The surface of a cube is painted red and blue, 3 sides of each. The cube is cut into 27 equal sized smaller cubes. Find both answers to the question, "How many cubes have at least 2 sides painted blue?" and "How many cubes have at least 1 side painted red?"

I have no idea how to do this at all what so ever.

Pick up a real cube and draw a 3x3 grid one each face.

Fill in the grids on 3 faces with R and 3 with B.

The very center cubelet has no painted faces.

Now count the others and total up the various color combinations.

Make sure that it makes no difference how the two colors are arranged on the cube.

To find the answers to the given questions, we need to understand the structure of the painted cube and how it is divided into smaller cubes. Let's break down the problem step by step.

Step 1: Understand the cube structure
The large cube is painted on its surface, and we know that 3 sides of each face are painted blue. We can visualize the cube as a smaller cube with edge length 3, where each smaller cube represents a painted face.

Step 2: Divide the cube
Since the larger cube has a volume of 27 smaller cubes, and each smaller cube represents a painted face, we can divide the large cube into 27 smaller cubes with edge lengths of 3.

Step 3: Determine the cubes with at least 2 blue sides
Now, let's count the smaller cubes that have at least 2 sides painted blue. To calculate this, we need to determine how many smaller cubes have painted blue faces on 2 or more adjacent sides.

To do this, we need to identify the smaller cubes located at the corners, edges, and center of each face of the large cube. These are the cubes that have a maximum potential of having at least 2 blue sides.

- Corners: In each corner of the large cube, there are 8 smaller cubes. Each of these cubes is adjacent to 3 faces of the larger cube, so they always have at least 2 blue sides. Therefore, there are 8 smaller cubes in the corners with at least 2 sides painted blue.
- Edges: Along each edge of the large cube, there are 12 smaller cubes. Each cube is adjacent to 2 faces of the larger cube, so it can have either 1 or 2 blue sides. However, we need to exclude the corner cubes that we have already counted, as they are counted in the corner cases. So, in total, there are 12 - 8 = 4 smaller cubes along each edge with at least 2 sides painted blue.
- Centers of each face: On each face of the large cube, there is a center cube. Each center cube is adjacent to only 1 face, so it can have either 0 or 1 blue side. Since we are counting the cubes with at least 2 sides painted blue, we can ignore these center cubes.

Therefore, the total number of cubes with at least 2 sides painted blue is 8 (corners) + (4 edges per face * 6 faces) = 8 + 24 = 32 cubes.

Step 4: Determine the cubes with at least 1 red side
To find the number of cubes with at least 1 side painted red, we need to consider all the smaller cubes in the large cube, as each cube represents a painted face.

Since all faces are painted red and blue, and we have already found the number of cubes with at least 2 blue sides, we can conclude that the remaining cubes must have at least 1 red side.

Since the total volume of the large cube is 27 smaller cubes, and we have already counted the 32 cubes with at least 2 blue sides, the remaining 27 - 32 = -5 cubes will have at least 1 red side.

However, a negative number of cubes is not possible, so the correct answer is that there are 0 cubes with at least 1 side painted red.

In summary, the answers are:
- How many cubes have at least 2 sides painted blue? 32 cubes
- How many cubes have at least 1 side painted red? 0 cubes