Logical Reasoning Concepts for Competitive Exams 2026

Logical Reasoning one liner - Non-Verbal Reasoning (Visual and Spatial Ability)

1. Mirror and Water Images

• What fundamental inversion occurs in a plane mirror image? - Lateral Inversion (Left becomes Right and Right becomes Left)
• What fundamental inversion occurs in a water reflection image? - Vertical Inversion (Top becomes Bottom and Bottom becomes Top)
• How many capital letters in the English alphabet have identical mirror images? - 11 Letters (A, H, I, M, O, T, U, V, W, X, Y)
• How many capital letters in the English alphabet have identical water images? - 8 Letters (C, D, E, H, I, K, O, X)
• How many capital letters have identical mirror AND water images? - 4 Letters (H, I, O, X)
• Which single digit from 0 to 9 has an identical mirror image in standard font? - 0 and 8
• What is the mirror image of the capital letter 'B'? - ᗺ (Curves face Left)
• What is the water image of the capital letter 'E'? - E (Remains unchanged)
• What is the mirror image of the word 'MOM'? - MOM
• What is the water image of the word 'KICK'? - KICK
• If a 12-hour clock displays 4:25, what time does its vertical mirror image show? - 7:35 ($11:60 - 4:25$)
• If a clock displays 9:50, what time is seen in its mirror reflection? - 2:10 ($11:60 - 9:50$)
• What is the formula to calculate the mirror image time for any given time ($H:M$)? - Subtract from 11:60 (or 23:60 for 24-hr format)
• What is the standard reference time used to find the water image of a clock? - Subtract from 18:30 (or 17:90 if minutes $> 30$)
• If a clock shows 8:20, what is its apparent water image time? - 10:10 ($18:30 - 8:20$)
• If a clock shows 2:45, what is its apparent water image time? - 3:45 ($17:90 - 2:45 = 15:45 \to 3:45$)
• When a mirror is placed horizontally below an object ($MN$ line at bottom), what does it form? - Water Image (Vertical Inversion)
• When a mirror is placed vertically to the right of an object ($MN$ line at right), what does it form? - Standard Mirror Image (Lateral Inversion)
• What happens to an arrow pointing towards the North-East in a standard vertical mirror image? - Points towards North-West
• What happens to an arrow pointing towards the North-East in a horizontal water reflection? - Points towards South-East
• What is the mirror image of the number '7'? - Lateral inversion of 7 (Horizontal top bar stays on top, stem slopes to left)
• What is the water image of the number '3'? - 3 (Remains visually unchanged in symmetric fonts)
• What is the water image of the letter 'M'? - W
• What is the water image of the letter 'N'? - Vertical inversion of N (Diagonal slopes upward from bottom-left to top-right)
• What is the mirror image of the letter 'L'? - Horizontal base extends to the left
• What is the mirror image of the letter 'J'? - Hook curves to the right
• What is the water image of the letter 'A'? - Apex points downward ($\forall$)
• What is the water image of the letter 'V'? - Apex points upward ($\Lambda$)
• Does the thickness or size of an object change in mirror or water reflection? - No (Size and proportions remain identical)
• In a mirror image, what happens to the distance of an object point from the mirror line? - Image point is at the same perpendicular distance behind the mirror
• What is the mirror image of the word 'SHIMLA'? - ALMIHS (with each letter individually laterally inverted)
• What is the reading order of letters in the mirror image of a word? - Reversed (from right to left)
• In a water image of a word, does the left-to-right order of the letters change? - No (Reading order stays left-to-right, only vertical orientation flips)
• What is the water image of the word 'CODE'? - CODE
• What is the water image of the word 'BOOK'? - BOOK
• What is the water image of the word 'HIKE'? - HIKE
• What is the mirror image of the word 'RADAR'? - RADAR (with letters R laterally inverted)
• What is the water image of the digit '6'? - Inverted to look like an upside-down loop (resembling '9' with different curvature)
• In a geometric figure with a shaded upper-left quadrant, where does the shade appear in a mirror image? - Upper-Right Quadrant
• In a geometric figure with a shaded upper-left quadrant, where does the shade appear in a water image? - Lower-Left Quadrant
• Where does the shade appear if the mirror is placed on the top of the object? - Lower-Left Quadrant (Equivalent to water image)
• What is the mirror image of a right-angled triangle with its hypotenuse on the right? - Right-angled triangle with hypotenuse on the left
• What is the water image of an upward-pointing equilateral triangle? - Downward-pointing equilateral triangle
• How does a circle with a dot at the center appear in both mirror and water images? - Identical (Perfect rotational symmetry)
• What is the mirror image of a clockwise rotating circular arrow? - Anti-clockwise rotating arrow
• What is the water image of a clockwise rotating circular arrow? - Anti-clockwise rotating arrow
• If a clock displays exactly 6:00, what time is seen in its vertical mirror reflection? - 6:00 ($11:60 - 6:00 = 5:60 = 6:00$)
• If a clock displays exactly 12:00, what time is shown in its vertical mirror reflection? - 12:00
• If a clock displays 12:30, what time does its mirror image display? - 11:30 ($23:60 - 12:30$)
• What is the mirror image of the letter 'P'? - Loop extends to the left (৭)
  
