🧠 Logical Reasoning
Logical sequences follow a hidden rule or pattern
▶ Play Door 22Mathematical Puzzles and Logical Sequences
What are Mathematical Puzzles?
Mathematical puzzles are problems that require logical thinking, pattern recognition, and mathematical skills to solve. They are like brain teasers that make you think creatively.
What are Logical Sequences?
Logical sequences are ordered arrangements of numbers, letters, or shapes that follow a specific rule or pattern. Identifying the rule helps you predict the next element in the sequence.
Common Types of Number Sequences:
TypeDescriptionExampleRule
ArithmeticConstant difference3, 7, 11, 15, ...Add 4
GeometricConstant ratio2, 6, 18, 54, ...Multiply by 3
Square numbersPerfect squares1, 4, 9, 16, 25, ...n²
Cube numbersPerfect cubes1, 8, 27, 64, 125, ...n³
FibonacciSum of two previous1, 1, 2, 3, 5, 8, 13, ...Add previous two
Prime numbersPrime numbers2, 3, 5, 7, 11, 13, ...Next prime
Tips for Solving Puzzles and Sequences:
• Look…
Worked example
Problem: Find the next three terms in the sequence: 4, 9, 16, 25, ...
- Step 1: Observe the sequence: 4, 9, 16, 25
- Step 2: Recognize these as squares: 2²=4, 3²=9, 4²=16, 5²=25
- Step 3: Pattern: squares of consecutive integers starting from 2
- Step 4: Next terms: 6²=36, 7²=49, 8²=64
Answer: Step 4: Next terms: 6²=36, 7²=49, 8²=64
Venn Diagrams and Critical Thinking
What are Venn Diagrams?
Venn diagrams are visual representations that show relationships between different sets of items. Circles (or other shapes) overlap to show common elements.
Basic Venn Diagram Components:
ComponentSymbolMeaning
Universal SetRectangleAll possible elements
Set ACircleElements belonging to group A
Set BCircleElements belonging to group B
IntersectionOverlap regionElements in BOTH A and B
UnionAll circles combinedElements in A OR B (or both)
ComplementOutside circlesElements NOT in the set
Venn Diagram Regions (Two Sets):
`
┌─────────────────────────────────────────────┐
│ Universal Set │
│ ┌─────────────┐ ┌─────────────┐ │
│ │ A │ │ B │ │
│ │ ┌─────┐ │ │ │ │
│ │ │A∩B │ │ │ │ │
│ │ └─────┘ │ │ │ │
│ └─────────────┘ └─────────────┘ │
│ │
│ Outside: Elements in neither A nor B │
└─────────────────────────────────────────────┘
`
Set…
Worked example
Problem: In a class of 40 students, 25 like mathematics, 20 like science, and 10 like both. How many students like neither subject?
- Step 1: Draw Venn diagram with two circles (Math, Science)
- Step 2: n(Math) = 25, n(Science) = 20, n(Math ∩ Science) = 10
- Step 3: Only Math = 25 - 10 = 15
- Step 4: Only Science = 20 - 10 = 10
- Step 5: Total liking at least one = 15 + 10 + 10 = 35
Answer: Step 5: Total liking at least one = 15 + 10 + 10 = 35
Coding-Decoding Basics
What is Coding-Decoding?
Coding-decoding is a method of converting a message into a secret code (encoding) and converting it back (decoding). It involves substituting letters, numbers, or symbols according to a specific rule.
Common Types of Coding:
TypeDescriptionExample
Letter ShiftingShift each letter by a fixed numberA→D, B→E, C→F (+3)
Reverse OrderWrite the word backwardsCAT → TAC
Position ValueUse alphabetical positionsA=1, B=2, C=3
Opposite LettersA↔Z, B↔Y, C↔X, etc.A→Z, B→Y
Pattern CodingCombination of operationsAdd, subtract, multiply
Alphabet Positions:
ABCDEFGHIJKLM
12345678910111213
NOPQRSTUVWXYZ
14151617181920212223242526
Opposite Letter Pairs (A=1, Z=26 → 1+26=27):
• A ↔ Z (1 + 26 = 27)
• B ↔ Y (2 + 25 = 27)
• C ↔ X (3 + 24 = 27)
• D ↔ W (4 + 23 = 27)
• ... and so on.
Coding-Decoding Steps:
• Identify the coding rule (shift, reverse, etc.)…
Worked example
Problem: If in a code, MADRAS is written as NZESBT, how is BOMBAY written in that code?
- Step 1: Compare MADRAS and NZESBT letter by letter:
- Step 2: M → N (+1), A → Z (-1 or +25), D → E (+1), R → S (+1), A → B (+1), S → T (+1)
- Step 3: Pattern: Except second letter (A→Z is -1), all others are +1
- Step 4: Actually A→Z is also +25 (cyclic shift)
- Step 5: Apply to BOMBAY:
Answer: Step 5: Apply to BOMBAY: