Triangle Counting Tricks

Duration: 1 hr 15 min

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Inside: a video lesson and guided study material.

Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: Concept and Objectives of Teaching / शिक्षण की अवधारणा एवं उद्देश्य, Levels of Teaching / शिक्षण के स्तर, Learner's Characteristics / शिक्षार्थी की विशेषताएँ, Learning: Nature and Theories / अधिगम: प्रकृति एवं सिद्धांत, Cognition, Memory and Metacognition / संज्ञान, स्मृति एवं अधिसंज्ञान, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Effective Teaching and the Teacher / प्रभावी शिक्षण एवं शिक्षक, Instructional Objectives and Lesson Planning / अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन, Methods of Teaching / शिक्षण विधियाँ, Offline and Online Teaching Methods / ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ, Teaching Support System / शिक्षण सहायक प्रणाली, Evaluation Systems / मूल्यांकन प्रणाली
  2. Paper - 1 | Unit - 2 | Research Aptitude: Research: Meaning, Features and Scientific Method / अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि, Research Paradigms and Approaches / अनुसंधान के प्रतिमान एवं उपागम, Types of Research / अनुसंधान के प्रकार, Variables and Measurement / चर एवं मापन, Methods of Research / अनुसंधान की विधियाँ, Steps of Research / अनुसंधान के चरण, Sampling / प्रतिचयन, Tools and Methods of Data Collection / आँकड़ा संग्रह के उपकरण एवं विधियाँ, Data Analysis and Statistics in Research / आँकड़ा विश्लेषण एवं शोध सांख्यिकी, Thesis and Article Writing / शोध-प्रबंध एवं लेख लेखन, Application of ICT in Research / अनुसंधान में ICT का अनुप्रयोग, Research Ethics / शोध नैतिकता
  3. Paper - 1 | Unit - 3 | Comprehension: Literal Comprehension, Vocabulary and Language in Context, Inferential Comprehension, Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions)
  4. Paper - 1 | Unit - 4 | Communication: Communication: Meaning, Nature and Process / संचार: अर्थ, प्रकृति एवं प्रक्रिया, Types and Levels of Communication / संचार के प्रकार एवं स्तर, Verbal and Non-verbal Communication / मौखिक एवं अमौखिक संचार, Semiotics and Meaning / संकेत विज्ञान एवं अर्थ, Models and Theories of Communication / संचार के मॉडल एवं सिद्धांत, Effective Communication / प्रभावी संचार, Barriers to Communication / संचार की बाधाएँ, Classroom Communication / कक्षा संचार, Mass Media: Types and Evolution / जनसंचार माध्यम: प्रकार एवं विकास, Mass Media and Society / जनसंचार एवं समाज, Digital and New Media / डिजिटल एवं नव माध्यम, General Awareness (Legacy Paper 1) / सामान्य जागरूकता (पुराना पेपर 1)
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: Number System and Simplification / संख्या पद्धति एवं सरलीकरण, Algebra and Ages / बीजगणित एवं आयु, Percentage / प्रतिशतता, Ratio, Proportion, Partnership and Mixtures / अनुपात, समानुपात, साझेदारी एवं मिश्रण, Averages / औसत, Profit, Loss and Discount / लाभ, हानि एवं बट्टा, Simple and Compound Interest / साधारण एवं चक्रवृद्धि ब्याज, Time and Work / समय एवं कार्य, Speed, Time and Distance / चाल, समय एवं दूरी, Mensuration and Geometry / क्षेत्रमिति एवं ज्यामिति, Series / श्रेणी, Coding-Decoding / कूटलेखन-कूटवाचन, Relationships, Directions and Ranking / संबंध, दिशा एवं क्रम, Arrangements, Puzzles, Clock and Calendar / व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर, Mathematical Foundations, Analysis
  6. Paper - 1 | Unit - 6 | Logical Reasoning: Structure of Arguments / युक्ति की संरचना, Terms, Definitions and Uses of Language / पद, परिभाषा एवं भाषा के उपयोग, Categorical Propositions / निरुपाधिक प्रतिज्ञप्तियाँ, Categorical Syllogism / निरुपाधिक न्यायवाक्य, Venn Diagrams / वेन आरेख, Deduction, Induction and Analogy / निगमन, आगमन एवं सादृश्यता, Fallacies / तर्कदोष, Indian Logic: Pramanas / भारतीय तर्कशास्त्र: प्रमाण, Anumana and Vyapti / अनुमान एवं व्याप्ति, Hetvabhasa / हेत्वाभास, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle / अरस्तू, Western Schools of Philosophy / पाश्चात्य दर्शन की विचारधाराएँ, Kant / कांट, Sadhana Paada / साधनपाद, Vaiśeṣika / वैशेषिक
  7. Paper - 1 | Unit - 7 | Data Interpretation: Data and Graphical Representation / आँकड़े एवं आलेखीय निरूपण, Data Interpretation: Tables / आँकड़ा निर्वचन: तालिका
  8. 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, Protection of Geographical Indications / भौगोलिक संकेतों का संरक्षण
