A cube `C={(x, y, z)|o le x, y, zle1}` is cut by a sharp knife along the plane `x=y, y=z, z=x`. If no piece is moved until all three cuts are made, the number of pieces is

Options

A. `6`
B. `7`
C. `8`
D. `27`

Answer

To solve the problem of how many pieces a cube is divided into when cut by the planes \(x = y\), \(y = z\), and \(z = x\), we can follow these steps:

### Step 1: Understand the Cube
The cube \(C\) is defined by the set of points \((x, y, z)\) such that \(0 \leq x, y, z \leq 1\). This means the cube has vertices at the points:
- \( (0, 0, 0) \)
- \( (1, 0, 0) \)
- \( (0, 1, 0) \)
- \( (0, 0, 1) \)
- \( (1, 1, 0) \)
- \( (1, 0, 1) \)
- \( (0, 1, 1) \)
- \( (1, 1, 1) \)

### Step 2: Analyze the Planes
The planes given are:
1. \(x = y\)
2. \(y = z\)
3. \(z = x\)

These planes will intersect the cube and divide it into different regions.

### Step 3: Visualize the Cuts
- **Cut along \(x = y\)**: This plane divides the cube into two regions where \(x\) is less than or equal to \(y\) and where \(x\) is greater than \(y\).
- **Cut along \(y = z\)**: This plane further divides the regions created by the first cut into smaller sections.
- **Cut along \(z = x\)**: This final cut will further divide the already divided regions.

### Step 4: Count the Regions
To visualize how many distinct regions are created by these three cuts, we can think of the cube being divided into smaller sections based on the relative sizes of \(x\), \(y\), and \(z\). Each of the cuts creates a new division in the cube.

1. The first cut \(x = y\) divides the cube into 2 parts.
2. The second cut \(y = z\) divides each of those parts into 2 more parts, resulting in \(2 \times 2 = 4\) parts.
3. The third cut \(z = x\) divides each of those 4 parts into 2 more parts, resulting in \(4 \times 2 = 8\) parts.

### Conclusion
Thus, after making all three cuts, the cube is divided into **8 distinct pieces**.