  

2. Cubes and Dice

• How many total faces (surfaces), edges, and vertices does a standard cube have? - 6 Faces, 12 Edges, and 8 Vertices
• What is the defining mathematical property of a Standard (Canonical) Dice? - The sum of numbers on any two opposite faces is always equal to 7 ($1+6=7, 2+5=7, 3+4=7$)
• On a standard dice, which number is always opposite to 1? - 6
• On a standard dice, which number is always opposite to 2? - 5
• On a standard dice, which number is always opposite to 3? - 4
• What is an Ordinary (General) Dice? - A dice where the sum of adjacent visible faces can be 7 and opposite faces do not necessarily sum to 7
• How many faces of a single cube can be seen simultaneously in an isometric 2D drawing? - At most 3 Adjacent Faces
• Can two opposite faces of a dice ever be visible together in a single view? - Never (Opposite faces are never adjacent)
• What is Rule 1 of Dice (Common Face Rule)? - If one face is common in two positions, write visible faces clockwise to find opposite pairs
• What is Rule 2 of Dice (Two Common Faces Rule)? - If two faces are common in two positions, the remaining unshared faces are opposite to each other
• In two views of a dice, faces show (1, 2, 3) and (1, 4, 5). Which face is opposite to 2? - 4 (Moving clockwise from common face 1: $1 \to 2 \to 3$ and $1 \to 4 \to 5$)
• In two views of a dice, faces show (1, 2, 3) and (1, 4, 5). Which face is opposite to 3? - 5
• In two views of a dice, faces show (1, 2, 3) and (1, 4, 5). Which face is opposite to 1? - 6 (The missing number)
• In two views of a dice, faces show (2, 3, 5) and (2, 3, 6). Which face is opposite to 5? - 6 (Two common faces rule)
• In an open unfolded dice cross-sheet, how are opposite faces determined? - Every alternate face along a straight strip is opposite to each other
• In an open dice sheet with a vertical strip of numbers 1-2-3-4, which face is opposite to 1? - 3
• In an open dice sheet with a vertical strip of numbers 1-2-3-4, which face is opposite to 2? - 4
• Can two faces that share an edge in an open dice sheet be opposite in the folded cube? - No (They become adjacent faces)
• If a large cube of side $N\text{ cm}$ is cut into smaller cubes of side $1\text{ cm}$, what is the value of $n$? - $n = \text{Side of Large Cube} / \text{Side of Smaller Cube}$
• What is the formula for the total number of smaller unit cubes formed from a large cube? - $n^3$
• If a painted cube of side $4\text{ cm}$ is cut into $1\text{ cm}$ cubes, what is the total number of small cubes? - 64 ($4^3$)
• Where are the 3-face painted smaller cubes always located on a painted large cube? - At the 8 Corners (Vertices)
• What is always the total number of 3-face painted small cubes in any size painted cube? - 8 (Constant for all cubes)
• Where are the 2-face painted smaller cubes located? - Along the 12 Edges (excluding corners)
• What is the mathematical formula for the number of 2-face painted small cubes? - $12(n - 2)$
• For a $4\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=4$), how many cubes have exactly 2 faces painted? - 24 cubes ($12(4-2) = 24$)
• Where are the 1-face painted smaller cubes located? - At the Center of the 6 Faces
• What is the mathematical formula for the number of 1-face painted small cubes? - $6(n - 2)^2$
• For a $4\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=4$), how many cubes have exactly 1 face painted? - 24 cubes ($6(4-2)^2 = 24$)
• Where are the 0-face painted (unpainted / colorless) inner core cubes located? - Inside the Interior Volume of the cube
• What is the mathematical formula for the number of unpainted (colorless) small cubes? - $(n - 2)^3$
• For a $4\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=4$), how many cubes have zero faces painted? - 8 cubes ($(4-2)^3 = 8$)
• For a $3\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=3$), how many cubes have 0 faces painted? - 1 cube ($(3-2)^3 = 1$)
• For a $5\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=5$), what is the total number of small cubes? - 125 ($5^3$)
• For a $5\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=5$), how many cubes have exactly 1 face painted? - 54 cubes ($6(5-2)^2 = 6 \times 9 = 54$)
• For a $5\text{ cm}$ painted cube cut into $1\text{ cm}$ cubes ($n=5$), how many cubes have exactly 2 faces painted? - 36 cubes ($12(5-2) = 36$)
• How many minimum cuts are required parallel to the faces to divide a cube into 64 identical unit cubes? - 9 cuts (3 cuts along each of the 3 axes: $3 \times (4-1) = 9$)
• What is the formula for minimum cuts required along 3 axes to get $n^3$ small cubes? - $3(n - 1)$
• Can any smaller unit cube have 4 or more painted faces when a large painted cube is cut? - No (Maximum painted faces for any cube is 3)
• What is the maximum number of cuts required to cut a cube into 27 smaller cubes? - 6 cuts ($3 \times (3-1) = 6$)
• If opposite faces of a cube are painted Red, Blue, and Green respectively, how many colors appear on corner cubes? - 3 Colors (One Red, One Blue, One Green)
• If only two opposite faces of a cube are painted, how many small unit cubes have 3 painted faces? - 0 (Corners can have at most 1 painted face)
• In an open dice sheet with 'T' shape, which faces form opposites? - The two extended lateral wings form opposite faces
• What is the total surface area of a cube of side $a$? - $6a^2$
• What is the volume of a cube of side $a$? - $a^3$
• If the sum of adjacent numbers on any visible pair of a dice is 7, can it be a standard dice? - No (It is an ordinary dice)
• In a single dice throw, what is the probability of rolling an even prime number? - $1/6$ (Only the number 2)
• If a cube is painted on all sides and cut into 216 unit cubes ($n=6$), how many cubes have 2 faces painted? - 48 cubes ($12(6-2) = 48$)
• If a cube is cut into 216 unit cubes ($n=6$), how many cubes have 0 faces painted? - 64 cubes ($(6-2)^3 = 64$)
• If 3 positions of a dice show common elements, how many distinct opposite pairs exist? - 3 pairs
  
