Question

27 cubes of edge 10 cm are arranged to create a larger cube. If the cubes at the eight corners are replaced with spheres of diameter 10 cm, what is the minimum number of cubes that do not touch any of the spheres?

Hint:

In spatial reasoning problems involving cubes, clearly classify the cubes by their position (corner, edge, face, center). The properties of each type are distinct. If a question uses an ambiguous term like "touch," consider different interpretations (sharing a face, edge, or vertex) to see which one fits the provided answer choices.

Answer 1

Correct Answer: 7

Step 1: Understanding the Question:
We have a large 3x3x3 cube made of 27 smaller cubes. The 8 corner cubes are replaced by spheres. We need to find how many of the remaining smaller cubes do not touch any of these 8 spheres. The key to this problem is the definition of "touch". In 3D geometry puzzles, "touching" can mean sharing a face, an edge, or a corner. A less common interpretation is sharing a face only. Let's analyze based on the "sharing a face" definition, as it leads to the correct answer.
Step 2: Analyzing the Cube Structure:
A 3x3x3 cube has different types of smaller cubes based on their position:

Corner cubes: 8 cubes (these are replaced by spheres).
Edge-center cubes: 12 cubes (located in the middle of each edge).
Face-center cubes: 6 cubes (located in the center of each face).
Central cube: 1 cube (at the very core of the large cube). Total cubes = 8 + 12 + 6 + 1 = 27. After replacing the corners, 19 cubes remain: 12 edge-center, 6 face-center, and 1 central cube.
Step 3: Detailed Explanation (Defining "Touch" as Sharing a Face):
We need to determine which of the remaining 19 cubes share a face with any of the original 8 corner cube positions.

Edge-center cubes: Each edge-center cube is positioned between two corner cubes on the same edge. It shares a face with both of these corner cubes. Therefore, all 12 edge-center cubes will touch the spheres.
Face-center cubes: A face-center cube is located at the center of a face. It shares faces with the central cube and the four edge-center cubes on that face. It does not share a face with any corner cube (it only touches them at the edges). Therefore, the 6 face-center cubes do not touch the spheres.
Central cube: The single cube at the core of the structure only shares faces with the 6 face-center cubes. It does not share a face with any corner cube. Therefore, the central cube does not touch the spheres. Step 4: Final Answer:
The number of cubes that do not touch any spheres is the sum of the face-center cubes and the central cube.
\[ \text{Number of non-touching cubes} = 6 (\text{face-center}) + 1 (\text{central}) = 7 \]