Cube and Cuboid

1. Core Concepts & Theoretical Blueprint

Cube and Cuboid reasoning questions present a large cube or cuboid, painted on its outer surface(s) with one or more colors, that is then cut into a specified number of smaller identical unit cubes — and ask you to determine how many smaller cubes have a specific number of painted faces (0, 1, 2, or 3), or a specific color combination, based purely on the smaller cube's ORIGINAL POSITION within the larger structure before cutting.

The underlying spatial logic rests on positional classification: every unit cube's number of painted faces is determined ENTIRELY by how many outer surfaces of the large cube/cuboid it touched before cutting — corner unit cubes touch 3 outer surfaces (3 painted faces), edge unit cubes (non-corner) touch 2 outer surfaces (2 painted faces), face-center unit cubes (non-edge) touch 1 outer surface (1 painted face), and fully interior unit cubes touch 0 outer surfaces (0 painted faces, completely unpainted).

Reference Table: Painted-Face Count Formulas for an N×N×N Cube Cut into N³ Unit Cubes

Painted Faces	Position Type	Count Formula (for N≥2)
3 faces painted	Corner cubes	Always exactly 8 (every cube has exactly 8 corners)
2 faces painted	Edge cubes (non-corner)	12 × (N−2)
1 face painted	Face-center cubes (non-edge)	6 × (N−2)²
0 faces painted	Fully interior cubes	(N−2)³
Total unit cubes	—	N³ (must equal the sum of all above categories)

The Universal Trap: (1) Students apply the standard N×N×N cube formulas directly to CUBOID (non-cube, i.e., unequal side lengths L×B×H) problems without adjusting — cuboid problems require separate edge-length-specific counting for each dimension rather than the single-variable N formulas, since the number of unit cubes along each edge differs. (2) Students forget that the formulas above ONLY apply when ALL SIX faces of the large cube are painted — if only SOME faces are painted (e.g., only 4 side faces, or only the top and bottom), the corner/edge/face-center counts must be recalculated by carefully considering which specific faces contribute paint to each unit cube position. (3) Students miscount by forgetting that "2 faces painted with 2 DIFFERENT colors" (if different faces are painted different colors) is a distinct sub-category requiring identification of which specific pair of adjacent painted faces (and therefore which specific edge) a unit cube sits on — always map out which large-cube faces are which color before counting color-specific combinations.

2. Exhaustive Question Typology

                            CUBE AND CUBOID
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      |               |                |                |               |
   Type 1          Type 2           Type 3           Type 4          Type 5
 Standard          Cuboid            Partially-        Multi-Color     Cross-
 N×N×N Cube,        (Unequal          Painted           (Different      Sectional/
 All Faces           Sides)            Cube (Some         Colors on      Layer-Based
 Painted,            Painted            Faces              Different     Counting
 Standard Count      Cube               Unpainted)         Faces)

Type 1 — Standard N×N×N Cube, All Faces Painted, Standard Count

Core Scenario: "A cube of side 4 cm is painted on all six faces and then cut into unit cubes of side 1 cm. How many unit cubes have exactly 2 faces painted?" Governing Rule/Logic: IF the cube is N×N×N with all six faces painted THEN apply the formula for 2-face-painted cubes: 12×(N−2) = 12×(4−2) = 12×2 = 24.

Type 2 — Cuboid (Unequal Sides) Painted Cuboid

Core Scenario: "A cuboid measuring 5×4×3 units is painted on all faces and cut into unit cubes. How many unit cubes have exactly 1 face painted?" Governing Rule/Logic: IF the shape is a cuboid with dimensions L×B×H THEN calculate face-center cubes separately for EACH pair of opposite faces: for the L×B faces (top/bottom): 2×(L−2)×(B−2); for the L×H faces (front/back): 2×(L−2)×(H−2); for the B×H faces (left/right): 2×(B−2)×(H−2); sum all three pairs for the total 1-face-painted count.

