The objective is to determine the count of smaller cubes possessing precisely two painted faces when a large, painted cube is partitioned into smaller, uniform cubes.
- The initial large cube is painted on all six exterior surfaces.
- This large cube is divided into 64 smaller cubes of identical dimensions.
- Given that \( 64 = 4^3 \), each edge of the large cube is segmented into 4 equal divisions.
- The number of painted faces on a smaller cube (0, 1, 2, or 3) is contingent upon its location within the original cube.
- The specific requirement is to identify the quantity of smaller cubes with exactly two painted faces.
- Cubes with three painted faces are exclusively located at the corners.
- Cubes with two painted faces are situated along the edges, excluding the corner cubes.
- Cubes with one painted face are found on the faces of the large cube, excluding those on the edges and corners.
- Cubes with zero painted faces are those entirely enclosed within the large cube, not exposed to any painted surface.
- Each edge of the large cube comprises 4 smaller cubes.
- The two cubes at either extremity of each edge are corner cubes, exhibiting three painted faces.
- The smaller cubes situated between these two corner cubes along an edge possess exactly two painted faces.
- The count of such cubes per edge is calculated as \( 4 - 2 = 2 \).
- A cube has a total of 12 edges.
- Therefore, the aggregate number of smaller cubes with exactly two painted faces is \( 12 \times 2 = 24 \).
The quantity of smaller cubes that exhibit exactly two painted faces is determined to be 24.