  

3. Paper Folding and Cutting

• What geometric principle governs the unfolding of a folded and punched sheet of paper? - Reflection Symmetry (Mirror image along the fold line)
• When a square paper folded from Left to Right is unfolded, what reflection is formed? - Lateral Mirror Reflection across the vertical fold line
• When a paper folded from Bottom to Top is unfolded, what reflection is formed? - Vertical Reflection (Water image across the horizontal fold line)
• If a circular hole is punched through a square paper folded twice into 4 layers, how many holes appear upon unfolding? - 4 Holes ($2^2$)
• What is the general rule for the total number of cutouts appearing on an unfolded sheet? - $\text{Number of cuts} \times 2^k$ (where $k$ is the number of folds)
• If a paper is folded 3 times in half and 1 triangle is cut out, how many triangles appear on the fully unfolded paper? - 8 Triangles ($1 \times 2^3$)
• In paper cutting problems, what is the reverse tracing method? - Unfolding the paper step-by-step in the exact reverse chronological order of folding
• When a paper folded along a diagonal line ($45^\circ$) is punched, how does the punch reflect? - Diagonally across the $45^\circ$ line of symmetry
• If a semicircular cut is made along the closed folded edge of a paper folded once, what shape appears when unfolded? - Full Circle
• If a triangular cut is made on the closed folded crease of a single-fold paper, what shape appears upon unfolding? - Rhombus (Diamond shape)
• What happens to a cutout made on the open (loose) edges of a folded paper upon unfolding? - It appears as separate, isolated notches on outer borders
• If a quarter-circle paper is punched with a square hole near the vertex, where do the 4 squares appear in the open sheet? - Around the central center point forming a diamond/cluster
• Does the shape of the paper change when unfolded completely? - No (The original boundary shape is restored)
• If a square sheet is folded into a triangle (1st fold) and then into a smaller triangle (2nd fold), how many paper layers are created? - 4 Layers
• If a circular paper is folded in half (semicircle) and then in half again (quadrant), how many layers are formed? - 4 Layers
• If a triangular cutout with apex pointing towards the crease is unfolded across the crease, where does the reflected apex point? - Away from the crease (Opposite direction)
• When unfolding across a vertical axis, do horizontal lines change orientation? - No (Horizontal stays horizontal, only left/right positions invert)
• When unfolding across a horizontal axis, do vertical lines change orientation? - Yes (Vertical direction flips upside down)
• If a sheet is folded into thirds (3 equal layers) and punched once, how many holes appear? - 3 Holes
• If a cut is made on a corner of a paper folded into a quadrant, how many corners of the unfolded sheet show that cut? - All 4 Corners
• If a hole is punched through an unfolded section of the sheet that was not in the fold path, does it multiply? - No (It remains a single hole)
• What is indicated by dotted lines in paper folding question figures? - The fold line / hidden folded portion
• What is indicated by arrows on dotted lines in question figures? - The direction in which the paper was folded
• When a transparent sheet with a pattern is folded along a center line, what happens to the two patterns? - They overlap directly onto one half via reflection
• If a transparent square has an arrow on the left pointing right and is folded left-over-right, how does it appear? - Arrow pointing right on the right half
• If a transparent square has a horizontal line on the top half and is folded top-over-bottom, where does the line appear? - Horizontal line on the bottom half
• If a circle on the left half and a triangle on the right half of a transparent sheet are folded together, what is the composite? - A circle overlapping/enclosing the triangle
• If two symmetrical identical patterns on opposite halves of a transparent sheet are folded along the line of symmetry, how many patterns appear? - Only 1 composite overlapping pattern
• What is the symmetry order of a square paper folded along both its horizontal and vertical midlines? - Four-fold symmetry ($D_4$)
• What is the symmetry order of a square paper folded along both diagonals? - Diagonal four-fold reflection symmetry
• If a strip of paper is folded in an accordion (Z-fold) style with 5 layers and a heart is cut out, how many hearts appear? - 5 Hearts
• In an accordion fold, do adjacent cutouts alternate in mirror orientation? - Yes (Alternate left and right reflections)
• If a corner of a paper is folded into a small triangle and the tip is snipped off, what shape is missing in the opened corner? - Diamond / Octagonal notch
• When evaluating options in paper cutting questions, what is the fastest elimination check? - Check if the original question cut is preserved in the corresponding quadrant of the option
• If a circular sheet is folded into 8 sectors ($45^\circ$ each) and a dot is punched, how many dots form a circle on unfolding? - 8 Dots
• If an elliptical hole is cut with its major axis vertical, how does it appear across a vertical fold? - Elliptical hole with major axis still vertical
• If an elliptical hole is cut with its major axis horizontal, how does it appear across a horizontal fold? - Elliptical hole with major axis still horizontal
• If an asymmetric letter 'F' is cut into the top layer of a folded paper, what does its horizontal reflection look like? - Upside-down F (Water image of F)
• If an asymmetric letter 'F' is cut into the top layer of a folded paper, what does its vertical reflection look like? - Laterally inverted F (Mirror image of F)
• Can paper folding and cutting create an odd number of identical symmetric cutouts from full binary folds? - No (Binary folds always produce powers of 2: 2, 4, 8, 16...)
• If a square sheet is folded 4 times in half, how many total layers are created? - 16 Layers ($2^4$)
• If a circle is punched at the exact central junction of all fold creases, how many holes appear when unfolded? - 1 Single Central Hole
• Why does a cut on the central intersecting vertex produce only 1 hole instead of 4? - Because the vertex represents the singular continuous origin point of all quadrants
• If a rectangular sheet is folded horizontally once and two distinct holes are punched, how many holes appear on unfolding? - 4 Holes ($2 \times 2$)
• If a star-shaped punch is made on a 4-layer folded paper, how many stars appear upon full unfolding? - 4 Stars
• Does the distance from the punched hole to the fold line change when unfolded? - No (Distance from the fold line remains equal on both sides)
• If a paper is folded diagonally from bottom-left to top-right, which axis serves as the reflection line? - $y = x$ line
• What is the critical rule for identifying invalid answer options in paper folding? - Any option lacking axis symmetry with respect to the fold lines is immediately invalid
• If a paper has 3 concentric circle punches in a quadrant fold, how many total circles appear on the full sheet? - 12 concentric circle arcs forming 3 complete concentric rings
• What visual skill is primarily evaluated by Paper Folding and Cutting tests? - Mental rotation and 2D spatial transformation ability
  
