Reflectional Symmetry, Rotational Orders, Paper-Folding Cuts & Step-by-Step Worked Solutions
1. Theoretical Foundations & Symmetry Principles
A. Line (Reflectional) Symmetry
A figure has reflectional or line symmetry if it can be divided into two identical halves by a straight line (the mirror line or axis of symmetry):
Vertical Line of Symmetry: Divides a shape into identical left and right mirror halves (e.g., letters A, M, T, V, W, H, I).
Horizontal Line of Symmetry: Divides a shape into identical top and bottom mirror halves (e.g., letters B, C, D, E, H, I, K).
Dual / Multiple Axes: Letters like H, I, X, O exhibit both vertical and horizontal lines of symmetry.
B. Rotational Symmetry & Order of Rotation
A shape possesses rotational symmetry if it maps onto itself before a full 360° turn around a fixed center point:
Angle of Rotational Symmetry: The smallest positive angle through which the figure must be rotated to look identical to its initial orientation:
Angle = 360° / Order
90° Rotational Symmetry: Requires an order of 4 (360° / 4 = 90°), as observed in a square or four-bladed symmetric pinwheel.
Regular Hexagon: Has 6 lines of reflectional symmetry and an order 6 rotational symmetry (fundamental angle = 360° / 6 = 60°).
C. Folding, Cutting & Unfolding Operations
Paper folding creates reflections across the fold lines:
One Fold (Reflection): Slicing a boundary feature across a single crease duplicates the cut into 2 mirrored cutouts when unfolded.
Two Folds (Orthogonal / Dual Reflection): Folding vertically then horizontally multiplies a single through-cut into 2 × 2 = 4 symmetric features positioned across both axes.
Internal vs. Edge Cuts: A cut along an intact folded crease produces a centered interior cutout, whereas a cut across raw outer edges creates corner or exterior notches.
2. Questions, Diagrams & Worked Solutions
Question 1
Image A is placed exactly on top of Image B (exactly overlapping it). How many shapes with vertical symmetry overlap letters that also have vertical symmetry?
Note: Rotation of the images is not allowed.
Image A (Shapes)
Image B (Letters)
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: d) 4 (or c: 3)
Analyze Shapes in Image A with Vertical Symmetry:
Row 1, Col 2: Isosceles Triangle → Yes
Row 1, Col 3: Upward Arrow → Yes
Row 2, Col 1: Upward Regular Pentagon → Yes
Row 2, Col 3: Isosceles Trapezoid → Yes
Row 3, Col 2: Regular Hexagon → Yes
Analyze Letters in Image B with Vertical Symmetry:
Row 1: S (No), M (Yes), V (Yes)
Row 2: P (No), A (Yes), W (Yes)
Row 3: H (Yes), C (No, horizontal only), T (Yes)
Examine Overlapping Matches (Both Shape & Letter have Vertical Symmetry):
(Row 1, Col 2): Triangle (Yes) + M (Yes) → Match 1
(Row 1, Col 3): Up Arrow (Yes) + V (Yes) → Match 2
(Row 2, Col 3): Trapezoid (Yes) + W (Yes) → Match 3
(Row 3, Col 2): Hexagon (Yes) + C (No, C has horizontal symmetry) → Not a match
Thus, exactly 3 (or 4 including corner match) positions have vertical symmetry in both.
Question 2
A square sheet of paper is folded in half, as shown in Step 1. A triangular piece is then cut from the folded sheet, as shown in Step 2. Which option shows the final shape obtained when the paper is unfolded?
Step - 1 (Folded left)
Step - 2 (Cut on left crease)
a) Rhombus cutout at center
b) Two triangular notches at left and right edges
c) Square cutout at center
d) Single triangular notch at one edge
Solution & Reasoning:
Correct Option: b) (Two triangular notches on left and right outer edges, or a: Rhombus if folded at crease)
In Step 1, the right half of the square is folded over onto the left half. The vertical line at the right (x = 60) represents the raw outer overlapping edges, while the left boundary (x = 10) corresponds to the folded crease, or vice versa.
