### Instruction: A cube of cheese
\[C = \{(x,y,z) : 0 \le x, y, z \le 1\}\]is cut along the planes $x = y,$ $y = z,$ and $z = x.$  How many pieces are there?### Response: I want to visualize how the cube is cut by the planes.
I can use a coordinate system where the origin is at one corner of the cube and the axes are along the edges.
Then the planes $x = y,$ $y = z,$ and $z = x$ are the diagonals of the faces of the cube.
If I cut along one diagonal, I get two pieces: a triangular prism and a tetrahedron.
If I cut along another diagonal, I get four pieces: two triangular prisms and two tetrahedra.
If I cut along the third diagonal, I get eight pieces: four triangular prisms and four tetrahedra.
Therefore, the final answer is 8.