  

4. Embedded Figures, Figure Completion and Matrix

• What is an 'Embedded Figure'? - A specific target geometric shape that is hidden/contained intact within a larger complex design
• In embedded figure questions, is rotation of the target figure allowed unless specified otherwise? - Usually No (The figure must appear in the exact same orientation unless rotation is explicitly allowed)
• What is the primary method to find an embedded figure in complex options? - Trace continuous straight lines and angle intersections matching the target shape
• If a target shape contains a right-angled triangle with a circle at the top vertex, what feature must the host figure contain? - A right-angled triangle with a coincident circle at that specific vertex
• What is 'Figure Completion'? - Identifying the missing quadrant or segment (usually $1/4\text{th}$) to complete a symmetric geometrical design
• In standard $2 \times 2$ figure completion matrices, how is the missing 4th quadrant related to the 1st quadrant? - Diagonally symmetric or opposite mirror-image
• How is the missing bottom-right quadrant related to the bottom-left quadrant in horizontally symmetric patterns? - Horizontal Mirror Image (Lateral reflection)
• How is the missing bottom-right quadrant related to the top-right quadrant in vertically symmetric patterns? - Vertical Mirror Image (Water reflection)
• In a $3 \times 3$ Non-Verbal Matrix, how is the logical pattern rule evaluated? - Row-wise (left to right) and/or Column-wise (top to bottom)
• What are the three most common transformation rules in $3 \times 3$ figure matrices? - Rotation (e.g., $+45^\circ, +90^\circ$), Element Addition/Subtraction, and Superimposition (XOR/Union)
• In a matrix row where Image 1 has 1 line, Image 2 has 2 lines, what must Image 3 have? - 3 Lines (Arithmetic progression of elements)
• What is the 'Union / Superimposition' rule in figure matrices? - Image 3 is formed by overlaying all lines of Image 1 and Image 2 together
• What is the 'Common Elements Only (Intersection)' rule in figure matrices? - Image 3 retains only the lines/shapes that appear in both Image 1 and Image 2
• What is the 'Cancellation / XOR' rule in figure matrices? - Lines common to Image 1 and Image 2 disappear, while unique lines are retained in Image 3
• If an arrow rotates $45^\circ$ clockwise in each step across a row (North $\to$ North-East $\to$ ?), what is the 3rd step? - East
• If an inner dot moves from top-left $\to$ top-right $\to$ bottom-right, where does it go next in a clockwise path? - Bottom-Left
• What is 'Figure Decomposition'? - Assembling individual 2D polygon pieces to form a target composite figure without overlap
• If 4 identical right-angled isosceles triangles are joined along their hypotenuses at a central point, what shape is formed? - Square
• If 2 identical equilateral triangles are joined along a common base, what polygon is formed? - Rhombus (Diamond shape)
• In a matrix, if black/white shading alternates (Black $\to$ White $\to$ Black), what is the next state? - White
• What is 'Visual Analogy'? - Finding a figure that shares the exact same transformation relationship with Figure C as Figure B shares with Figure A ($A : B :: C : D$)
• If $A$ is an outer circle with an inner triangle, and $B$ is an outer triangle with an inner circle, what rule is applied? - Inversion of Outer and Inner shapes
• If a 3-sided polygon (triangle) becomes a 4-sided polygon (square), what does a 5-sided polygon (pentagon) become? - 6-sided polygon (Hexagon / $+1$ side rule)
• If a shape scales down and multiplies into 3 smaller identical copies, what must happen to the analog shape? - It scales down and forms 3 small copies
• What is 'Visual Classification' (Odd One Out in figures)? - Identifying the single figure that does not follow the common geometric rule shared by the others
• If 4 options contain shapes with odd numbers of sides (3, 5, 7) and 1 option has 6 sides, which is the odd one out? - The 6-sided shape (Even number of sides)
• If 4 figures are rotated versions of the same asymmetric shape, how do you find the odd one out? - The odd one is a mirror-flipped (laterally inverted) version that cannot be matched by pure 2D rotation
• What is the chirality test in Non-Verbal reasoning? - Checking whether an asymmetric shape is "right-handed" or "left-handed" (mirror image vs rotation)
• If all shapes in a set have an axis of symmetry except one, which is the odd one out? - The asymmetric shape
• If 4 figures have lines intersecting at $90^\circ$ (perpendicular) and 1 has lines intersecting at $60^\circ$, which is odd? - The $60^\circ$ intersecting figure
• In a $3 \times 3$ grid where each row contains one Circle, one Triangle, and one Square, what rule is operating? - Latin Square / Exhaustive Category Rule
• If Row 3 has a Triangle and a Square, what missing shape must fill the 3rd slot under the Latin Square rule? - Circle
• What does 'Point Inversion' (Rotation by $180^\circ$) do to an asymmetric figure? - Flips it both horizontally and vertically simultaneously
• In figure completion, what should you do if the outer boundary is a circle divided by concentric arcs? - Continue the circular concentric arcs through the missing quadrant
• If diagonals cross the center in quadrants 1, 2, and 3, what must appear in quadrant 4? - A matching diagonal segment meeting at the center
• What indicates that an element addition pattern is progressive across columns? - Elements increase vertically down each column by a constant step ($+1, +2\dots$)
• If a figure has a plus sign ($+$) moving $90^\circ$ anti-clockwise in each frame, where does it move from West? - South
• What happens to a line segment in a matrix governed by 'reflection across diagonal'? - Its horizontal and vertical coordinates interchange
• If 3 pieces consist of a semicircle and two quarter circles of the same radius, what complete shape do they form? - One Full Circle
• If 6 identical equilateral triangles are assembled around a single point, what regular polygon is formed? - Regular Hexagon
• In visual classification, if all figures are divided into equal area fractions except one, which is odd? - The figure with unequal area divisions
• If 4 figures have an arrow and a dot pointing in the same direction, and 1 has them pointing in opposite directions, which is odd? - The opposite-pointing figure
• What is a 'Closed Figure vs Open Figure' classification rule? - Separating shapes whose perimeters are fully enclosed from those with open endpoints
• If a pattern has 3 concentric squares with alternating shading (Dark, Light, Dark), what is the outermost shade? - Dark
• In figure completion, what is the fastest visual checking technique? - Match the orientation of corner vertices and diagonal lines passing into the central quadrant
• How do you identify a hidden letter 'Z' embedded in a complex mesh? - Look for two horizontal parallel lines connected by a diagonal transversal
• How do you identify a hidden Swastika ($\text{卐}$) shape in a complex design? - Search for a central cross with orthogonal perpendicular arms extending clockwise
• If an embedded target figure has a specific curved arc, can an option containing only straight segments be the answer? - No (An arc cannot be formed by straight segments)
• In a $3 \times 3$ matrix, if number of shaded sectors is 2, 4, 6 in Row 1 and 1, 3, 5 in Row 2, what is the pattern in Row 3 starting with 3, 5? - 7 shaded sectors ($+2$ progression)
• What core cognitive skill is measured by embedded figures and matrix tests? - Disembedding ability and field-independent visual perception

