Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples - Aptitude Questions & Answers
Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples is one of the most important topics in Logical Reasoning Explained: Concepts, Tricks, and Practice Questions. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.
Mastering Cube concepts is essential for aptitude tests and competitive exams. This comprehensive guide covers everything from basic cube properties to advanced painted cube problems, with clear formulas, examples, and visual diagrams.
Cube Geometry: Complete Guide with Formulas & Examples
What is a Cube?
A cube is a three-dimensional solid object with six equal square faces, twelve equal edges, and eight vertices (corners). All faces are perpendicular to adjacent faces, and all edges are equal in length.
Basic Properties of a Cube
Faces
Flat surfaces of the cube
Edges
Line segments where faces meet
Corners/Verices
Points where edges meet
Dimensions
Length = Breadth = Height
1. Maximum/Minimum Number of Cubes from Cuts
Key Formulas
Minimum Number of Pieces
When all cuts are parallel
Maximum Number of Cubes
Where:
nl = cuts in length direction
nw = cuts in width direction
nh = cuts in height direction
Important Note
For a cube, dimensions are equal: Length = Width = Height
Total cuts should be divided into 3 equal or consecutive parts for maximum cubes
Example Problems
Example 1: Even Division
Problem: How many minimum and maximum pieces can be obtained when 9 cuts are made on a cube?
Solution:
- Minimum pieces: Cuts + 1 = 9 + 1 = 10 pieces
- Division of cuts: Since it\\\'s a cube, divide cuts equally in 3 directions:
nl = nw = nh = 9 ÷ 3 = 3 cuts in each direction - Maximum cubes: (3+1) × (3+1) × (3+1) = 4 × 4 × 4 = 64 cubes
Example 2: Uneven Division
Problem: How many maximum pieces can be obtained when 19 cuts are made on a cube?
Solution:
Since 19 is not divisible by 3, distribute cuts consecutively:
- Let cuts in three directions be as equal as possible
- 19 ÷ 3 = 6 remainder 1, so distribute as: 6, 6, 7
- nl = 6, nw = 6, nh = 7
- Maximum cubes = (6+1) × (6+1) × (7+1) = 7 × 7 × 8 = 392 cubes
2. Maximum/Minimum Cuts Required
Key Formulas
Minimum Cuts Required
Where cuts are made parallel to faces
Maximum Cuts Required
When cuts are not necessarily parallel
Example Problem
Example 3: Cuts for Given Pieces
Problem: How many minimum/maximum cuts are required to obtain 216 pieces from a cube?
Solution:
- Find cube dimensions: 216 = 6 × 6 × 6 = (5+1) × (5+1) × (5+1)
- So nl = 5, nw = 5, nh = 5
- Minimum cuts: nl + nw + nh = 5 + 5 + 5 = 15 cuts
- Maximum cuts: 216 - 1 = 215 cuts
3. Painted Cubes Problems
Case 1: All Sides Painted (Standard Case)
Formulas for n × n × n Cube
3 Sides Painted
Corner cubes
2 Sides Painted
Edge cubes (excluding corners)
1 Side Painted
Face cubes (excluding edges)
No Side Painted
Internal cubes
Visual Representation (n = 4)
Layer 1 (Top)
Layer 2
Layer 3
Layer 4 (Bottom)
Example 4: Standard Painted Cube
Problem: 64 small cubes form a large cube painted on all faces. Find:
- No face painted
- Exactly one face painted
- Exactly two faces painted
- Three faces painted
Solution:
- Find n: n³ = 64 ⇒ n = 4
- No face painted: (n-2)³ = (4-2)³ = 2³ = 8 cubes
- One face painted: 6 × (n-2)² = 6 × 2² = 6 × 4 = 24 cubes
- Two faces painted: 12 × (n-2) = 12 × 2 = 24 cubes
- Three faces painted: Always 8 cubes
2. 24 cubes (1 face)
3. 24 cubes (2 faces)
4. 8 cubes (3 faces)
Case 2: Two Adjacent Faces Same Color
Example 5: Two Adjacent Pink Faces
Problem: 125 cubes form a large cube. Two adjacent faces are pink, others different colors. Find:
- Cubes with only pink color
