In a number or letter series, find the rule linking consecutive terms — a constant difference, a ratio, squares/cubes, or alternating patterns — then extend it.
In coding–decoding, letters or digits are shifted or substituted by a rule (for example +1 in the alphabet); discover the rule from the given pair and apply it to the new word.
Example 1: Next term: 2, 6, 12, 20, …?
Differences 4, 6, 8 grow by 2, so next gap is 10 → 30.
Example 2: If CAT → DBU, then DOG → ?
Each letter moves +1: D→E, O→P, G→H, giving EPH.
Quick recap
Series: spot the rule (difference, ratio, squares) and extend.
Coding: find the shift/substitution from the example, then apply it.
✓ Quick check
In the sequence P₁ = √3, P₂ = √12, P₃ = √27, P₄ = √48, the next term P₅ is:
The terms are 1√3, 2√3, 3√3, 4√3, so the next is 5√3 = √(25 × 3) = √75.
Find the next term in the pattern of interior angles of a regular polygon with n sides, starting from n = 3, 4, 5... If a quadrilateral has three acute angles each equal to 75°, what is the logical value of its fourth angle?
Sum of angles of a quadrilateral is 360°. Sum of three angles = 75° + 75° + 75° = 225°. Fourth angle = 360° − 225° = 135°.
Direction, Distance and Blood Relations
Direction and distance problems track movements (turns are left/right relative to facing); sketch each step and use Pythagoras for the straight-line distance from start to finish.
Blood relations are solved by drawing a small family tree from the clues and reading off the required relationship.
Example 1: Walk 3 km East then 4 km North. Distance from start?
√(3² + 4²) = 5 km.
Example 2: A's mother is B's sister. How is B related to A?
B is A's aunt (mother's sister) or uncle — B is A's aunt.
Quick recap
Sketch directions; straight-line gap via Pythagoras.
Draw a family tree to decode blood relations.
✓ Quick check
In a logical sequence of figures, the first circle is divided into 2 sectors, the second into 4 sectors, and the third into 8 sectors. Following this pattern, how many sectors will the fifth circle have?
The number of sectors follows the pattern 2¹, 2², 2³. The fourth circle will have 2⁴ = 16. The fifth circle will have 2⁵ = 32.
Find the missing term in the sequence: 2, √5, √6, √7, 2√2, …
Writing each term under a root: √4, √5, √6, √7, √8. The next term is √9 = 3.
Venn Diagrams, Dice and Calendar/Clock
Three-set Venn diagrams show overlaps; the central region belongs to all three sets. Dice and cube problems use the fact that opposite faces of a standard die total 7, and a painted cube cut into smaller cubes has predictable painted-face counts.
Clock: the minute hand moves 6° per minute, the hour hand 0.5° per minute. Calendar: each ordinary year shifts the weekday by 1, a leap year by 2.
Example 1: Angle between the hands at 3:00?
The hour hand is at 90° from 12, the minute at 0°, so 90°.
Example 2: A die shows 2 on top. What is on the bottom?
Opposite faces sum to 7, so 5.
Quick recap
Venn centre = common to all three sets; opposite die faces sum to 7.
Minute hand 6°/min, hour hand 0.5°/min; ordinary year shifts weekday by 1.
✓ Quick check
In a histogram, the width of the rectangles is proportional to:
The width of a rectangle in a histogram corresponds to the size of the class interval on the x-axis.
In a logical diagram, if a rectangle has its diagonals intersecting at point O such that angle BOC = 60°, what is the value of angle OAD?
In a rectangle, diagonals are equal and bisect each other, so OA = OB = OC = OD. Angle AOD = angle BOC = 60° (vertically opposite). In triangle AOD, OA = OD, which implies angle OAD = angle ODA. Since angle AOD = 60°, angle OAD + angle ODA = 180° − 60° = 120° → angle OAD = 60°.
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