5. Counting of Figures

• What is the shortcut formula to count triangles in a large triangle divided into $n$ vertical strips from one vertex? - $\frac{n(n+1)}{2}$
• A triangle has its base divided into 4 smaller segments by lines from the opposite vertex ($n=4$). How many total triangles exist? - 10 Triangles ($\frac{4 \times 5}{2} = 10$)
• A triangle has its base divided into 5 vertical segments ($n=5$). What is the total number of triangles? - 15 Triangles ($\frac{5 \times 6}{2} = 15$)
• A triangle has its base divided into 6 vertical segments ($n=6$). What is the total number of triangles? - 21 Triangles ($\frac{6 \times 7}{2} = 21$)
• If a triangle with $n=4$ vertical base segments is divided by 2 horizontal lines into 3 tiers, what is the total number of triangles? - 30 Triangles ($10 \times 3 = 30$)
• What is the general formula for triangles in a base-divided triangle with $H$ horizontal tiers? - $\frac{n(n+1)}{2} \times H$
• What is the shortcut formula to count triangles in a square/rectangle with diagonals and medians intersecting at center with $N$ basic triangular compartments? - $2N$
• A square has both diagonals drawn, creating 4 basic small triangles ($N=4$). How many total triangles exist? - 8 Triangles ($2 \times 4 = 8$)
• A square has both diagonals and one horizontal line through center, creating 6 basic triangles ($N=6$). Total triangles? - 12 Triangles ($2 \times 6 = 12$)
• A square has both diagonals, one horizontal, and one vertical line through center ($N=8$). Total triangles? - 16 Triangles ($2 \times 8 = 16$)
• If two squares with diagonals (8 triangles each) are joined side-by-side, how many additional large triangles are formed at the junction? - 2 Extra Triangles (Total = $8 + 8 + 2 = 18$)
• If 3 squares with diagonals (8 each) are joined in a linear row, what is the total number of triangles? - 28 Triangles ($8 \times 3 + 2 \times 2 = 28$)
• If 4 squares with diagonals (8 each) are joined in a straight line, what is the total number of triangles? - 38 Triangles ($8 \times 4 + 2 \times 3 = 38$)
• If 4 squares with diagonals (8 each) are joined in a $2 \times 2$ square block, what is the total number of triangles? - 44 Triangles ($32 + 8 \text{ junction} + 4 \text{ large composite} = 44$)
• In an inverted triangular inscribed pyramid with $n$ base layers, what is the formula for total triangles when $n$ is even? - $T(n) = \frac{n(n+2)(2n+1)}{8}$
• In a triangular grid with $n=3$ base triangles, what is the total number of triangles? - 13 Triangles ($\lfloor \frac{3 \times 5 \times 7}{8} \rfloor = \lfloor 13.125 \rfloor = 13$)
• In a triangular grid with $n=4$ base triangles, what is the total number of triangles? - 27 Triangles ($\frac{4 \times 6 \times 9}{8} = 27$)
• In a triangular grid with $n=5$ base triangles, what is the total number of triangles? - 48 Triangles ($\lfloor \frac{5 \times 7 \times 11}{8} \rfloor = \lfloor 48.125 \rfloor = 48$)
• What is the total number of triangles in a standard 5-pointed Star ($\bigstar$)? - 10 Triangles (5 small outer points $+ 5$ large composite triangles at vertices)
• What is the total number of triangles in a 6-pointed Star formed by two intersecting equilateral triangles ($\✡$)? - 8 Triangles (6 small outer triangles $+ 2$ large base triangles)
• In a triangle where lines from two vertices divide opposite sides into $n$ equal segments, what is the total number of triangles? - $n^3$
• If lines from two vertices to opposite sides divide the triangle into $n=2$ segments, how many triangles exist? - 8 Triangles ($2^3 = 8$)
• If lines from two vertices divide the opposite sides into $n=3$ segments, how many triangles exist? - 27 Triangles ($3^3 = 27$)
• What is the shortcut formula to count total Squares in an $m \times n$ grid? - $\sum (m \times n) + (m-1)(n-1) + (m-2)(n-2) \dots \text{ until one dimension reaches } 1$
