Question

A solid cube, 8 inches x 8 inches x 8 inches is painted red on all the faces. The cube is then sliced into smaller equal sized cubes of 1 inch x 1 inch x 1 inch. How many of these small cubes will have exactly two red faces?

Hint:

To find the cubes with exactly two red faces, focus on the edge cubes, excluding the corner cubes.

Answer 1

Step 1: Understanding the structure.
The cube has dimensions of 8 inches x 8 inches x 8 inches. After slicing it into smaller cubes of 1 inch x 1 inch x 1 inch, the total number of small cubes will be: \[ \text{Total small cubes} = 8 \times 8 \times 8 = 512 \, \text{cubes}. \]
Step 2: Identifying cubes with two red faces.
Cubes with exactly two red faces are located along the edges of the cube, excluding the corner cubes. Each edge of the large cube has 8 small cubes, but the corner cubes (which have three red faces) are excluded. Thus, there are 6 cubes with two red faces on each edge of the cube. Since the cube has 12 edges: \[ \text{Cubes with two red faces} = 12 \times 6 = 72. \]
Step 3: Conclusion.
Thus, there are 48 cubes with exactly two red faces.
\[ \boxed{48} \]