- Cubes with both pink and blue
- Cubes with blue color
Solution:
- Find n: n³ = 125 ⇒ n = 5
- Only pink cubes:
- One face pink cubes (excluding edges): (n-2)² = (5-2)² = 3² = 9
- Two adjacent faces: 2 × 9 = 18
- Edge between pink faces (counted once): n-2 = 5-2 = 3
- Total only pink = 18 + 3 = 21 cubes
- Pink and blue cubes:
- Edges where pink and blue meet: 3 edges
- Cubes on these edges (excluding corners): (n-2) × 3 = 3 × 3 = 9
- Corner cubes: 2 (top and bottom of common edge)
- Total = 9 + 2 = 11 cubes (or use formula: 3n - 4 = 15 - 4 = 11)
- Blue cubes:
- One blue face: n² = 25 cubes
- Two blue faces: 2 × 25 = 50
- Subtract common edge counted twice: 50 - n = 50 - 5 = 45 cubes
2. 11 cubes (pink & blue)
3. 45 cubes (blue)
Case 3: Three Adjacent Faces Same Color
Example 6: Three Adjacent Green Faces
Problem: Cube painted with three adjacent faces green, opposite three faces yellow. Cut into 64 cubes. Find:
- Cubes with only green color
- Cubes with both green and yellow
Solution:
- Find n: n³ = 64 ⇒ n = 4
- Only green cubes:
- One face green: 3 × (n-2)² = 3 × 2² = 3 × 4 = 12
- Two faces green (edges between green faces): 3 × (n-2) = 3 × 2 = 6
- Three faces green (corner): 1
- Total = 12 + 6 + 1 = 19 cubes
- Green and yellow cubes:
- Edges between green and yellow: 6 edges
- Cubes on these edges (excluding corners): 6 × (n-2) = 6 × 2 = 12
- Corner cubes (where green and yellow meet): 6
- Total = 12 + 6 = 18 cubes
2. 18 cubes (green & yellow)
Quick Reference Formulas
| Problem Type | Formula | Explanation |
|---|---|---|
| Maximum cubes from cuts | (nl+1)×(nw+1)×(nh+1) | Divide total cuts into 3 equal/consecutive parts |
| Minimum cuts for cubes | nl + nw + nh | Where nl, nw, nh come from (nl+1)×(nw+1)×(nh+1) |
| 3 sides painted | 8 (always) | Corner cubes |
| 2 sides painted | 12 × (n-2) | Edge cubes excluding corners |
| 1 side painted | 6 × (n-2)² | Face cubes excluding edges |
| No side painted | (n-2)³ | Internal cubes |
Frequently Asked Questions
Why are there always 8 cubes with 3 sides painted?
Because a cube has 8 corners, and corner cubes always have 3 faces exposed and painted (assuming the cube is painted on all faces).
How to divide cuts when number is not divisible by 3?
Distribute cuts consecutively. For example, for 10 cuts: 3, 3, 4 or 3, 4, 3. The maximum cubes will be the same for any order.
What\\\'s the difference between pieces and cubes?
\\\"Pieces\\\" refers to any shape after cutting. \\\"Cubes\\\" specifically means the pieces are cube-shaped. Maximum cubes formula gives cube-shaped pieces.
How to visualize painted cubes problems?
Think in terms of layers. Outer layer has paint, inner layers may not. Corners always have 3 painted faces, edges have 2, face centers have 1.
Frequently Asked Questions
What is Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples?
Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.
Is Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples important for competitive exams?
Yes, Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.
How to prepare Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples easily?
Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples.
What are the important formulas in Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples?
Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples.
How many questions come from Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples?
Typically 5-10 questions come from Cubes in Logical Reasoning: Tips, Tricks, and Solved Examples in most competitive exams, making it a high-scoring section.
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