• How many total squares are in a symmetric $3 \times 3$ grid ($n=3$)? - 14 Squares ($3^2 + 2^2 + 1^2 = 9 + 4 + 1 = 14$)
• How many total squares are in a symmetric $4 \times 4$ chessboard grid ($n=4$)? - 30 Squares ($4^2 + 3^2 + 2^2 + 1^2 = 16 + 9 + 4 + 1 = 30$)
• How many total squares are in an $8 \times 8$ standard international chessboard? - 204 Squares ($\frac{8 \times 9 \times 17}{6} = 204$)
• What is the formula for the sum of squares of first $n$ natural numbers? - $\frac{n(n+1)(2n+1)}{6}$
• How many total squares are in an asymmetric $4 \times 3$ rectangular grid? - 20 Squares ($4 \times 3 + 3 \times 2 + 2 \times 1 = 12 + 6 + 2 = 20$)
• How many total squares are in an asymmetric $5 \times 3$ rectangular grid? - 26 Squares ($5 \times 3 + 4 \times 2 + 3 \times 1 = 15 + 8 + 3 = 26$)
• How many total squares are in an asymmetric $6 \times 4$ grid? - 50 Squares ($6 \times 4 + 5 \times 3 + 4 \times 2 + 3 \times 1 = 24 + 15 + 8 + 3 = 50$)
• If a plus sign ($+$) is inscribed inside a square, how many total squares are formed? - 5 Squares (4 small unit squares $+ 1$ outer bounding square)
• If a square has a plus sign ($+$) and an inscribed diamond square inside it, how many squares exist? - 6 Squares ($5 + 1 = 6$)
• If 3 concentric squares each have a plus sign ($+$) inscribed inside them, how many total squares exist? - 15 Squares ($5 \times 3 = 15$)
• What is the shortcut formula to count total Rectangles (including squares) in an $m \times n$ grid? - $\frac{m(m+1)}{2} \times \frac{n(n+1)}{2}$
• How many total rectangles exist in a symmetric $2 \times 2$ grid? - 9 Rectangles ($\frac{2 \times 3}{2} \times \frac{2 \times 3}{2} = 3 \times 3 = 9$)
• How many total rectangles exist in a symmetric $3 \times 3$ grid? - 36 Rectangles ($\frac{3 \times 4}{2} \times \frac{3 \times 4}{2} = 6 \times 6 = 36$)
• How many total rectangles exist in an $8 \times 8$ standard chessboard? - 1,296 Rectangles ($36 \times 36 = 1296$)
• How many rectangles that are NOT squares exist in a $3 \times 3$ grid? - 22 Rectangles ($\text{Total Rectangles } (36) - \text{Total Squares } (14) = 22$)
• How many total rectangles exist in an asymmetric $4 \times 3$ grid? - 60 Rectangles ($\frac{4 \times 5}{2} \times \frac{3 \times 4}{2} = 10 \times 6 = 60$)
• How many total rectangles exist in a single straight row of $n$ adjacent unit squares? - $\frac{n(n+1)}{2}$
• In a single horizontal strip of 5 unit squares ($n=5$), how many total rectangles are formed? - 15 Rectangles ($\frac{5 \times 6}{2} = 15$)
• What are the three categories of straight lines counted in complex geometric figures? - Horizontal lines ($H$), Vertical lines ($V$), and Slant/Diagonal lines ($S$)
• What is the minimum number of straight lines required to make a closed polygon? - 3 Straight Lines (forming a Triangle)
• How many straight lines are in a regular Hexagon with all 3 main diagonals connecting opposite vertices? - 9 Straight Lines (6 perimeter lines $+ 3$ internal diagonals)
• How many straight lines make up a standard 5-pointed Star ($\bigstar$)? - 5 Continuous Straight Lines
• How many straight lines make up a standard 6-pointed Star ($\✡$)? - 6 Straight Lines (3 lines per triangle $\times 2$)
• What is the shortcut to count total Parallelograms formed by a set of $m$ parallel lines intersecting $n$ parallel lines? - $\binom{m}{2} \times \binom{n}{2} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2}$
• How many parallelograms are formed when 4 horizontal parallel lines intersect 3 vertical parallel lines? - 18 Parallelograms ($\frac{4 \times 3}{2} \times \frac{3 \times 2}{2} = 6 \times 3 = 18$)
• How many total circles are in a cluster of 1 central circle surrounded by 6 tangent circles enclosed in a large outer circle? - 8 Circles ($1 + 6 + 1 = 8$)
  