  9. Paper - 1 | Unit - 9 | People Development and Environment: Ecosystems and Biodiversity / पारितंत्र और जैव विविधता, Human and Environment Interaction / मानव और पर्यावरण की अंतःक्रिया, Development and Environment / विकास और पर्यावरण, Natural Resources – Forests, Water and Soil / प्राकृतिक संसाधन: वन, जल और मृदा, Energy Resources / ऊर्जा संसाधन, Air Pollution / वायु प्रदूषण, Water Pollution and Treatment / जल प्रदूषण और उपचार, Noise Pollution / ध्वनि प्रदूषण, Waste Management / अपशिष्ट प्रबंधन, Environmental Health and Toxicology / पर्यावरणीय स्वास्थ्य और विषविज्ञान, Climate Change and Global Environmental Issues / जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ, Natural Hazards and Disaster Management / प्राकृतिक प्रकोप और आपदा प्रबंधन, Indian Environmental Laws and Policies / भारतीय पर्यावरण कानून और नीतियाँ, International Environmental Agreements / अंतर्राष्ट्रीय पर्यावरण समझौते, Environment / पर्यावरण, Geography / भूगोल, Factory and Industry Systems / कारखाना और उद्योग व्यवस्थाएँ, Earth Materials, Surface Features and Processes, Communicable Diseases and Immunization / संक्रामक रोग और प्रतिरक्षण
  10. Paper - 1 | Unit - 10 | Higher Education System: Education in Ancient India / प्राचीन भारत में शिक्षा, Ancient Centres of Higher Learning / प्राचीन भारत के उच्च शिक्षा केंद्र, Colonial Education Policies and Commissions / औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग, Colonial Colleges, Universities and Reformers / औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक, Education Commissions and Policies after Independence / स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ, National Education Policy 2020 / राष्ट्रीय शिक्षा नीति 2020, Types of Higher Education Institutions / उच्च शिक्षा संस्थानों के प्रकार, Learning Programmes and Academic Credits / अधिगम कार्यक्रम और अकादमिक क्रेडिट, Professional, Technical and Skill-Based Education / व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा, Value Education and Environmental Education / मूल्य शिक्षा और पर्यावरण शिक्षा, Governance, Administration and Financing / शासन, प्रशासन और वित्तपोषण, Regulation, Accreditation and Ranking / उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग, Indian Polity and Governance (Legacy Unit 10) / भारतीय राजव्यवस्था और शासन (पुराना यूनिट 10)
  11. Paper 2 | Unit 1 | Discrete Structures and Optimization: Propositional and Predicate Logic, Set Theory, Relations, Functions, Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Graph Theory, Group Theory, Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Optimization, Markov Chains
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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, Cloud Computing, Mobile Computing
  20. 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
  21. Full Notes: Paper 1 Full Notes
  22. Paper 1 | Full Mock Tests:
  23. Paper 2 | Full Mock Tests:
  24. Paper 1 | Previous Year Papers:
  25. Paper 2 | Previous Year Papers:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video is a comprehensive tutorial on solving triangle counting problems, a common topic in competitive exams. The instructor systematically introduces different types of triangle configurations and teaches a set of mathematical formulas to count them efficiently. The lesson begins with simple cases, such as a triangle divided by lines from the apex, and progresses to more complex patterns, including a large grid of small triangles and a square with diagonals. For each type, the instructor demonstrates a methodical approach, often using summation formulas like n(n+1)/2 to calculate the total number of triangles. The video uses a digital whiteboard to draw and label diagrams, and the instructor provides worked examples to illustrate the application of the formulas. The overall goal is to equip students with a reliable strategy to solve these problems quickly and accurately under exam conditions.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card reading "TRIANGLE COUNTING TRICKS". The instructor, Yash Jain Sir, introduces the topic, explaining that the session will cover various types of triangle counting problems. He presents the first example, a large triangle divided into smaller triangles by lines drawn from the apex to the base, and begins to explain the method for counting them.

