2: Calculate the total number of smaller cubes.
To calculate the total number of smaller cubes, we need to cube the number of smaller cubes on each edge. In this case, 4^3 = 64 smaller cubes in total.

Step 3: Identify the cubes with no paint on any of their faces.
For a cube to have no paint on any of its faces, it means that all six faces of the smaller cube should not have any paint.

Step 4: Calculate the number of cubes with no paint on any face.
To find the number of cubes with no paint on all their faces, we need to determine how many cubes have paint on at least one face and subtract that number from the total number of cubes calculated in Step 2.

In this case, each face of the larger cube has been painted, so the smaller cubes on the outermost layer will have paint on at least one face.
There are three layers in each dimension of the larger cube. The total number of smaller cubes in the outermost layer of each dimension would be (4-2) * 4 = 8 cubes.

So, the number of cubes with no paint on any face would be 64 - (8^3) = 64 - 512 = -448.

However, since we cannot have a negative number of cubes, it means that all the smaller cubes will have paint on at least one face. Therefore, there are no smaller cubes with no paint on any of their faces.