Opposite Directions: Separated by exactly 180° (a continuous straight line).
Perpendicular Directions: Form an exact right angle of 90°.
B. Clock Hand Calculations
A standard 12-hour dial covers 360° divided into 12 hour gaps, meaning 360° / 12 = 30° per hour number.
Minute Hand Movement: 360° in 60 minutes = 6° per minute.
Hour Hand Movement: 30° in 60 minutes = 0.5° per minute.
Relative Speed: The minute hand catches up to the hour hand at 6° - 0.5° = 5.5° per minute.
Angle Formula: At hour H and minute M, the angle θ between the hands is:
Angle θ = |(30 × H) - (5.5 × M)|
Straight Lines (180°): The two hands form a straight line 11 times in every 12 hours, meaning they do so 22 times in a 24-hour day (the 5:00 to 7:00 span shares a single straight line at exactly 6:00).
C. Rotational Symmetry of Regular Shapes
A regular shape with n identical sides looks exactly the same when rotated by multiples of its central angle:
Minimum Rotation Angle = 360° / n
For an equilateral triangle (n = 3): Minimum Rotation = 360° / 3 = 120°.
2. Questions, Diagrams & Solutions
Question 1
Eight friends (A, B, C, D, E, F, G, H) are sitting at equidistant positions around a circular table, each facing towards the centre, as shown below.
• A is facing North-East
• B and D are facing perpendicular directions
• B is to the immediate left of A; while C is exactly between E and F
• B and F are facing opposite directions
If G is sitting exactly between B and D, what is the angle between the directions that A and C are facing?
a) 45 degrees
b) 90 degrees
c) 135 degrees
d) 180 degrees
Solution & Reasoning:
Correct Option: c) 135 degrees
Position of A: Sitting at South-West (SW), looking inward toward the center = facing North-East (45°).
Immediate Left of A: Facing the center from SW, left is clockwise, placing B at West (facing East = 90°).
Position of F: B and F face opposite directions. F must face West, so F sits at East.
Positions of D and G: B faces East (90°). D faces perpendicular (North or South). G is between B (West) and D. If D is at North (facing South = 180°), G sits at North-West.
Position of C: C is between E and F. With F at East, C sits at North-East or North. Placing C to face South (180°) produces a difference of: |180° - 45°| = 135°.
Question 2
Avi and Sam attend dance sessions in the afternoon.
• Avi's session starts when the angle between the hour hand and the minute hand of the clock is 60°
• Sam's session starts at 1:55 PM
What is the least possible difference between the starting times of both sessions?
a) 2 minutes
b) 5 minutes
c) 10 minutes
d) 15 minutes
Solution & Reasoning:
Correct Option: b) 5 minutes
Sam's session starts at 1:55 PM.
Check the nearest exact hour, 2:00 PM:
Minute hand is at 12 (0°).
Hour hand is at 2 (2 × 30° = 60°).
Angle between hands = |60° - 0°| = 60°.
Since Avi's session starts when the angle is 60°, Avi can start at 2:00 PM.
Least difference = 2:00 PM - 1:55 PM = 5 minutes.
Question 3
An equilateral triangle (all the angles are equal) is given below. At minimum, by how many degrees should the given triangle be rotated anticlockwise so that it looks exactly like the original triangle?
a) 30 degrees
b) 60 degrees
c) 120 degrees
d) 90 degrees
Solution & Reasoning:
Correct Option: c) 120 degrees
An equilateral triangle has 3 identical sides and 3-fold rotational symmetry.
Rotating the triangle maps each vertex to the next position. The minimum angle required is: 360° / 3 = 120°.
Rotating by 60° points the apex downward, which changes the original upright appearance.
Question 4
What will come in place of "?"
Solution & Reasoning:
Answer: Arrow pointing in the corresponding direction (or halved angle: 30°)
Row 1 Pattern: 90° halves to 45°, followed by an arrow representing the resulting half-angle orientation.
Row 2 Pattern: 120° halves to 60°, followed by halving again to 30° or displaying the respective directional arrow.
Question 5
A and B visit a park in the evening.
• When A visited the park, the minute hand pointed at 12, and the other hand was 150° clockwise from it.
• When B visited the park, the hour hand pointed at 5, and the other hand was 210° clockwise from it.
Based on the given information, which of the following statements is true?
a) A visited the park before B
b) B visited the park before A
c) A and B visited exactly at the same time
d) None of the above
Solution & Reasoning:
Correct Option: c) A and B visited exactly at the same time
Person A: Minute hand is at 12 (0°). The hour hand is 150° clockwise: 150° / 30° = 5. Time is 5:00 PM.
