Recursion Practice Question

Duration: 4 min

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Inside: a video lesson and guided study material.

Module outline

  1. Course Overview: About the Course
  2. Paper - 1 | Unit - 1 | Teaching Aptitude: शिक्षण की अवधारणा एवं उद्देश्य / Concept and Objectives of Teaching, शिक्षण के स्तर / Levels of Teaching, शिक्षार्थी की विशेषताएँ / Learner's Characteristics, अधिगम: प्रकृति एवं सिद्धांत / Learning: Nature and Theories, संज्ञान, स्मृति एवं अधिसंज्ञान / Cognition, Memory and Metacognition, शिक्षण को प्रभावित करने वाले कारक / Factors Affecting Teaching, प्रभावी शिक्षण एवं शिक्षक / Effective Teaching and the Teacher, अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन / Instructional Objectives and Lesson Planning, शिक्षण विधियाँ / Methods of Teaching, ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ / Offline and Online Teaching Methods, शिक्षण सहायक प्रणाली / Teaching Support System, मूल्यांकन प्रणाली / Evaluation Systems, Sāṃkhya / सांख्य, व्यक्तिगत व्यवहार / Individual Behaviour, मानव संसाधन प्रबंधन / Human Resource Management, बाल्यावस्था में विकास के क्षेत्र / Domains of Child Development, संवेग का जैविक आधार / Biological Basis of Emotion, Upanishads and Bhagavad Gita / उपनिषद और भगवद्गीता, केन्द्रीय और परिधीय तंत्रिका तंत्र / Central and Peripheral Nervous Systems
  3. Paper - 1 | Unit - 2 | Research Aptitude: अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि / Research: Meaning, Features and Scientific Method, अनुसंधान के प्रतिमान एवं उपागम / Research Paradigms and Approaches, अनुसंधान के प्रकार / Types of Research, चर एवं मापन / Variables and Measurement, अनुसंधान की विधियाँ / Methods of Research, अनुसंधान के चरण / Steps of Research, प्रतिचयन / Sampling, आँकड़ा संग्रह के उपकरण एवं विधियाँ / Tools and Methods of Data Collection, आँकड़ा विश्लेषण एवं शोध सांख्यिकी / Data Analysis and Statistics in Research, शोध-प्रबंध एवं लेख लेखन / Thesis and Article Writing, अनुसंधान में ICT का अनुप्रयोग / Application of ICT in Research, शोध नैतिकता / Research Ethics
  4. Paper - 1 | Unit - 3 | Comprehension: प्रत्यक्ष बोध / Literal Comprehension, गद्यांश में शब्दावली और भाषा / Vocabulary and Language in Context, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Restoration and Eighteenth-Century Drama, Modern British Fiction, William Shakespeare, Mid-Twentieth-Century and Contemporary British Fiction, American Drama
  5. Paper - 1 | Unit - 4 | Communication: संचार: अर्थ, प्रकृति एवं प्रक्रिया / Communication: Meaning, Nature and Process, संचार के प्रकार एवं स्तर / Types and Levels of Communication, मौखिक एवं अमौखिक संचार / Verbal and Non-verbal Communication, संकेत विज्ञान एवं अर्थ / Semiotics and Meaning, संचार के मॉडल एवं सिद्धांत / Models and Theories of Communication, प्रभावी संचार / Effective Communication, संचार की बाधाएँ / Barriers to Communication, कक्षा संचार / Classroom Communication, जनसंचार माध्यम: प्रकार एवं विकास / Mass Media: Types and Evolution, जनसंचार एवं समाज / Mass Media and Society, डिजिटल एवं नव माध्यम / Digital and New Media, Static General Knowledge, Morphology and Syntax, Public Health Nutrition / जन स्वास्थ्य पोषण, Newtonian Motion and Stability, GK- Current Affairs, Indian Economy, Art and Architecture, India after Independence
