Painted Cube Questions Follow the Position of Each Small Cube

Duration: 7 min

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Module outline

  1. English Comprehension: Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Reading Comprehension, Vocabulary, Idioms and Phrases, Narrative Sequencing, Critical Reasoning
  2. Reasoning and Aptitude: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Seating Arrangements, Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Data Sufficiency, Selection Decision Table (Decision Making), Calendar, Clock, Cubes & Dices, Data Arrangement, Venn Diagram, Analytical Reasoning (Counting Figures Reasoning), Number System, HCF LCM, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Speed Time and Distance, Mensuration and Geometry, Arithmetic, Mixture and Alligation, Age Problems, Data Interpretation, Alphabet and Word Tests, Numerical Reasoning, Number System, Averages, Mixtures and Alligation, Percentages, Series Completion, Analogy and Classification, Patterns in Two-Dimensional Figures, Coding and Decoding, Mathematical Operations, Direction Sense, Logical Deduction and Induction, Simplification and Approximation, Assembling and Grouping of Shapes, Transformation of Shapes, Three-Dimensional Patterns and Visualization, Cubes and Dice, Algebra, Traverse Surveying, Clocks and Calendars, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Ratio and Proportion, Mensuration, Permutations and Combinations, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Sequences and Series, Powers, Surds and Logarithms, Subject-Verb Agreement, Sentence Completion, Geometry, Probability, Trigonometry, Measurement and Quantity Take-Off, Statistics, Data Sufficiency, Blood Relations, Speed, Time and Distance, Time and Work, Data Interpretation, Sequence and Series (Series), Algebra, Permutation and Combination, Combinatorics
  3. General Awareness: Physics, Chemistry, Biology, Geography, History, Polity, Indian Economy
  4. Computer Fundamentals: Basics of COA, Number System, Number Representation, Semiconductor Memories and Programmable Logic, Timing, Hazards and Testing, Processor Organization and Instructions, Number Systems and Codes, Combinational Logic Design, Combinational Logic Circuits, Number Systems and Codes
  5. Operating System: Introduction to OS, Unix OS, Linux OS
  6. C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  7. OOPS with C++ & Java: OOPS with C++, Java
  8. Database Management System: Basics of DBMS, SQL
  9. MS Office: MS Office Applications
  10. Internet & Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services
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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