Cube

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Cube — Key Facts for SSC CGL • A cube has 6 faces, 12 edges, and 8 vertices. • A cube made of small unit cubes (cubelets) has side length $n$. It contains:

• Total small cubes = $n^3$
• Cubes with no face painted (interior only) = $(n-2)^3$ (when $n > 1$)
• Cubes with 1 face painted = $6 \times (n-2)^2$
• Cubes with 2 faces painted = $12 \times (n-1)$
• Cubes with 3 faces painted = 8 (corner cubes, always 8) • For a single cut through a cube’s face: the number of new surfaces created = $2 \times \text{area of cut}$ (each cut exposes two new faces). • A cube’s diagonal (space diagonal, corner to corner): $a\sqrt{3}$ where $a$ = side length. • A cube’s surface area = $6a^2$; volume = $a^3$.

⚡ Exam Tip: Memorise the painting formula for $n^3$ cubes. The most common SSC CGL question: “A cube is painted on all faces, cut into $n^3$ small cubes, and then asked: how many have 0/1/2/3 faces painted?” Draw a 3×3×3 cube mentally and label the positions (corners, edge-centres, face-centres, interior) to verify the formula before applying it.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Cube — SSC CGL Study Guide

Core Concept: Cube questions in SSC CGL typically involve a large cube made of smaller unit cubes, where some or all faces are painted before cutting. You must determine how many small cubes have paint on 0, 1, 2, or 3 faces. The formulas derive from identifying where small cubes are located within the larger structure.

Location-Based Classification of Small Cubes:

Position	Number of Faces Painted	Formula
8 corners	3	Always 8
Edge-centre cubes (not corners)	2	$12 \times (n-2)$
Face-centre cubes (not edges/corners)	1	$6 \times (n-2)^2$
Completely interior cubes	0	$(n-2)^3$

Worked Example 1 (Standard Painting): A 5×5×5 cube is painted red on all faces, then cut into 125 unit cubes. How many cubes have paint on exactly 2 faces?

• Using formula: $12 \times (n-2) = 12 \times (5-2) = 12 \times 3 = 36$
• Answer: 36 cubes

Worked Example 2 (Verification for 3×3×3): A 3×3×3 cube is painted. Find all categories.

• Total = 27
• 3 faces painted: 8 corners = 8
• 2 faces painted: $12 \times (3-2) = 12 \times 1 = 12$ (these are the 12 edge-centre cubes, 1 per edge, since each edge has exactly 1 cube that is not a corner)
• 1 face painted: $6 \times (3-2)^2 = 6 \times 1 = 6$ (the 6 face-centre cubes)
• 0 faces painted: $(3-2)^3 = 1$ (the single central cube)
• Check: 8 + 12 + 6 + 1 = 27 ✓

Worked Example 3 (Partial Painting): A cube of side 4 cm is painted on the outside and then cut into 1 cm³ cubes. How many have paint on exactly 1 face and exactly 2 faces?

• n = 4
• 1 face painted: $6 \times (4-2)^2 = 6 \times 4 = 24$
• 2 faces painted: $12 \times (4-2) = 12 \times 2 = 24$

Common Student Mistakes: Using $(n-1)$ instead of $(n-2)$ for interior/face-centre calculations, forgetting that the formula $(n-2)^3$ gives 0 when $n=2$ (which is correct — a 2×2×2 cube has no interior unpainted cubes), and confusing edge cubes with corner cubes in partially painted scenarios.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Cube — Comprehensive SSC CGL Notes

Theoretical Foundation: Cube problems appear in both the Quantitative Aptitude (Mensuration) and Non-Verbal Reasoning sections of SSC CGL. The painting-and-cutting problem is the most common variant. Understanding the spatial geometry — that cubes on edges have 2 painted faces only if they are not corners, and cubes on faces have 1 painted face only if they are not on edges or corners — is essential.

Variants of Cube Painting Problems:

1. Two Opposite Faces Painted: If only two opposite faces of the large cube are painted (instead of all 6):

   • Cubes with 2 painted faces: The two corner cubes at the intersection of the painted faces = 2
   • Cubes with 1 painted face: All cubes along the two painted faces excluding corners = $2 \times (n-2)^2$ + edge adjustments
   • Actually: The two painted opposite faces each contribute $(n-2)^2$ face-centre cubes (1-paint). The shared edges contribute 0 (they have 2 painted faces which are opposite faces — wait, opposite faces painted means those edges lie on both painted faces… → they have 2 painted faces).
   • Let the painted faces be front and back. Edge cubes on the 4 edges between these faces: $4 \times (n-2)$ cubes, each with 2 painted faces.
   • Corner cubes shared: 2 (each corner has 3 faces painted but only 2 painted since we only painted 2 faces total… actually corners on the painted faces that don’t touch the third painted face → they have 2 painted faces from our two painted faces).

2. Three Alternate Faces Painted (e.g., top, front, right):

   • Corner cube at intersection of these 3 faces → 3 painted faces (1 cube)
   • Edges: Each of the 3 painted faces shares 2 edges. At each edge, $n-2$ cubes have 2 painted faces. Total: $3 \times 2 \times (n-2) = 6(n-2)$
   • Face centres: $(n-2)^2$ per painted face = $3(n-2)^2$
   • Remaining cubes: Unpainted

3. One Face Painted:

   • Cubes with 1 painted face: $(n-2)^2$ per face painted = $6(n-2)^2$ if all 6 faces painted
   • If only 1 face painted: $(n-2)^2$ cubes (face-centre only)

4. Cube Cut by Multiple Planes: When a cube is cut by planes parallel to faces:

   • Count pieces by multiplying cut divisions: $(a+1)(b+1)(c+1)$ pieces if cut into $a \times b \times c$ smaller cuboids
   • Surface area of all pieces: each small cuboid has 2 new faces per cut

Worked Example — Edge Cases: A 4×4×4 cube has its two opposite faces (left and right) painted. How many unit cubes have paint on exactly 1 face?

• The left and right faces are painted.
• Each face has $(n-2)^2 = (4-2)^2 = 4$ face-centre cubes with exactly 1 painted face
• The 4 edge cubes on each painted face (shared between painted and unpainted faces) are NOT included here since edges have 2 painted faces
• So: 4 + 4 = 8 cubes have exactly 1 painted face

SSC CGL PYQ Pattern (2019-2023):

• 2023 Tier-I: 1 cube painting question (standard all-6-faces painted, 5×5×5)
• 2022 Tier-I: 1 cube cutting and counting question
• Most common variant: Fully painted cube cut into $n^3$ cubes → find number with 0, 1, 2, 3 painted faces
• Less common: Two adjacent faces painted, or single face painted
• Always verify: corner cubes always have 3 painted faces (8 corners), edge cubes always have 2 painted faces (12 edges × (n-2)), face cubes always have 1 painted face ($6 \times (n-2)^2$), interior always 0 painted ($(n-2)^3$)

Formula Quick Reference:

Cubes with	Formula	For n=3
3 faces	8	8
2 faces	$12(n-2)$	12
1 face	$6(n-2)^2$	6
0 faces	$(n-2)^3$	1
Total	$n^3$	27

Practice Strategy: Start with n=3 and n=4 cubes and draw them out. Label each cube’s position. This builds intuition so you don’t have to memorize formulas blindly. Then apply formulas for larger n (n=5,6,7) and verify.

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