When a larger cube is sliced into smaller cubes, the cubes with no painted faces are those located in the innermost core of the larger cube. These form a smaller cube within the original. For a cube sliced into N x N x N smaller cubes, the number of unpainted cubes is (N-2) x (N-2) x (N-2). In this case, the larger cube is sliced into 4x4x4 smaller cubes, so N=4. Therefore, the number of cubes with no painted faces is (4-2) x (4-2) x (4-2) = 2 x 2 x 2 = 8. However, the question states the outer surface is painted and it is sliced to yield 64 small cubes (4x4x4). The number of cubes with exactly one face painted is 6*(N-2)^2 = 6*(4-2)^2 = 6*4 = 24. The number of cubes with exactly two faces painted is 12*(N-2) = 12*(4-2) = 12*2 = 24. The number of cubes with three faces painted is 8 (the corners). The total number of cubes is 4*4*4 = 64. The cubes with no painted faces are those in the center, forming a (N-2)x(N-2)x(N-2) cube. So, (4-2)x(4-2)x(4-2) = 2x2x2 = 8. The cubes with at least one face painted are 64 - 8 = 56. The question asks for the number of small cubes that do not have painted faces. These are the cubes that are not on the surface of the original larger cube. If the large cube is divided into n x n x n smaller cubes, the number of cubes with no painted faces is (n-2) x (n-2) x (n-2). Here, n=4. So, (4-2) x (4-2) x (4-2) = 2 x 2 x 2 = 8. Reconsidering the options and the correct answer being B (16), there might be a misunderstanding in the interpretation or a flaw in the question/options provided. Assuming the standard interpretation where a 4x4x4 cube sliced yields cubes with 0, 1, 2, or 3 painted faces, the calculation for 0 painted faces is (4-2)^3 = 8. If the correct answer is indeed 16, it implies a different problem structure or interpretation. Let's re-examine the problem: 4cm cube sliced into 1cm cubes results in 4x4x4=64 cubes. Cubes with no paint are the inner core. The dimensions of the inner core are (4-2)x(4-2)x(4-2) = 2x2x2 = 8 cubes. This strongly points to 8. If 16 is the correct answer, it's likely there's a misinterpretation of the question or an error in the provided correct answer. However, adhering strictly to the common understanding of such problems and the provided correct answer (B=16), it suggests a calculation that leads to 16. This is not derivable from the standard cube slicing problem for unpainted faces. Let's assume there's a mistake in my understanding of the problem or the expected outcome. The number of cubes with exactly one face painted is 6 * (n-2)^2 = 6 * (4-2)^2 = 6 * 2^2 = 6 * 4 = 24. The number of cubes with exactly two faces painted is 12 * (n-2) = 12 * (4-2) = 12 * 2 = 24. The number of cubes with three faces painted is 8 (corners). Total painted cubes = 24 + 24 + 8 = 56. Total cubes = 64. Cubes with no painted faces = 64 - 56 = 8. It appears there might be an error in the known correct answer.