  6. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: संख्या पद्धति एवं सरलीकरण / Number System and Simplification, बीजगणित एवं आयु / Algebra and Ages, प्रतिशतता / Percentage, अनुपात, समानुपात, साझेदारी एवं मिश्रण / Ratio, Proportion, Partnership and Mixtures, औसत / Averages, लाभ, हानि एवं बट्टा / Profit, Loss and Discount, साधारण एवं चक्रवृद्धि ब्याज / Simple and Compound Interest, समय एवं कार्य / Time and Work, चाल, समय एवं दूरी / Speed, Time and Distance, क्षेत्रमिति एवं ज्यामिति / Mensuration and Geometry, श्रेणी / Series, कूटलेखन-कूटवाचन / Coding-Decoding, संबंध, दिशा एवं क्रम / Relationships, Directions and Ranking, व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर / Arrangements, Puzzles, Clock and Calendar, Mathematical Foundations, Polity, जीवनी, Deconstruction, Gender and Queer Theory, Modern British and Irish Poetry, Central Force and Scattering, Chemical thermodynamics, Microscopic techniques, संविधान का दर्शन / Philosophy of the Constitution, Waves, Optics and Charged-Particle Motion, Membrane structure and function, Analysis
  7. Paper - 1 | Unit - 6 | Logical Reasoning: युक्ति की संरचना / Structure of Arguments, पद, परिभाषा एवं भाषा के उपयोग / Terms, Definitions and Uses of Language, निरुपाधिक प्रतिज्ञप्तियाँ / Categorical Propositions, निरुपाधिक न्यायवाक्य / Categorical Syllogism, वेन आरेख / Venn Diagrams, निगमन, आगमन एवं सादृश्यता / Deduction, Induction and Analogy, तर्कदोष / Fallacies, भारतीय तर्कशास्त्र: प्रमाण / Indian Logic: Pramanas, अनुमान एवं व्याप्ति / Anumana and Vyapti, हेत्वाभास / Hetvabhasa, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle / अरस्तू, पाश्चात्य दर्शन की विचारधाराएँ / Western Schools of Philosophy, Kant / कांट, Sadhana Paada, Vaiśeṣika / वैशेषिक
  8. Paper - 1 | Unit - 7 | Data Interpretation: आँकड़े एवं आलेखीय निरूपण / Data and Graphical Representation, आँकड़ा निर्वचन: तालिका / Data Interpretation: Tables
  9. Paper - 1 | Unit - 8 | Information and Communication Technology: Computer Fundamentals and Hardware, Memory, Storage and Data Representation, Computer Software, Office Applications, Multimedia and File Formats, Computer Networks, Internet and World Wide Web, Email and Online Communication, Cyber Security, Digital Ethics, IPR and Society, Cloud and Distributed Computing, Emerging Technologies, ICT in Education and Governance, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Electronic Data Interchange and E-Commerce, Internet Fundamentals, Web Design & Development, Web Publishing & Hosting, Emerging Technologies, Terms & Abbreviations, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, भौगोलिक संकेतों का संरक्षण / Protection of Geographical Indications
  10. Paper - 1 | Unit - 9 | People Development and Environment: पारितंत्र और जैव विविधता / Ecosystems and Biodiversity, मानव और पर्यावरण की अंतःक्रिया / Human and Environment Interaction, विकास और पर्यावरण / Development and Environment, प्राकृतिक संसाधन: वन, जल और मृदा / Natural Resources – Forests, Water and Soil, ऊर्जा संसाधन / Energy Resources, वायु प्रदूषण / Air Pollution, जल प्रदूषण और उपचार / Water Pollution and Treatment, ध्वनि प्रदूषण / Noise Pollution, अपशिष्ट प्रबंधन / Waste Management, पर्यावरणीय स्वास्थ्य और विषविज्ञान / Environmental Health and Toxicology, जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ / Climate Change and Global Environmental Issues, प्राकृतिक प्रकोप और आपदा प्रबंधन / Natural Hazards and Disaster Management, भारतीय पर्यावरण कानून और नीतियाँ / Indian Environmental Laws and Policies, अंतर्राष्ट्रीय पर्यावरण समझौते / International Environmental Agreements, Environment, Geography, कारखाना और उद्योग व्यवस्थाएँ / Factory and Industry Systems, Earth Materials, Surface Features and Processes, संक्रामक रोग और प्रतिरक्षण / Communicable Diseases and Immunization
