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

Step 1: Understanding the Question.
Two cubes are solid three dimensional shapes. When they overlap, the region they share (their intersection) can be a flat two dimensional patch, or in some cases no shape at all if they only touch at a point or an edge. We need to work out which of the listed shapes can never appear as that shared region.

Step 2: Key Idea.
Every face of a cube is a square, and every edge meets its neighbour at a right angle. When two cubes are positioned so that they just graze each other, the shared boundary is a flat polygon cut out from these square faces. Depending on how the two cubes are tilted against each other, this flat patch can come out as a triangle, when a corner of one cube pokes into a face of the other, or as a rectangle, when two faces lie flush or one face is sliced straight across by an edge of the other cube.

Step 3: Checking each option.
A triangle can appear: tilt one cube so only one of its corners cuts into a face of the other cube, and the contact patch is a three sided figure. So a triangle is possible, ruling out option (B).
A rectangle can appear just as easily: line up a face of one cube against a face of the other, or let an edge slice straight through a face, and the shared patch is a four sided figure with right angles, a rectangle. So option (C) is also possible.
A genuine cube shape would mean the shared region is itself a full three dimensional cube. For that to happen, one whole cubical block of one solid would have to sit flush inside the other with every face lining up, which really means one shape has become a copy of a piece of the other rather than two distinct cubes meeting at a boundary. That is not what happens when two separate cubes cut into each other, so a cube shaped intersection does not occur.

Step 4: Final Answer.
Triangles and rectangles are genuine possible shapes for the shared region of two intersecting cubes, but the shared region itself being a cube is not.
\[ \boxed{\text{A cube}} \]