Jenny made a block using small cubes, each having a volume of 9 cubic cm. To…
2024
Jenny made a block using small cubes, each having a volume of 9 cubic cm. To make the block, she used 4 small cubes along the length, 8 small cubes along the width, and 16 small cubes along the depth. She realised she had used more small cubes than necessary — the block would look exactly the same from outside even if it were hollow, with only the outer shell of cubes present. What is the minimum number of cubes she needs to take out so that the block becomes hollow?
- A.
344
- B.
512
- C.
168
- D.
342
Attempted by 4 students.
Show answer & explanation
Correct answer: C
Concept: When a solid cuboid built from unit cubes (outer dimensions L × W × D cubes) is hollowed out so that only a single-cube-thick outer shell remains, the cubes removed are exactly the inner cuboid left after peeling off one layer of cubes from every face. So the inner core has dimensions (L−2) × (W−2) × (D−2), and the minimum number of cubes that must be removed is exactly this inner-core volume — not the outer shell that is left behind.
Application:
Total cubes used to build the solid block = length × width × depth = 4 × 8 × 16 = 512.
To hollow the block while keeping its outer look and dimensions unchanged, only a one-cube-thick shell on the outside must remain; the entire inner core (excluding the outer layer on every side) has to be removed.
Inner core dimensions = (4 − 2) × (8 − 2) × (16 − 2) = 2 × 6 × 14.
Inner core volume, i.e. the number of cubes to remove = 2 × 6 × 14 = 168.
Cross-check: Cubes remaining in the shell after removal = total − removed = 512 − 168 = 344, which independently matches the shell structure, confirming that 168 is the minimum number of cubes to be removed.