  2. 2:00 – 5:00 02:00-05:00

    The instructor demonstrates the first method for counting triangles. He labels the small triangles along the base from 1 to 4. He then explains that the total number of triangles is the sum of the numbers from 1 to 4, which is 1+2+3+4=10. He writes the formula n(n+1)/2, where n is the number of base segments, and shows that 4(4+1)/2 = 10. He then moves to a second example with 6 base segments, calculating 6(6+1)/2 = 21.

  3. 5:00 – 10:00 05:00-10:00

    The instructor introduces a new type of problem, a triangle divided into smaller triangles by horizontal lines. He labels the sections from 1 to 3. He explains that the total number of triangles is the sum of the first 3 natural numbers, which is 1+2+3=6. He then shows a more complex example with 6 horizontal lines, calculating 1+2+3+4+5+6=21. He also demonstrates a method for counting triangles in a figure with a horizontal line, writing 6+6=12.

  4. 10:00 – 15:00 10:00-15:00

    The instructor presents a new problem type: a square with its diagonals drawn, forming 4 triangles. He explains that there are 4 small triangles and 4 larger triangles formed by combining two small ones, totaling 8. He then shows a more complex figure with a square divided into a 3x3 grid, and explains that the total number of triangles is 8x2=16. He then shows a 4x4 grid, calculating 8x2=16 and then 16x2=32 triangles.

  5. 15:00 – 20:00 15:00-20:00

    The instructor introduces a complex problem: a large triangle composed of many small equilateral triangles. The question asks for the total number of triangles. He explains that the figure is a 14-level triangle, with the base divided into 14 segments. He then shows a table with the number of triangles of different sizes, starting with 1 triangle of size 1, 5 of size 2, 13 of size 3, and so on, up to 170 of size 8.

  6. 20:00 – 25:00 20:00-25:00

    The instructor explains the pattern for counting triangles in a large grid. He shows a 3-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, and 6 of size 3. He writes the sum 1+3+6=10. He then shows a 4-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, and 10 of size 4, summing to 20. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  7. 25:00 – 30:00 25:00-30:00

    The instructor continues with the large grid problem. He shows a 4-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, and 10 of size 4, summing to 20. He then shows a 5-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, and 15 of size 5, summing to 35. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  8. 30:00 – 35:00 30:00-35:00

    The instructor shows a 6-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, and 21 of size 6, summing to 56. He then shows a 7-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, and 28 of size 7, summing to 84. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  9. 35:00 – 40:00 35:00-40:00

    The instructor shows a 8-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, and 36 of size 8, summing to 120. He then shows a 9-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, and 45 of size 9, summing to 165. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  10. 40:00 – 45:00 40:00-45:00

    The instructor shows a 10-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, and 55 of size 10, summing to 220. He then shows a 11-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, and 66 of size 11, summing to 286. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  11. 45:00 – 50:00 45:00-50:00

    The instructor shows a 12-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, and 78 of size 12, summing to 364. He then shows a 13-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, and 91 of size 13, summing to 455. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  12. 50:00 – 55:00 50:00-55:00

    The instructor shows a 14-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, and 105 of size 14, summing to 560. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  13. 55:00 – 60:00 55:00-60:00

    The instructor shows a 15-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, and 120 of size 15, summing to 680. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  14. 60:00 – 65:00 60:00-65:00

    The instructor shows a 16-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, and 136 of size 16, summing to 816. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  15. 65:00 – 70:00 65:00-70:00

    The instructor shows a 17-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, 136 of size 16, and 153 of size 17, summing to 969. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  16. 70:00 – 75:00 70:00-75:00

    The instructor shows a 18-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, 136 of size 16, 153 of size 17, and 171 of size 18, summing to 1140. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  17. 75:00 – 75:11 75:00-75:11

    The video concludes with a final screen that says "THANKS FOR WATCHING". The instructor has finished explaining the methods for counting triangles in various configurations and has provided a comprehensive set of formulas and examples for students to use in their exam preparation.

The video provides a structured and methodical approach to solving triangle counting problems. It begins by establishing a foundational formula, n(n+1)/2, for counting triangles in a simple configuration where lines are drawn from the apex to the base. The instructor then systematically builds upon this by introducing more complex patterns, such as horizontal divisions and a large grid of small triangles. For each new type of figure, he demonstrates a clear counting strategy, often using summation to find the total. The core of the lesson is the application of a general formula for counting triangles in a large grid: the number of triangles of size k is the sum of the first (n-k+1) natural numbers, where n is the number of base segments. This allows for a quick calculation of the total number of triangles in any such configuration. The video is designed to equip students with a reliable and efficient strategy for tackling these common exam questions.

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