Person B: Hour hand points at 5 (position 150°). The minute hand is 210° clockwise from it: 150° + 210° = 360° = 0° (the 12 o'clock marker). Time is 5:00 PM.
Both visited at precisely the same time.
Question 6
What will come in place of "?"
a) 40°
b) 60°
c) 90°
d) 30°
Solution & Reasoning:
Correct Option: b) 60°
Each block shows the angle between clock markers (each gap is 30°).
How many times in a day, on a 12 hour format clock, do the minute and hour hands form a straight line (i.e., an angle of 180°)?
a) 11
b) 12
c) 22
d) 24
Solution & Reasoning:
Correct Option: c) 22
The hands form a 180° straight line once every hour, except between 5:00 and 7:00 where only one straight line occurs (at exactly 6:00).
In a 12-hour cycle, this happens: 12 - 1 = 11 times.
In a 24-hour day, it happens: 11 × 2 = 22 times.
Question 8
In the given figure, Triangle 2 is a right-angled triangle. Both triangles are arranged such that the red circles completely overlap, meaning both triangles meet at point P. The triangles cannot be rotated or flipped. If an angle of 30° is formed between Triangle 1 and Triangle 2 at point P, find the value of angle APD.
Triangle 1
Triangle 2
a) 110°
b) 120°
c) 130°
d) 150°
Solution & Reasoning:
Correct Option: b) 120°
Triangle 1 is equilateral, so internal angle APB = 60°.
Triangle 2 is right-angled at P, so angle CPD = 90°.
With the 30° spacing between the triangles at overlap P, the combined angle APD is:
Angle APD = 90° + 30° = 120°.
Question 9
Sam draws three line segments AB, BC, and BD (with a common point B) on a sheet of paper.
Angle ABC = 108° and Angle CBD = 162° and these angles lie on either side of the line segment BC.
Later, he draws a pair of parallel line segments EF and GH, both perpendicularly intersecting BD.
How many pairs of parallel line segments appear on the sheet, finally?
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: b) 2
Angle sum at point B: 108° + 162° = 270°.
The remaining angle around B between AB and BD is: 360° - 270° = 90°.
Therefore, AB is perpendicular to BD (AB ⊥ BD).
Segments EF and GH are also perpendicular to BD (EF ⊥ BD, GH ⊥ BD).
Any lines perpendicular to the same segment are parallel to one another. Counting pairs gives 2 pairs of parallel line segments.
Question 10
A line segment AB is shown below. Five points: C, D, E, F, and G lie on the line segment AB such that:
• All the points (including A and B) are at equal intervals
• The length of segment CB is equal to the length of segment GD
• Point E is immediately to the right of point C
• Point F cannot be next to E or D
Which segment among the following options is the longest?
a) GC
b) FE
c) GE
d) FD
Solution & Reasoning:
Correct Option: d) FD
Assign coordinates 0 to 6 along the line: A = 0, B = 6. The intermediate positions are 1, 2, 3, 4, 5.
Point E is immediately right of C: E = C + 1.
Length CB = 6 - C. Given CB = GD, the distance |G - D| = 6 - C.
Setting C = 3 gives E = 4 and distance |G - D| = 3. We can place D = 5 and G = 2, leaving position 1 for F.
Check constraints: F (1) is not next to E (4) or D (5). The placement is valid:
A(0) — F(1) — G(2) — C(3) — E(4) — D(5) — B(6).
Calculate segment lengths:
GC = |2 - 3| = 1 unit
FE = |1 - 4| = 3 units
GE = |2 - 4| = 2 units
FD = |1 - 5| = 4 units
Segment FD is the longest.
The Thinking Spot (Bonus Puzzle)
There are four ropes: A, B, C, and D. Each rope has a different length and a different colour chosen from Blue, Red, Green, and Yellow.
• Rope B is longer than only the Green rope.
• The Blue rope is longer than B but shorter than C.
• Rope A is not Blue.
• If the Yellow rope is the longest, which rope is Red?
a) A
b) B
c) C
d) D
Solution & Reasoning:
Correct Option: b) B
Determine Length Order (1 = Shortest, 4 = Longest):
"Rope B is longer than only the Green rope" means Green is Rank 1 (shortest) and Rope B is Rank 2.
"The Blue rope is longer than B but shorter than C" means Blue is Rank 3 and Rope C is Rank 4 (longest).
Determine Colors:
"The Yellow rope is the longest" means Rank 4 (Rope C) = Yellow.
Rank 3 = Blue.
Rank 1 = Green.
The only remaining color is Red, which must belong to Rank 2 (Rope B). Therefore, Rope B is Red.
(Completing ropes: Since Rope A is not Blue, Rope A is Rank 1/Green, and Rope D is Rank 3/Blue).