Painted Cube Questions Follow the Position of Each Small Cube

Duration: 7 min

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  1. About Course: About the Course
  2. Paper 2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  3. Paper 2 | Computer Organization And Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Control Unit Design, Instr Formats & Modes, Programming the Basic Computer
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  11. Paper 2 | Information Systems and Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  12. Paper 2 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics
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  14. Paper 2 | Cyber Security And Emerging Technologies: Emerging Tech Trends, Internet of Things
  15. Paper 2 | Cloud Technology: Cloud Computing
  16. Paper 2 | Probability and statistics: Probability, Statistics
  17. Paper-1 English: Vocabulary, Preposition, Articles, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Cloze Test (Cloze Passage), Sentence Construction
  18. Paper 1 General Mental Ability & Logical reasoning: Ratio and Proportion (Ratios), Divisibility Rules, Data Interpretation, Mixture and Alligation, Permutation and Combination, Logarithm, Number System, HCF LCM, Sequence and Series (Series), Speed Time and Distance, Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Percentage, Average, Powers and Exponents (Surds and Indices), Algebra, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Pipe and Cistern, Mensuration and Geometry, Mental Ability, Age Problems, Partnership, Arithmetic, Speed Mathematics and Basic Mathematics, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Seating Arrangements, Blood Relations, Directions (Direction Test), Analytical Reasoning (Counting Figures Reasoning), Puzzle Solving (Puzzles), Calendar, Clock, Cubes & Dices, Selection Decision Table (Decision Making), Logical Sequence Of Words, Venn Diagram, Analogy
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  20. Full Length Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lesson explains painted cube questions by linking each small cube's position to its number of painted faces. The instructor uses a table and 3D diagram to show that corner cubes have 3 painted faces, edge cubes (excluding corners) have 2, face-interior cubes have 1, and completely interior cubes have 0. Formulas are given: 8 corners; 12(n-2) edge cubes; 6(n-2)^2 face-interior cubes; and (n-2)^3 interior cubes. A useful result is derived: small cubes with at least two painted faces equal 8 + 12(n-2) = 12n - 16. The lesson then applies this to a word problem: an 18 cm cube painted on all faces is cut into 3 cm cubes. Since n = 18/3 = 6, the number of small cubes with exactly two painted faces is 12(6-2) = 48. The solution is verified by summing all categories: 8 + 48 + 96 + 64 = 216, which equals 6^3.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video introduces painted cube questions with a table categorizing small cubes by position and number of painted faces. The instructor highlights the 'Painted faces' column, underlines 'On an edge, excluding corners', and circles the value 2 and formula 12(n - 2). A large cube diagram is annotated with red arrows pointing to edges and corners, visually connecting the table's abstract categories to specific locations on a 3D model. The full table is shown: corner (3 faces, 8), edge excluding corners (2 faces, 12(n-2)), inside a face excluding edges (1 face, 6(n-2)^2), and completely inside (0 faces, (n-2)^3).

  2. 2:00 – 5:00 02:00-05:00

    The instructor progressively highlights each row of the table with red circles and boxes, moving from corner cubes (3 faces) through edge cubes (2 faces, formula 12(n-2)) to face-interior cubes (1 face, formula 6(n-2)^2) and completely interior cubes (0 faces). A 'Useful result' is derived: small cubes with at least two painted faces equal 8 + 12(n-2) = 12n - 16. A verification step shows that all four formulas sum to n^3, confirming the partition is complete. The segment transitions to a new slide titled 'Small Cubes with Exactly Two Painted Faces', presenting a word problem: an 18 cm cube painted yellow on all faces is cut into smaller cubes of side 3 cm each.

  3. 5:00 – 7:04 05:00-07:04

    The instructor solves the word problem by first calculating n = 18/3 = 6, the number of small cubes along one edge. The formula for exactly two painted faces is applied: 12(n-2) = 12(6-2) = 48. A summary slide titled 'Two Painted Faces Means Edge Cubes' confirms the answer of 48. An independent verification sums all categories: 8 corners + 48 edge cubes + 96 face-interior cubes + 64 interior cubes = 216, which equals 6^3, confirming the solution is consistent with the total number of small cubes.

The core teaching progression moves from general classification to specific application. First, the instructor establishes a complete taxonomy of small cube positions within a larger divided cube, assigning each position a fixed number of painted faces and a counting formula. The key insight is that exactly two painted faces corresponds uniquely to edge cubes excluding corners, counted by 12(n-2). The 'at least two painted faces' result (12n - 16) combines corner and edge cubes. The worked example demonstrates the standard method: compute n from side lengths, identify the relevant position category, apply its formula, and verify by summing all categories to n^3. This verification step is a critical exam technique that catches arithmetic or categorization errors.

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