Critical Thinking & Problem Solving

Overview & Fundamental Concept

Comprehensive AISSEE Class 6 Intelligence guide for Critical Thinking & Problem Solving. Covers Direction Sense (8 compass points, sunrise/sunset shadow laws, Pythagorean distance), 10 Venn Diagram types, Ranking formulas (T = L + R - 1), BODMAS coded operations, solved models, and 10 drills.

• 1. The 8 Compass Directions Framework:
• • 4 Cardinal Directions: North (Up / $0^\circ$), East (Right / $90^\circ$), South (Down / $180^\circ$), West (Left / $270^\circ$).
• • 4 Intercardinal (Ordinal) Directions: North-East (between N and E at $45^\circ$), South-East (between S and E at $135^\circ$), South-West (between S and W at $225^\circ$), North-West (between N and W at $315^\circ$).
• 2. Clockwise & Anticlockwise Angular Turns:
• • A Right Turn is ALWAYS a $90^\circ$ rotation in the Clockwise direction.
• • A Left Turn is ALWAYS a $90^\circ$ rotation in the Anticlockwise direction.
• • Facing North: Turn Right $\to$ East; Turn Left $\to$ West.
• • Facing South: Turn Right $\to$ West; Turn Left $\to$ East (Crucial reversal!).
• • Facing East: Turn Right $\to$ South; Turn Left $\to$ North.
• • Facing West: Turn Right $\to$ North; Turn Left $\to$ South.
• 3. Net Angular Rotation Shortcut: If a person makes multiple angular turns, sum all clockwise (CW) angles and all anticlockwise (ACW) angles separately. Net Rotation $= |\text{CW} - \text{ACW}|$ in the direction of the larger sum!
  • Example: Facing East $\to 45^\circ$ CW, then $90^\circ$ ACW, then $135^\circ$ CW $\implies$ Total CW $= 180^\circ$, Total ACW $= 90^\circ \implies$ Net $= 90^\circ$ CW from East $\implies$ Facing South!
• 4. Total Distance Traveled vs Shortest Displacement:
• • Total Distance: Scalar sum of all path segments walked ($d_{\text{total}} = d_1 + d_2 + d_3 + \dots$).
• • Shortest Distance (Displacement): Straight line distance from start point to end point, calculated via the Pythagorean Theorem: $d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$.
• • Essential Pythagorean Triplets for AISSEE: $3-4-5, 5-12-13, 6-8-10, 8-15-17, 9-12-15, 7-24-25$.

• 1. Fundamental Optical Axiom: A shadow is always cast directly OPPOSITE to the position of the light source (the Sun).
• 2. Morning Condition (Sunrise & Morning Hours):
• • Sun is in the EAST $\implies$ All shadows fall strictly towards the WEST.
• • Person Facing North: West is to their LEFT $\implies$ Shadow falls on their LEFT.
• • Person Facing South: West is to their RIGHT $\implies$ Shadow falls on their RIGHT.
• • Person Facing East: West is behind them $\implies$ Shadow falls BEHIND them.
• • Person Facing West: West is in front of them $\implies$ Shadow falls IN FRONT of them.
• 3. Evening Condition (Sunset & Late Afternoon Hours):
• • Sun is in the WEST $\implies$ All shadows fall strictly towards the EAST.
• • Person Facing North: East is to their RIGHT $\implies$ Shadow falls on their RIGHT.
• • Person Facing South: East is to their LEFT $\implies$ Shadow falls on their LEFT.
• • Person Facing East: East is in front $\implies$ Shadow falls IN FRONT.
• • Person Facing West: East is behind $\implies$ Shadow falls BEHIND.
• 4. Exactly at 12:00 Noon: The Sun is directly overhead at zenith. Rays fall perpendicularly $\implies$ NO lateral shadow is formed horizontally on the ground.