Type 3 — Partially-Painted Cube (Some Faces Unpainted)

Core Scenario: "A cube of side 5 units has only its TOP and BOTTOM faces painted (the four side faces remain unpainted), then cut into unit cubes. How many unit cubes have exactly 1 face painted?" Governing Rule/Logic: IF only specific faces are painted THEN count only the unit cubes touching those SPECIFIC painted faces (excluding edges shared with unpainted faces, which would give the unit cube 0 painted faces from that shared edge) — for top/bottom-only painting, cubes touching ONLY the top or ONLY the bottom face (not also touching a side edge, since sides are unpainted) get 1 painted face: this equals the full top face's unit cubes MINUS those also on an outer edge (which would still only have 1 painted face here, since edges of the cube touching unpainted side faces don't add extra paint) — so effectively the entire top layer's N² cubes each get exactly 1 painted face (from the top), and similarly for the bottom layer.

Type 4 — Multi-Color (Different Colors on Different Faces)

Core Scenario: "A cube is painted Red on top and bottom, Blue on front and back, and Green on left and right, then cut into unit cubes. How many unit cubes have exactly one Red face and one Blue face (2 faces painted, specifically Red+Blue)?" Governing Rule/Logic: IF different face-pairs have different colors THEN a unit cube with a Red+Blue combination must sit on the specific EDGE where a Red face (top or bottom) meets a Blue face (front or back) — identify exactly which 4 of the cube's 12 total edges are Red-Blue edges (top-front, top-back, bottom-front, bottom-back) and count unit cubes along those 4 specific edges only.

Type 5 — Cross-Sectional/Layer-Based Counting

Core Scenario: "A cube of side 6 units is painted on all faces and cut into unit cubes. How many unit cubes in the MIDDLE horizontal layer (layer 3, counting from the bottom) have exactly 1 painted face?" Governing Rule/Logic: IF the question asks about a SPECIFIC layer (a horizontal cross-section) rather than the whole cube THEN analyze that single N×N layer independently: for a middle layer (not the top or bottom layer, so it has no painted horizontal face), only the outer boundary cubes of that layer (touching the 4 vertical side faces) have exactly 1 painted face, following the same edge/corner logic as a 2D square's perimeter — specifically, the layer's 4 corner cubes have 2 painted faces (touching two side faces), while the layer's remaining boundary cubes (non-corner) have exactly 1 painted face: 4×(N−2) for that single layer.

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Standard N×N×N Cube, All Faces Painted, Standard Count

Q1. A cube of side 5 cm is painted on all faces and cut into unit cubes of side 1 cm. How many unit cubes have exactly 3 faces painted? (A) 6 (B) 8 (C) 12 (D) 24

Correct Answer: (B) 8 Solution: Corner cubes (3 faces painted) are always exactly 8, regardless of the cube's size (N≥2), since every cube has exactly 8 corners.

Q2. A cube of side 6 cm is painted on all faces and cut into unit cubes of side 1 cm. How many unit cubes have exactly 1 face painted? (A) 48 (B) 64 (C) 72 (D) 96

Correct Answer: (D) 96 Solution: Apply the formula: 6×(N−2)² = 6×(6−2)² = 6×16 = 96.

Q3. A cube of side 4 cm is painted on all faces and cut into unit cubes of side 1 cm. How many unit cubes have 0 faces painted (completely unpainted)? (A) 4 (B) 8 (C) 16 (D) 27

Correct Answer: (B) 8 Solution: Apply the formula: (N−2)³ = (4−2)³ = 2³ = 8.

Type 2 — Cuboid (Unequal Sides) Painted Cuboid

Q1. A cuboid measuring 4×3×2 units is painted on all faces and cut into unit cubes. How many unit cubes have exactly 3 faces painted (corner cubes)? (A) 6 (B) 8 (C) 12 (D) 24

Correct Answer: (B) 8 Solution: Corner cubes are always exactly 8 for any rectangular cuboid (or cube), since every rectangular solid has exactly 8 corners, regardless of its specific dimensions.

Q2. A cuboid measuring 5×4×3 units is painted on all faces and cut into unit cubes. How many unit cubes have exactly 0 faces painted (completely interior)? (A) 3 (B) 6 (C) 9 (D) 12

Correct Answer: (B) 6 Solution: Interior cubes formula for a cuboid: (L−2)×(B−2)×(H−2) = (5−2)×(4−2)×(3−2) = 3×2×1 = 6.

Q3. A cuboid measuring 6×4×2 units is painted on all faces and cut into unit cubes. How many unit cubes have exactly 2 faces painted (edge cubes, non-corner)? (A) 24 (B) 28 (C) 32 (D) 36

Correct Answer: (C) 32 Solution: For a cuboid, edge cubes must be counted per edge-length category (a cuboid has 4 edges of each of its 3 distinct lengths). Edges of length 6 (4 such edges): (6−2)×4=16. Edges of length 4 (4 such edges): (4−2)×4=8. Edges of length 2 (4 such edges): (2−2)×4=0 (since length 2 means no non-corner cubes exist along that edge — the entire edge is just 2 corner cubes). Total edge cubes = 16+8+0 = 24 — recompute: this gives 24, not 32. Correct Answer: (A) 24.

Type 3 — Partially-Painted Cube (Some Faces Unpainted)

Q1. A cube of side 4 units has only its top face painted (all other 5 faces unpainted), then cut into unit cubes. How many unit cubes have exactly 1 face painted? (A) 4 (B) 8 (C) 16 (D) 32

Correct Answer: (C) 16 Solution: Since only the top face is painted, every one of the N² unit cubes in the top layer gets exactly 1 painted face (from the top), regardless of whether they're also on a side edge (since the sides are unpainted, touching an unpainted side face doesn't add any paint). Total = N² = 4² = 16.

Q2. A cube of side 5 units has its top and bottom faces painted (the four side faces unpainted), then cut into unit cubes. How many unit cubes have exactly 0 faces painted? (A) 25 (B) 50 (C) 75 (D) 100

Correct Answer: (C) 75 Solution: Since only top and bottom are painted, the 3 middle layers (out of 5 total horizontal layers) are entirely unpainted (0 faces painted for every cube in these layers, since the sides are also unpainted). Each middle layer has N²=25 unit cubes, and there are (5−2)=3 such middle layers. Total 0-painted cubes = 25×3=75.

Q3. A cube of side 3 units has all four SIDE faces painted (top and bottom unpainted), then cut into unit cubes. How many unit cubes have exactly 2 faces painted? (A) 4 (B) 8 (C) 12 (D) 16

Correct Answer: (A) 4 Solution: With only the 4 side faces painted, a unit cube gets 2 painted faces only if it sits on a VERTICAL edge where two side faces meet (a corner of the cube, viewed from the top, in any horizontal layer). For a 3×3×3 cube, each horizontal layer has 4 such vertical-edge positions (the 4 vertical corners), and since all 3 layers (top, middle, bottom) have side faces painted at that position (as the painting is on the full height of the side faces), each of the 3 layers contributes 4 such 2-painted-face cubes — but wait, since top/bottom are unpainted, a cube in the TOP layer sitting on a vertical edge still only touches 2 side faces (both painted), giving it exactly 2 painted faces regardless of layer. With 3 layers × 4 vertical-edge positions = 12 — recheck: for a 3-unit cube, the middle layer's 4 corner-equivalent positions plus top and bottom layers' same 4 positions each layer = 3×4=12 cubes with 2 painted faces. Correct Answer: (C) 12.

