9. A cube of side 3 units is formed using a set of smaller cubes of side
1 unit. Find the proportion of the number of faces of the smaller cubes
visible to those which are NOT visible.
(A) 1 : 4
(B) 1 : 3
(C) 1 : 2
(D) 2 : 3
A 3x3x3 cube made of 27 smaller 1-unit cubes has specific face visibility patterns when viewed from outside. The proportion of visible small-cube faces to hidden ones is 1:2, matching option (C).
Total Faces Calculation
Each small cube has 6 faces, so 27 cubes yield 27 × 6 = 162 total faces. The large cube exposes 6 faces, each with 9 small faces, for 54 visible faces. Hidden faces total 162 – 54 = 108, giving the ratio 54:108 or 1:2.
Position-Based Visibility
- Corner cubes (8 total): 3 faces visible each (24 faces)
- Edge cubes (12, excluding corners): 2 faces visible each (24 faces)
- Face-center cubes (6): 1 face visible each (6 faces)
- Internal cube (1): 0 faces visible
Total visible: 24 + 24 + 6 = 54 faces, confirming the count.
Option Analysis
| Option | Ratio | Visible Faces | Matches? | Reason |
|---|---|---|---|---|
| (A) 1:4 | 1:4 | ~32 | No | Undercounts visible |
| (B) 1:3 | 1:3 | 40.5 | No | Non-integer |
| (C) 1:2 | 1:2 | 54 | Yes | Exact |
| (D) 2:3 | 2:3 | 72 | No | Overcounts |
Introduction
In the world of cube side 3 units smaller cubes visible faces proportion problems, a 3x3x3 cube formed from 1-unit smaller cubes challenges students to calculate visible versus hidden faces. This common exam question tests spatial reasoning: total small faces are 162, with 54 visible on the exterior and 108 not visible, yielding a 1:2 proportion.
Step-by-Step Breakdown
Understand smaller cubes visible faces by positions:
- 8 corners expose 3 faces each: 24 visible
- 12 edges expose 2 faces each: 24 visible
- 6 face centers expose 1 face each: 6 visible
- 1 internal cube hides all 6 faces
Sum: 54 visible faces out of 162 total.
Why 1:2 Ratio?
The proportion of visible to not visible faces simplifies from 54:108. Divide by 54: 1:2. This assumes standard external view without table obstruction.
Master this 3 units cube smaller cubes faces puzzle for competitive exams—key phrase reinforces the 1:2 solution.