  11. Paper - 1 | Unit - 10 | Higher Education System: प्राचीन भारत में शिक्षा / Education in Ancient India, प्राचीन भारत के उच्च शिक्षा केंद्र / Ancient Centres of Higher Learning, औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग / Colonial Education Policies and Commissions, औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक / Colonial Colleges, Universities and Reformers, स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ / Education Commissions and Policies after Independence, राष्ट्रीय शिक्षा नीति 2020 / National Education Policy 2020, उच्च शिक्षा संस्थानों के प्रकार / Types of Higher Education Institutions, अधिगम कार्यक्रम और अकादमिक क्रेडिट / Learning Programmes and Academic Credits, व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा / Professional, Technical and Skill-Based Education, मूल्य शिक्षा और पर्यावरण शिक्षा / Value Education and Environmental Education, शासन, प्रशासन और वित्तपोषण / Governance, Administration and Financing, उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग / Regulation, Accreditation and Ranking
  12. Paper 2 | Unit 1 | Discrete Structures and Optimization: Propositional and Predicate Logic, Set Theory, Relations, Functions, क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, Graph Theory, Group Theory, Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Optimization, Markov Chains
  13. Paper 2 | Unit 2 | Computer System Architecture: Logic Gates & Hardware, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Basics of COA, Register Transfer and Microoperations, Programming the Basic Computer, Instr Formats & Modes, Control Unit Design, Pipelining, Input Output Organisation, Cache Memory Organization, Multiprocessors
  14. Paper 2 | Unit 3 | Programming Languages and Computer Graphics: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, HTML Basics, XML, JavaScript-Basics, Java, Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, OOPS with C++, Java Fundamentals, Array, String, IO & References, OOPS with Java
  15. Paper 2 | Unit 4 | Database Management Systems: 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, ORDBMS, Database Security & Authorization, Query Processing & Optimization, Enhanced Data Models, Data Warehousing & Mining, Big Data Systems, NoSQL
  16. Paper 2 | Unit 5 | System Software and Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Windows OS, Linux OS, Security, Distributed Systems, Virtual Machines
  17. Paper 2 | Unit 6 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  18. Paper 2 | Unit 7 | Data Structures and Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  19. Paper 2 | Unit 8 | Theory of Computation and Compilers: 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, 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
  20. Paper 2 | Unit 9 | Data Communication and 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, Network Security, Mobile Technology, Cloud Computing and IoT, Cloud Computing, Mobile Computing
  21. Paper 2 | Unit 10 | Artificial Intelligence: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms
  22. Live Classes: NTA UGC NET 2025 Live Class
  23. Paper 1 | Full Mock Tests:
  24. Paper 2 | Full Mock Tests:
  25. Paper 1 | Previous Year Papers:
  26. Paper 2 | Previous Year Papers:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The user wants a JSON summary of the provided video lecture. The video is about finding the output of a recursive pseudo-code. The video is split into two main parts based on the timeline windows provided: 00:00-02:00 and 02:00-03:49. **Window 1 (00:00 - 02:00):** - The instructor introduces a problem: "Find the output of the following pseudo code?" - The code is: Void main() { fun(4); } Void fun(int x) { if (x > 0) { fun(x - 1); Printf("%d", x); } } - He starts tracing the recursion. He draws a tree structure. - `fun(4)` calls `fun(3)`. - `fun(3)` calls `fun(2)`. - `fun(2)` calls `fun(1)`. - `fun(1)` calls `fun(0)`. - `fun(0)` hits the base case