Type 4 — Multi-Color (Different Colors on Different Faces)

Q1. A cube of side 4 units is painted Red on top and bottom, and Blue on all 4 side faces, then cut into unit cubes. How many unit cubes have exactly 1 Red face and 1 Blue face (i.e., exactly 2 faces painted, one of each color)? (A) 8 (B) 12 (C) 16 (D) 24

Correct Answer: (C) 16 Solution: A Red+Blue combination occurs on unit cubes sitting on the edge where the top/bottom (Red) meets a side face (Blue) — this is the standard "2 faces painted" edge category, but restricted specifically to the TOP and BOTTOM edges (not the 4 vertical side-to-side edges, which would be Blue+Blue, same color, not counted as a distinct Red+Blue combination). There are 4 such Red-Blue edges at the top (where top meets each of the 4 sides) and 4 at the bottom, totaling 8 relevant edges. Each edge (of length 4) has (4−2)=2 non-corner cubes qualifying with exactly 2 painted faces. Total = 8 edges × 2 = 16.

Q2. Using the same cube (side 4, top/bottom Red, sides Blue), how many unit cubes have exactly 3 Blue faces painted (all-Blue corners)? (A) 0 (B) 4 (C) 8 (D) 16

Correct Answer: (A) 0 Solution: Every corner of the cube touches 3 faces: in this coloring scheme, every corner touches either the top or bottom (Red) PLUS two adjacent side faces (Blue) — meaning every corner cube has exactly 1 Red face and 2 Blue faces (never 3 Blue, since top/bottom is always involved at any corner). Zero corners have all-3-Blue.

Q3. Using the same cube (side 4, top/bottom Red, sides Blue), how many unit cubes have exactly 1 Red face and 2 Blue faces (corner cubes)? (A) 4 (B) 8 (C) 12 (D) 16

Correct Answer: (B) 8 Solution: As established, every corner cube (all 8 corners of any cube) touches 1 Red face (top or bottom) and 2 Blue faces (two adjacent sides) in this specific coloring scheme — all 8 corner cubes fall into this exact category.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1: The Fixed-8-Corners Constant Application: For ANY cube or cuboid problem (regardless of size, as long as each dimension is at least 2 units), immediately note that the "3 faces painted" (fully-painted, all-faces-colored scenario) count is ALWAYS exactly 8 — never calculate this one; just state it directly. Mental Model: The number of corners on any rectangular solid is a fixed geometric constant (8), completely independent of the solid's specific dimensions; recognizing this immediately eliminates one full calculation step from every standard cube/cuboid problem, and serves as a quick sanity check (if your calculated corner count isn't 8 for a standard fully-painted problem, you've made an error).

Shortcut 2: The Per-Dimension Formula Adaptation for Cuboids Application: For cuboid problems (unequal L×B×H), never apply the single-variable N formulas directly — instead, adapt each formula to use the THREE separate dimensions: interior cubes = (L−2)(B−2)(H−2); edge cubes require summing over the three distinct edge-length groups separately: 4×[(L−2)+(B−2)+(H−2)]; face-center cubes require summing over the three distinct face-pair types: 2×[(L−2)(B−2)+(B-2)(H-2)+(L-2)(H-2)]. Mental Model: A cuboid's three dimensions are generally different, so the elegant single-variable cube formulas (which rely on all edges being equal) must be decomposed into their three-dimension-specific components; treating cuboid problems as "three separate cube-like calculations combined" rather than force-fitting the cube formula prevents a very common, systematic error.

5. Deep-Dive: Most Frequently Asked Questions (Exam-Style Walkthroughs)

Problem 1 (SSC/RRB Level): A cube of side 6 cm, painted on all six faces, is cut into unit cubes of side 1 cm each. How many unit cubes have AT LEAST one face painted?

Traditional Method (Slow) — approx. 40-50 seconds: A slow solver separately calculates the 3-face, 2-face, and 1-face painted counts using each formula individually, then manually adds them together, potentially making an arithmetic slip in one of the three separate calculations or in the final addition.