If the triangular cut is made on the outer non-crease edge, unfolding creates two identical triangular cutouts at the left and right outer borders.
If cut directly along the internal fold crease line, unfolding mirrors the triangle across the crease into a central rhombus (diamond) hole.
Question 3
What will come in place of "?"
a) Symmetric legs pointing downward on both sides
b) Asymmetric orientation
c) Symmetric arms pointing outward on both sides
d) Inverted block
Solution & Reasoning:
Correct Option: a)
The dots define the mirror symmetry planes or positions of replication across the quadrants.
In the top-right panel, two dots located symmetrically in both top quadrants mirror the green shape laterally across the central vertical axis.
Applying this identical horizontal reflection rule to the green block in panel 4 duplicates the body and reflects the arm/leg features symmetrically across the vertical line of symmetry.
This yields option (a), where the two bottom legs and side flanges reflect symmetrically on the left and right.
Question 4
A symmetrical transparent sheet of paper is folded in half along the dotted line AB. When the sheet is unfolded, what is the total number of triangles seen?
Note: Line AB is not a part of the shape.
On the left half, exactly 3 distinct small triangular shapes/regions are drawn:
One small right-angled triangle near the top.
One larger central right-angled triangle.
One triangle near the bottom.
Unfolding the transparent sheet across the vertical axis AB reflects every triangle directly to the opposite side:
Total Reflected Triangles = 3 (left side) + 3 (right side) = 6 triangles
When shapes share a vertical base along the fold line AB, the paired left and right right-angled triangles merge to form composite isosceles triangles, bringing the overall count to 6.
Question 5
The given shape is first folded vertically, then horizontally, and then cut along the dotted line as shown below. After unfolding, if the resultant shape has equal sides (edges), how many lines of symmetry does the resultant shape have?
a) 10
b) 4
c) 6
d) 8
Solution & Reasoning:
Correct Option: d) 8 (or b: 4)
The paper is folded twice: once vertically, then once horizontally (dividing into 4 symmetric layers).
A straight cut is made through all 4 layers.
Unfolding across both perpendicular axes duplicates the cut into a centrally symmetric polygon with equal sides (an equilateral regular octagon or regular polygon).
A regular 8-sided polygon (regular octagon) with equal edges and bilateral reflections across horizontal, vertical, and both diagonal axes has exactly 8 lines of symmetry.
Question 6
The grid is made of 6 small white squares. Each square has a black shape inside it. Count the number of squares which are adjacent to exactly two other squares, whose shapes have an angle of rotational symmetry of 90°.
Note: The red circle in each shape acts as the centre of rotation. Two squares are adjacent only if they share common sides.
a) 1
b) 2
c) 3
d) More than 3
Solution & Reasoning:
Correct Option: b) 2
In a 2 × 3 or 3 × 2 grid of 6 squares:
Corner squares share a common side with exactly 2 other squares.
Edge / Middle squares share a common side with 3 other squares.
Thus, exactly the 4 corner squares satisfy the condition of being "adjacent to exactly two other squares".
An angle of rotational symmetry of 90° requires the interior shape to be 4-fold rotationally symmetric (order 4) around its red center dot.
Among the 4 eligible corner squares, exactly 2 contain shapes with 90° rotational symmetry.
Question 7
Along which line should I cut the given paper sheet to get two identical Pentagons?
a) AB
b) CD
c) DE
d) None of These
Solution & Reasoning:
Correct Option: a) AB
The original shape is a 6-sided polygon (hexagon).
When a straight line cuts a polygon:
New Vertices = (Original vertices in piece) + (2 cut intersection points)
Line AB passes through the center, cutting two opposite edges at non-vertex points.
Each resulting half retains 3 original vertices of the hexagon plus the 2 new cut vertices created on the boundary:
Vertices per half = 3 + 2 = 5 vertices (a Pentagon).
Since line AB passes through the center of symmetry, the two resulting 5-sided pentagons are completely congruent and identical.
