Correct Answer:
A. 8
This is a classic cube-cutting problem. A large cube is painted on all sides and then cut into 64 smaller, identical cubes. We need to find how many of these smaller cubes have no paint.
- Determine the dimensions: If a large cube is cut into 64 smaller cubes, then the number of smaller cubes along each edge of the large cube is the cube root of 64. The cube root of 64 is 4 (since 4 * 4 * 4 = 64). So, the large cube is a 4x4x4 arrangement of smaller cubes.
- Identify unpainted cubes: Cubes with no paint are those located entirely in the interior of the large cube, not touching any of its original outer surfaces.
- Calculate inner dimensions: To find the dimensions of this unpainted inner core, we subtract 2 from each dimension of the large cube (1 for the layer removed from the front and 1 for the layer removed from the back, similarly for top/bottom and left/right). So, the inner cube's dimensions are (4-2) x (4-2) x (4-2) = 2x2x2.
- Count unpainted cubes: The number of cubes in this inner core is 2 * 2 * 2 = 8.
Options B (27), C (36), and D (32) are incorrect. 27 would be the answer for a 5x5x5 cube (3x3x3 inner core).