  

6. Visual Pattern Series, Analogy and Classification

• What is a 'Non-Verbal Series'? - A sequence of figures where successive images change systematically according to a fixed logical rule
• In a clockwise rotational series, what is the next heading angle after North ($0^\circ$), North-East ($45^\circ$), and East ($90^\circ$)? - South-East ($135^\circ$)
• If an arrow alternates direction in successive frames ($\uparrow, \downarrow, \uparrow, \downarrow$), what is the 5th frame? - $\uparrow$ (Upward)
• What is a 'Continuous Step Progression' rule? - An element shifts by 1 step in step 1, 2 steps in step 2, 3 steps in step 3 ($+1, +2, +3\dots$)
• If an inner dot moves around the 4 corners of a square in a clockwise direction, how many steps return it to start? - 4 Steps
• If an element moves by $+90^\circ$ in step 1, $+180^\circ$ in step 2, and $+270^\circ$ in step 3, what is the rotation in step 4? - $+360^\circ$ (Full rotation / Identical position)
• What is an 'Element Increment Series'? - Figures add one new line segment or symbol in each successive frame ($1 \to 2 \to 3 \to 4\dots$)
• If Frame 1 is a Line (1), Frame 2 is an Angle (2), Frame 3 is a Triangle (3), what is Frame 4? - Quadrilateral / 4-sided Polygon (4 lines)
• What is a 'Shading Alternation Series'? - Sectors alternate between fully shaded (black), unshaded (white), and cross-hatched in a cyclic pattern
• If a shaded sector of a circle divided into 8 parts shifts by 2 sectors clockwise in each frame, what is the angular shift? - $90^\circ$ Clockwise ($2 \times 45^\circ$)
• If an outer shape sheds 1 side while an inner shape adds 1 side in each frame, what happens to a Hexagon with an inner Triangle? - Becomes a Pentagon with an inner Square
• What is an 'Element Inversion Series'? - Elements flip upside down or laterally on alternate steps
• In an analogy $A : B :: C : D$, if Figure $B$ is the vertical water image of Figure $A$, what must Figure $D$ be? - The vertical water image of Figure $C$
• In an analogy $A : B :: C : D$, if Figure $B$ is Figure $A$ rotated by $90^\circ$ clockwise, how is $D$ derived? - Rotate Figure $C$ by $90^\circ$ clockwise
• If Figure $A$ is an open arc and Figure $B$ is a fully closed circle, what does an open 3-line shape become in analogy? - A closed Triangle
• If Figure $A$ is a large circle containing 3 small squares, and $B$ is a large square containing 3 small circles, what is the rule? - Interchange of geometry between outer container and inner contents
• If a single solid arrow becomes two parallel dashed arrows in Figure $B$, what does a single solid line become in $D$? - Two parallel dashed lines
• What is the 'Number of Intersections' classification rule? - Identifying the odd figure whose line segments form a different number of vertex junctions
• What is the 'Curved vs Straight Edges' classification rule? - Separating figures formed purely of straight lines from figures containing curved arcs
• If 4 options contain shapes with an even number of lines of symmetry and 1 option has an odd number, which is odd? - The shape with an odd number of symmetry axes
• If 4 options show clockwise-directed arrows and 1 shows an anti-clockwise arrow, which is the odd one out? - The anti-clockwise directed arrow
• What is the 'Adjacent vs Opposite Shading' rule? - Differentiating figures where shaded sectors share a boundary from those where they sit across each other
• In a series of 5 figures, if elements enter at the top-left, move along a 'Z' path, and exit at bottom-right, what is the trajectory? - A fixed path-shift sequence
• What is a 'Head-to-Tail Orientation' rule? - Checking whether sequential arrows connect in a continuous loop or collide head-to-head
• If an arrow and a circle swap positions while the arrow flips direction in each frame, what happens in Frame 3 from start? - They return to their original positions with the arrow restored to initial direction