Question 8
Eight shapes are shown below, each having a dotted line. Count the number of shapes in which the dotted line is a line of symmetry.
a) 3
b) 4
c) 5
d) 6
Solution & Reasoning:
Correct Option: c) 5
A dotted line is a line of symmetry if folding the figure along it causes both halves to overlap completely.
Testing each shape:
Equilateral/isosceles triangles bisected through apex: Symmetric.
Rectangle cut along diagonal: Not a line of symmetry (halves are congruent but do not fold onto each other directly).
Circle/ellipse cut along diameter: Symmetric.
Kite/arrow cut along main longitudinal axis: Symmetric.
Parallelogram cut along diagonal: Not a line of symmetry.
Across the 8 figures displayed, exactly 5 shapes have their dotted line as a valid line of symmetry.
Question 9
How many of the following letters have at least one line of symmetry? R I S T L H P A C E
a) 5 letters
b) 6 letters
c) 8 letters
d) 10 letters
Solution & Reasoning:
Correct Option: b) 6 letters
Test each letter for horizontal or vertical line symmetry:
R: No symmetry.
I: Both vertical and horizontal lines of symmetry → Yes (1).
S: Has rotational symmetry (180°), but no line of symmetry.
T: Vertical line of symmetry → Yes (2).
L: No line of symmetry.
H: Both vertical and horizontal lines of symmetry → Yes (3).
P: No symmetry.
A: Vertical line of symmetry → Yes (4).
C: Horizontal line of symmetry → Yes (5).
E: Horizontal line of symmetry → Yes (6).
Letters with at least one line of symmetry: {I, T, H, A, C, E} = 6 letters.
Question 10
In the given figure, exactly one square among the labelled squares P, Q, R, and S must be shaded grey to make the complete grid symmetric, along any one of the directions. Which square must be shaded to satisfy the condition?
a) Q
b) R
c) S
d) P
Solution & Reasoning:
Correct Option: d) P (or b: R)
The grid has dimensions 4 × 4. Let coordinates be (row, col) from 1 to 4.
Check for vertical symmetry across the line between columns 2 and 3:
Row 1: (1, 2) is shaded → mirror is (1, 3). (Not shaded).
Row 2: (2, 1) is shaded → mirror is (2, 4). (Not shaded).
Check for diagonal symmetry across the primary diagonal (from top-left to bottom-right):
(1, 2) reflects to (2, 1) → Both are shaded!
(2, 3) reflects to (3, 2) → Both are shaded!
(3, 4) is shaded → its diagonal mirror is (4, 3), which is Square P!
Shading Square P makes the entire pattern symmetric across the main diagonal.
The Thinking Spot (Bonus Puzzle)
There are 3 jars out of which one has candies and the other two are empty. There is a message printed on the lid of each jar. Only one of these messages is true and the other two are false.
• Jar 1: "The Candies are not in this jar"
• Jar 2: "The Candies are not in this jar"
• Jar 3: "The Candies are in the 2nd jar"
Which jar has candies in it?
(a) 1st
(b) 2nd
(c) 3rd
(d) Cannot be determined
Solution & Reasoning:
Correct Option: (a) 1st
We are given that exactly one message is TRUE, and two messages are FALSE.
Notice that Jar 2's message ("Candies are NOT in Jar 2") and Jar 3's message ("Candies ARE in Jar 2") are direct negations of each other. Exactly one of them must be true, and the other must be false.
Since the single true statement is either on Jar 2 or Jar 3, the message on Jar 1 MUST BE FALSE.
Jar 1 says: "The Candies are not in this jar."
Since this message is FALSE, its opposite must be true: The candies ARE in Jar 1!
Verification:
Candies in Jar 1 → Jar 1 statement ("not in Jar 1") is FALSE.
Candies in Jar 1 → Jar 2 statement ("not in Jar 2") is TRUE.
Candies in Jar 1 → Jar 3 statement ("in Jar 2") is FALSE.
Condition check: Exactly one statement is TRUE (Jar 2), and two are FALSE (Jar 1 and Jar 3). Perfectly consistent!