A cube is cut in two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other is coloured green on two smaller adjacent faces and red on the remaining. Each is then cut into 32 cubes of same size and mixed up.
15.
How many cubes have two red and one green face on each 

To solve this, let's analyze the problem step-by-step.

### Step 1: Understanding the Initial Cube and Cuts
1. We start with a cube, which is divided into two equal parts along a plane parallel to one of its faces.
2. Each half of the cube will therefore have two larger faces (the cut faces) and four smaller faces.

### Step 2: Coloring the Halves
1. **First half**: Colored red on its two larger (cut) faces and green on the remaining four smaller faces.
2. **Second half**: Colored green on two adjacent smaller faces and red on the remaining four faces (including the two cut faces).

### Step 3: Dividing Each Half into Smaller Cubes
Each half is cut into 32 smaller cubes, so we have a total of \( 32 + 32 = 64 \) smaller cubes.

### Step 4: Finding Cubes with Two Red and One Green Face
1. To have **two red and one green face**, a cube must:
   - Be located at the edge between two red faces and one green face.

2. Let's analyze each half separately:
   - **First half**: The two large red faces are opposite each other, and the green faces are on the four sides. This arrangement gives exactly **8 edge cubes** with two red faces and one green face.
   - **Second half**: The two red faces are the larger faces from the cut, and they share edges with the two adjacent green faces. Similarly, this arrangement also results in **8 edge cubes** with two red faces and one green face.

3. Summing these, we find:
   \[
   8 + 8 = 16
   \]

### Conclusion
Thus, there are **16 cubes with two red faces and one green face**.

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### Related Questions

1. How many cubes have exactly three green faces?
2. How many cubes have exactly one red and two green faces?
3. What is the number of cubes that have all six faces unpainted?
4. If the second half had one more red face, how would the number of cubes with two red and one green face change?
5. How many cubes have exactly two green faces and no red faces?

### Tip
To solve similar painted cube problems, always start by visualizing or drawing the coloring on each face after each cut.

A cube is cut in two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other is coloured green on two smaller adjacent faces and red on the remaining. Each is then cut into 32 cubes of same size and mixed up. How many cubes have two red and one green face on each?

Solve a painted cube problem involving cutting a cube into smaller pieces with different colored faces. This problem requires understanding geometry, spatial visualization, and combinatorial reasoning to count the number of cubes with two red and one green face. Ideal for students in Grades 9-11.

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