Question

The intersection of two cubes cannot be:

• A. A cube
• B. A triangle
• C. A rectangle
• D. None of these

Hint:

Think about the flat boundary between two overlapping cubes: can that shared patch ever be an entire three dimensional cube itself?

Answer 1

The Correct Option is A

This question checks whether you can picture what shape appears where two solid cubes cut into each other. It helps to think about what happens right at the boundary where the two solids meet, since that boundary is exactly what decides the shape of the intersection the options describe.

1. Cube: for the shared part of two cubes to be a cube itself, one of the cubes would need to sit flush inside the other over an entire cubical region with matching flat faces on every side. Two genuinely distinct cubes meeting or cutting into each other from an angle never do this, so a cube shaped overlap does not occur.
2. Triangle: if one cube is tilted so just a single corner pokes into a flat face of the other cube, the boundary of that overlap is a three sided patch, a triangle. This is achievable, so it is not the answer.
3. Rectangle: if a face of one cube is sliced straight across by an edge of the other, or two faces sit flush against each other, the shared patch is a four sided shape with right angles, a rectangle. This is also achievable.
4. None of these: since one of the three listed shapes (cube) genuinely cannot occur, this option does not apply.

Only the cube option describes something that can never be the shape of the overlap between two cubes, since triangles and rectangles are both ordinary results of one cube cutting into another at an angle, while a cube shaped overlap would need the two solids to stop being two separate cubes.

Let's summarize:

• Cutting a corner of one cube into a face of another gives a triangular patch.
• Lining up faces or slicing an edge through a face gives a rectangular patch.
• A cube shaped overlap would need one cube to occupy an entire cubical block that is also part of the other cube, which is not a genuine intersection of two separate cubes.

So the shape that the intersection of two cubes cannot be is a cube, option (A).