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The user wants a JSON summary of the provided video lecture. The video is about finding the output of a recursive pseudo-code. The video is split into two main parts based on the timeline windows provided: 00:00-02:00 and 02:00-03:49. **Window 1 (00:00 - 02:00):** - The instructor introduces a problem: "Find the output of the following pseudo code?" - The code is: Void main() { fun(4); } Void fun(int x) { if (x > 0) { fun(x - 1); Printf("%d", x); } } - He starts tracing the recursion. He draws a tree structure. - `fun(4)` calls `fun(3)`. - `fun(3)` calls `fun(2)`. - `fun(2)` calls `fun(1)`. - `fun(1)` calls `fun(0)`. - `fun(0)` hits the base case (implied by `if (x > 0)` failing). - Then the printing happens on the way back up. - He writes down the output sequence: 1, 2, 3, 4. - He explains that the recursive call happens *before* the print statement, so it's a post-order traversal logic (though he doesn't use that term explicitly, he explains the stack unwinding). **Window 2 (02:00 - 03:49):** - The problem changes. - New code: Void main() { fun(3); } Void fun(int x) { if (x > 0) { Printf("%d", x); fun(x - 1); Printf("%d", x); fun(x - 1); Printf("%d", x); } } - He labels the lines of code inside `fun` as 1, 2, 3, 4, 5. - He starts drawing a complex tree for `fun(3)`. - `fun(3)` prints 3. - Calls `fun(2)` (line 2). - Inside `fun(2)`, it prints 2. - Calls `fun(1)` (line 2). - Inside `fun(1)`, it prints 1. - Calls `fun(0)` (line 2) -> returns. - Prints 1 (line 3). - Calls `fun(0)` (line 4) -> returns. - Prints 1 (line 5). - Back in `fun(2)`, prints 2 (line 3). - Calls `fun(1)` (line 4). - Inside `fun(1)`, prints 1. - Calls `fun(0)` -> returns. - Prints 1. - Calls `fun(0)` -> returns. - Prints 1. - Back in `fun(2)`, prints 2 (line 5). - Back in `fun(3)`, prints 3 (line 3). - Calls `fun(2)` (line 4). - This part gets very messy on the board. He draws a huge tree. - He writes the final output sequence at the bottom: 3 2 1 1 1 2 1 1 1 2 3. Wait, let me re-watch carefully. - Actually, looking at the final board writing in the last few frames: - He writes `3 2 1 1 1 2 1 1 1 2 3`. - Let's trace `fun(3)` again. - `fun(3)` -> Print 3. - `fun(2)` -> Print 2. - `fun(1)` -> Print 1. - `fun(0)` -> return. - Print 1. - `fun(0)` -> return. - Print 1. - Print 2. - `fun(1)` -> Print 1. - `fun(0)` -> return. - Print 1. - `fun(0)` -> return. - Print 1. - Print 2. - Print 3. - `fun(2)` -> Print 2. - `fun(1)` -> Print 1. - `fun(0)` -> return. - Print 1. - `fun(0)` -> return. - Print 1. - Print 2. - `fun(1)` -> Print 1. - `fun(0)` -> return. - Print 1. - `fun(0)` -> return. - Print 1. - Print 2. - Print 3. - The sequence seems to be: 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3. - Wait, looking at the board in the last frame (03:49), he writes `3 2 1 1 1 2 1 1 1 2 3`. This looks like the output for `fun(3)` but maybe he stopped or simplified? - Let's look at the board at 03:45. He writes `3 2 1 1 1 2 1 1 1 2 3`. - Let's re-examine the code. - `Printf("%d", x);` (Line 1) - `fun(x - 1);` (Line 2) - `Printf("%d", x);` (Line 3) - `fun(x - 1);` (Line 4) - `Printf("%d", x);` (Line 5) - Trace `fun(3)`: - Print 3. - `fun(2)`: - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - Print 3. - `fun(2)`: - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - Print 3. - The full output should be: 3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3. - The board at 03:45 shows `3 2 1 1 1 2 1 1 1 2 3`. This is only the first half. - Wait, looking closely at the board at 03:45, he writes `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, he writes `3 2 1 1 1 2 1 1 1 2 3` again? No, he underlines it. - Actually, looking at the very last frame, the text at the bottom is `3 2 1 1 1 2 1 1 1 2 3`. - Wait, let me look at the tree drawing. - Top node `fun(3)`. - Branches: 3, `fun(2)`, 3, `fun(2)`, 3. - The `fun(2)` branches: 2, `fun(1)`, 2, `fun(1)`, 2. - The `fun(1)` branches: 1, `fun(0)`, 1, `fun(0)`, 1. - So the structure is symmetric. - The output he writes at the bottom seems to be just the first part? Or maybe he made a mistake? - Let's look at the board at 03:45 again. - He writes `3 2 1 1 1 2 1 1 1 2 3`. - Then he writes `3 2 1 1 1 2 1 1 1 2 3` again? No, he underlines the first sequence. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Looking at crop 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he might have stopped or the video cuts off before he finishes the full trace for the second `fun(2)` call? - No, looking at the tree, he draws the whole tree. - Let's re-read the code. - `fun(3)` calls `fun(2)` (line 2). - Then prints 3 (line 3). - Then calls `fun(2)` (line 4). - Then prints 3 (line 5). - So the output is `Output(fun(2))` + `3` + `Output(fun(2))`. - `Output(fun(2))` is `Output(fun(1))` + `2` + `Output(fun(1))`. - `Output(fun(1))` is `Output(fun(0))` + `1` + `Output(fun(0))`. - `Output(fun(0))` is empty. - So `Output(fun(1))` = `1 1 1`. - `Output(fun(2))` = `1 1 1` + `2` + `1 1 1` = `1 1 1 2 1 1 1`. - `Output(fun(3))` = `1 1 1 2 1 1 1` + `3` + `1 1 1 2 1 1 1`. - Wait, the first print is inside `fun(3)`. - So `fun(3)` prints 3. - Then calls `fun(2)`. - `fun(2)` prints 2. - Then calls `fun(1)`. - `fun(1)` prints 1. - Then calls `fun(0)` (returns). - Prints 1. - Calls `fun(0)` (returns). - Prints 1. - Back in `fun(2)`, prints 2. - Calls `fun(1)`. - `fun(1)` prints 1. - Calls `fun(0)` (returns). - Prints 1. - Calls `fun(0)` (returns). - Prints 1. - Back in `fun(2)`, prints 2. - Back in `fun(3)`, prints 3. - Calls `fun(2)`. - `fun(2)` prints 2. - ... same pattern ... - Prints 2. - Back in `fun(3)`, prints 3. - So the full output is: 3, (1 1 1 2 1 1 1), 3, (1 1 1 2 1 1 1), 3. - Wait, the code is: 1. Print x 2. fun(x-1) 3. Print x 4. fun(x-1) 5. Print x - So for `fun(3)`: - Print 3. - `fun(2)`: - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - Print 3. - `fun(2)`: - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - `fun(1)`: - Print 1. - `fun(0)` (return). - Print 1. - `fun(0)` (return). - Print 1. - Print 2. - Print 3. - So the sequence is: 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3. - The board at 03:45 shows `3 2 1 1 1 2 1 1 1 2 3`. This is exactly the first half (up to the middle 3). - Then he writes `3 2 1 1 1 2 1 1 1 2 3` again? No, looking at 03:49, he writes `3 2 1 1 1 2 1 1 1 2 3` under the previous one? - Actually, looking at the very last frame, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Wait, let me look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then he writes `3 2 1 1 1 2 1 1 1 2 3` again? - No, looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he might have made a mistake or I am misinterpreting the board. - Let's look at the tree drawing. - Top: `fun(3)`. - Branches: 3, `fun(2)`, 3, `fun(2)`, 3. - The `fun(2)` branches: 2, `fun(1)`, 2, `fun(1)`, 2. - The `fun(1)` branches: 1, `fun(0)`, 1, `fun(0)`, 1. - So the output is indeed symmetric. - The text at the bottom `3 2 1 1 1 2 1 1 1 2 3` corresponds to the first `fun(3)` call's execution up to the middle print. - Wait, `fun(3)` prints 3, then `fun(2)`, then prints 3, then `fun(2)`, then prints 3. - So the output is `3` + `Output(fun(2))` + `3` + `Output(fun(2))` + `3`. - `Output(fun(2))` is `2` + `Output(fun(1))` + `2` + `Output(fun(1))` + `2`. - `Output(fun(1))` is `1` + `Output(fun(0))` + `1` + `Output(fun(0))` + `1` = `1 1 1`. - So `Output(fun(2))` = `2 1 1 1 2 1 1 1 2`. - So `Output(fun(3))` = `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - The board at 03:45 shows `3 2 1 1 1 2 1 1 1 2 3`. This is `3` + `Output(fun(2))` + `3`. - Then he writes `3 2 1 1 1 2 1 1 1 2 3` again? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the first half, then underlined it, then wrote the second half? - No, looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote the same sequence twice? Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3`? - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But

  2. 2:00 – 3:49 02:00-03:49

    the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2 1 1 1 2 1 1 1 2 3` next to it? - Looking at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It looks like he wrote `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Wait, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's look at the crop 03:45 again. - The text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Actually, looking at the crop 03:45, the text is `3 2 1 1 1 2 1 1 1 2 3`. - Then at 03:49, the text is `3 2 1 1 1 2 1 1 1 2 3`. - It seems he wrote the sequence `3 2 1 1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2

1 1 2 1 1 1 2 3` and then underlined it. - Let's assume the final output is the full sequence `3 2 1 1 1 2 1 1 1 2 3 2 1 1 1 2 1 1 1 2 3`. - But the board only shows `3 2 1 1 1 2 1 1 1 2 3`. - Maybe he stopped there? - Or maybe he wrote `3 2 1 1 1 2 1 1 1 2 3` and then `3 2

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