• A Venn Diagram uses closed geometric figures (typically circles) to represent logical classes, categories, and their set-theoretic intersections.
• The 10 Universal Three-Set Structural Types for AISSEE:
• • Type 1: Concentric Inclusions ($A \subset B \subset C$): Every member of A is in B, and every member of B is in C.
  Example: Seconds, Minutes, Hours; Jaipur, Rajasthan, India; Carnivores, Mammals, Animals.
• • Type 2: Mutually Disjoint ($A \cap B = \emptyset, B \cap C = \emptyset, A \cap C = \emptyset$): Three completely separate circles with zero overlap.
  Example: Dogs, Horses, Elephants; Tables, Chairs, Curtains; Copper, Gold, Silver.
• • Type 3: Partial Three-Way Intersections (Borromean Overlap): Some elements are common to pairs, and some belong to all three.
  Example: Authors, Teachers, Men (A man can be both author and teacher; an author can be neither teacher nor man).
• • Type 4: Two Mutually Disjoint Sets inside a Larger Super-Set: Two non-overlapping circles enclosed within a master circle.
  Example: Vegetables (Potatoes, Cabbage); Stationery (Pens, Erasers); States (Bihar, Punjab) inside India.
• • Type 5: One Set Enclosed in Another, with a Third Disjoint Set: One concentric pair and one completely separate circle.
  Example: Mammals, Whales, Ships (Whales are Mammals; Ships are totally separate).
• • Type 6: Two Sets Partially Overlap, and Both are Enclosed in a Third Set: Two overlapping circles inside a large circle.
  Example: Animals (Dogs, Pets) — Some dogs are pets, some pets are dogs, and all are animals.
• • Type 7: One Subset inside Another, with a Third Partially Overlapping Both: Concentric circles cut across by a third circle.
  Example: Females, Mothers, Doctors (All mothers are females; doctors can be mothers, non-mother females, or males).
• • Type 8: Two Disjoint Sets Overlapping with a Third Common Set: Two separate circles that both intersect a third middle circle without touching each other.
  Example: Men, Women, Teachers (Teachers can be men or women, but men and women are disjoint).
• • Type 9: One Class Encloses Another, and Partially Overlaps the Third: A master set contains a subset, and only the master set intersects a third set.
  Example: Engineers, Software Engineers, Human Beings.
• • Type 10: Quantitative Region Reading (Venn Geometries): Extracting specific numbers from intersecting triangles, squares, and circles representing specific professions or traits.

• 1. Finding Total Persons ($T$) from Both Ranks of a Single Individual: If the rank of an individual $X$ is given from both ends (Left/Top and Right/Bottom):
  $$\mathbf{T = L + R - 1}$$ (We subtract $1$ because person $X$ was counted twice: once from the left and once from the right!).
• 2. Finding Individual Rank from the Opposite End:
• • Rank from Right ($R$): $R = T - L + 1$.
• • Rank from Left ($L$): $L = T - R + 1$.
• 3. Position Interchanging Problems (The Symmetrical Shift Rule):
• • When Person A and Person B swap their seats in a row:
  $$\mathbf{T = (\text{New Rank of A}) + (\text{Old Rank of B}) - 1}$$ • Number of Persons Between A and B: $|\text{New Rank of A} - \text{Old Rank of A}| - 1$.
  • New Rank of B: $\text{Old Rank of B} + (\text{New Rank of A} - \text{Old Rank of A})$.
• 4. Overlapping vs Non-Overlapping Rows:
• • Non-Overlapping Case (No Crossing): When $T > L_A + R_B$, persons between them $= T - (L_A + R_B)$.
• • Overlapping Case (Crossing): When $T < L_A + R_B$, persons between them $= (L_A + R_B) - T - 2$ (Subtract $2$ because A and B were mutually counted across each other!).
• 5. Coded Mathematical Operations with BODMAS:
• • Exam questions substitute standard arithmetic symbols: e.g., 'If $+$ means $\times$, $-$ means $\div$, $\times$ means $-$, and $\div$ means $+$'.
• • Strict 3-Step Execution:
  1. Rewrite the entire expression replacing each symbol with its decoded real operator.
  2. DO NOT calculate left to right! Strictly follow BODMAS: Brackets $\to$ Orders $\to$ Division $\to$ Multiplication $\to$ Addition $\to$ Subtraction.
  3. Compute division and multiplication before touching addition or subtraction.