Painted Cube Questions Follow the Position of Each Small Cube

Duration: 7 min

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Module outline

  1. About Course: About the Course
  2. Paper 1 | Quantitative Aptitude: Speed Mathematics and Basic Mathematics, Percentage, Ratio and Proportion (Ratios), Powers and Exponents (Surds and Indices), Logarithm, Sequence and Series (Series), Mensuration and Geometry, Data Interpretation, Average, Arithmetic
  3. Paper 1 | Logical Reasoning: Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Analogy, Cubes & Dices, Syllogisms
  4. Paper 1 | English Language: Vocabulary, Noun, Pronoun, Verb, Adjectives, Adverb, Preposition, Conjunction, Interjection, Articles, Tenses, Subject Verb Agreement (Verb Noun Agreement), Modals, Determiner, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Construction
  5. Paper 1 | Unit Tests:
  6. Paper 1 | Full-Length Mock Tests:
  7. Paper 2 | Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Permutation and Combination, Probability, Statistics
  8. Paper 2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  9. Paper 2 | Computer Architecture: Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design
  10. Paper 2 | Programming and Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  11. Paper 2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  12. Paper 2 | Theory Of Computation/Automata Theory: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  13. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  14. Paper 2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  15. Paper 2 | DataBase Management System/DBMS: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery
  16. Paper 2 | Computer Networks: Introduction to CN, 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
  17. Paper 2 | CS/IT Sectional Mock Tests:
  18. Complete Paper Full 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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