• In figure classification, if 4 figures consist of concentric shapes that decrease in size and 1 has intersecting shapes, which is odd? - The intersecting shapes figure
• If 4 figures have an even ratio of shaded-to-unshaded area ($1:1$) and 1 has a $1:3$ ratio, which is odd? - The figure with $1:3$ ratio
• If a triangle has a circle attached to its right side, and in Figure $B$ it reflects to the left, what transformation occurred? - Lateral Reflection (Mirror image)
• In a 5-frame series, if lines increase by $+1, +2, +3, +4$, how many lines are added to Frame 4 to get Frame 5? - $+4$ Lines
• What is 'Spatial Decomposition Analogy'? - Figure $A$ is a solid shape, and Figure $B$ shows the same shape sliced into constituent geometric pieces
• If 4 figures are identical shapes rotated at various angles ($30^\circ, 60^\circ, 120^\circ, 210^\circ$) and 1 is a mirror reflection, which is odd? - The mirror reflection figure (Chirality mismatch)
• What is a 'Perimeter Attachment Rule'? - Small symbols (dots, crosses) move sequentially along the perimeter vertices of a main polygon
• If a dot travels along the vertices of a regular pentagon in clockwise order, how many frames complete a full cycle? - 5 Frames
• If a triangle rotates $120^\circ$ in each step around its centroid, how does its orientation appear in Step 3? - Identical to Step 0 (Original orientation, $3 \times 120^\circ = 360^\circ$)
• What is the 'Convex vs Concave Polygon' classification rule? - Identifying the single shape with an interior angle $> 180^\circ$ (reflex angle / indent) among convex shapes
• If a square series shows shaded sectors moving $1 \to 2 \to 3 \to 4$, what sequence does it represent? - Clockwise quadrant traversal
• If elements at opposite diagonal corners swap places while central elements remain static, what is the transformation? - Corner-swap involution
• What is the 'Line-Intersection Parity Rule'? - Odd one out is determined by whether internal intersection points sum to an even or odd integer
• In a progressive series, if a symbol changes from Circle $\to$ Triangle $\to$ Square $\to$ Pentagon, what is the 5th symbol? - Hexagon ($n=6$)
• If an arrow alternates between pointing North and pointing South while translating horizontally East, what trajectory is created? - A zigzag vector waveform
• What is the 'Number of Enclosed Regions' classification rule? - Counting the total number of isolated, water-tight internal compartments in each figure
• If 4 options contain shapes with exactly 2 enclosed loops (like 'B' or '8') and 1 option has 1 loop (like 'P' or '0'), which is odd? - The 1-loop figure
• In a visual analogy, if a solid 3D isometric cube becomes a flat 2D square, what does a solid 3D cylinder become? - A 2D rectangle (Orthographic front view)
• If a solid 3D sphere becomes a 2D circle, what does a solid 3D cone become in 2D silhouette? - An Isosceles Triangle
• What is the 'Centroidal Symmetry Rule'? - Checking whether an entire figure remains invariant under $180^\circ$ point reflection through its central center point
• In an alternating series, if Odd frames follow Rule X and Even frames follow Rule Y, how do you find Frame 6? - Apply Rule Y to Frame 4
• If an asymmetric polygon is reflected horizontally in step 1, reflected vertically in step 2, what is the net transformation from start? - $180^\circ$ Point Rotation
• What is the most critical elimination technique in Non-Verbal series questions? - Track one single element or corner dot across all frames before analyzing the whole composite
• Why do competitive exams include visual Non-Verbal reasoning tests? - To evaluate abstract fluid intelligence, visuospatial reasoning, and non-linguistic logic
• What is the fundamental requirement for mastering all Non-Verbal reasoning questions? - Systematic pattern tracking, symmetry recognition, and rigid mental rotation without guesswork