Exam Shortcut (Fast) — approx. 15-20 seconds: Apply a complementary approach instead of direct addition: "at least one face painted" is the complement of "zero faces painted." Total unit cubes = N³ = 6³=216. Zero-painted (fully interior) cubes = (N−2)³ = 4³=64. At least one face painted = Total − Zero-painted = 216−64 = 152. Answer: 152 unit cubes have at least one face painted. The complementary approach avoids calculating and summing three separate categories (3-face, 2-face, 1-face) individually.

Problem 2 (UPSC/Banking Advanced Level): A cuboid measuring 8×6×4 units is painted Red on the two 8×6 faces (top and bottom), Blue on the two 8×4 faces (front and back), and Green on the two 6×4 faces (left and right). It is then cut into unit cubes. How many unit cubes have exactly 2 painted faces of two DIFFERENT colors, and specifically, how many have a Red-Green combination?

Step-by-step derivation:

1. Recognize that "2 painted faces of different colors" occurs exclusively at unit cubes sitting on an EDGE where two differently-colored large faces meet (as opposed to an edge where the same color meets itself, which cannot happen here since each color is used on a genuinely different pair of opposite faces, so every edge in this cuboid is inherently a two-different-color edge).
2. Identify the three distinct edge-length categories and which color pair each represents: edges of length 8 (where Red top/bottom meets Blue front/back) — these are the 4 edges running along the 8-unit dimension where the 8×6 and 8×4 faces meet; edges of length 6 (where Red top/bottom meets Green left/right) — the 4 edges along the 6-unit dimension where 8×6 and 6×4 faces meet; edges of length 4 (where Blue front/back meets Green left/right) — the 4 edges along the 4-unit dimension where 8×4 and 6×4 faces meet.
3. Specifically for the Red-Green combination (asked in the question): this occurs on the edges of length 6 (where the Red top/bottom faces meet the Green left/right faces). There are 4 such edges (one at each of the 4 vertical corners where these two colors' faces meet along the 6-unit dimension).
4. Calculate non-corner cubes per edge: for an edge of length 6, non-corner (2-face-painted) cubes = (6−2) = 4 per edge.
5. Total Red-Green combination cubes = 4 edges × 4 cubes per edge = 16.
6. For completeness, calculate the OTHER two color-combination categories using the same method: Red-Blue combination occurs on the 4 edges of length 8: (8−2)×4 = 6×4=24. Blue-Green combination occurs on the 4 edges of length 4: (4−2)×4=2×4=8.
7. Sum all three combinations for the TOTAL "2 different colors painted" count: Red-Green(16) + Red-Blue(24) + Blue-Green(8) = 48.
8. Cross-verify using the general edge-cube formula: total edge (2-face-painted) cubes for a cuboid = 4×[(L−2)+(B−2)+(H−2)] = 4×[(8−2)+(6−2)+(4−2)] = 4×[6+4+2]=4×12=48 ✓, matching the sum from step 7 exactly, confirming the color-specific breakdown is internally consistent with the general formula.

Final Answer: 16 unit cubes have a Red-Green combination specifically; 48 unit cubes total have exactly 2 painted faces of two different colors (16 Red-Green + 24 Red-Blue + 8 Blue-Green).

6. Chapter Checklist for Students

• I immediately state "8" for any fully-painted cube/cuboid's corner (3-face-painted) count, using the Fixed-8-Corners Constant without recalculating.
• I adapt formulas to three separate dimensions (L, B, H) for any cuboid problem, never force-fitting the single-variable N cube formula.
• I use the complementary approach ("at least one" = Total − "zero painted") whenever a question asks for "at least" rather than an exact count.
• I carefully map out which specific large-cube faces are which color before attempting any multi-color combination question, identifying exactly which edges represent each color pair.
• I verify partially-painted cube answers by explicitly confirming which specific faces are painted and which are not, rather than assuming all six faces are painted by default.