10.3 Figure Sequences and Analogies
Key Takeaways
- Analyze every figure with five features: shape, count, position, shading, and orientation
- A 90° clockwise rotation is the single most common movement rule in figure sequences
- For analogies A : B :: C : ?, build a one-sentence bridge describing the A-to-B change, then apply that exact bridge to C
- Elimination — testing options against one feature at a time — is faster than staring at the sequence waiting for insight
- A mirrored (flipped) figure is not the same as a rotated figure; never confuse the two transformations
Why Figure Reasoning Matters for the PUPCET
The Abstract Reasoning portion of the PUPCET includes non-verbal items: rows of shapes that change step by step, and figure analogies that ask you to complete a relationship. These items test pure pattern recognition rather than memorized knowledge, which is why students who never practiced them freeze while prepared students finish in seconds. Since this chapter is text-based, we will describe each figure in words — the same skill you will use on exam day when you describe each frame to yourself.
The Five-Feature Checklist
Analyze every figure by asking five questions, one feature at a time. Remember them as Shape, Count, Position, Shading, Orientation:
| Feature | Question to ask | Common changes |
|---|---|---|
| Shape | What shapes appear? | Triangle → square → pentagon (sides increase by 1) |
| Count | How many dots, lines, or elements? | +1 each step, or alternating between two values |
| Position | Where is the key element? | A dot hopping from corner to corner clockwise |
| Shading | Which region is shaded? | The shaded part rotates, alternates, or fills in order |
| Orientation | Which way does it point or face? | Rotating 45°, 90°, or 180° clockwise |
Run the checklist in order and track one feature across the whole sequence before moving to the next. Most PUPCET figure items change only one or two features per step, so the checklist usually cracks the pattern on the first or second pass.
Rotation: The Most Common Rule
The single most frequent transformation is clockwise rotation by a fixed angle — usually 45°, 90°, or 180°. An arrow pointing up, then right, then down is rotating 90° clockwise, so the next frame points left. A shaded quarter of a circle sitting at top-right, then bottom-right, then bottom-left is also moving 90° clockwise, so next it moves to top-left. A quick way to check direction: imagine a clock hand sweeping from 12 to 3 to 6 to 9 — that is clockwise.
Counting sides and elements is the second pillar. Triangle (3 sides), square (4), pentagon (5), hexagon (6) — when the outer shape gains a side each frame, the missing figure is usually the next polygon. Dots, lines, and inner shapes often climb at the same rate, or deliberately move in the opposite direction (sides increase while dots decrease) to test whether you track features independently.
The Figure Analogy Format A : B :: C : ?
An analogy reads as A is to B as C is to what? Solve it with the bridge method:
- Build the bridge: write one sentence describing exactly what changed from A to B — for example, the figure doubled in size and kept its shading.
- Apply the identical bridge to C, changing nothing more and nothing less.
- Match the option that satisfies every part of the bridge — size, shape, shading, and orientation all at once.
If A : B changed only the shading, the correct answer changes only the shading of C. An option that also rotates or resizes C breaks the bridge and must be rejected.
Worked Examples
Example 1 — Rotation sequence
A small flag on a pole points east in frame 1, south in frame 2, and west in frame 3. The flag rotates 90° clockwise each frame (east → south → west follows the clock hand). Next frame: the flag points north.
Example 2 — Sides and dots climbing together
Frame 1: a triangle containing one dot. Frame 2: a square containing two dots. Frame 3: a pentagon containing three dots. Two features climb in parallel: sides go 3, 4, 5 and dots go 1, 2, 3. Next frame: a hexagon containing four dots.
Example 3 — Shading on the move
A circle is divided into four quarters. The top-left quarter is shaded in frame 1, the top-right in frame 2, and the bottom-right in frame 3. The shaded quarter jumps 90° clockwise each frame. Next frame: the bottom-left quarter is shaded.
Example 4 — Addition of elements
Frame 1: a plain square. Frame 2: the square with one diagonal. Frame 3: the square with both diagonals. Frame 4: both diagonals plus a vertical midline. Each frame adds exactly one line while keeping everything drawn before. Next frame: the same figure plus a horizontal midline.
Example 5 — Analogy with a size bridge
Small unshaded triangle : large unshaded triangle :: small shaded circle : ? Bridge: from A to B, the figure grows larger while shape and shading stay the same. Apply it to the small shaded circle: it grows larger, stays a circle, stays shaded. Answer: a large shaded circle. Reject any option that changes the shading or the shape — the bridge touched only size.
Example 6 — Analogy with a rotation bridge
An arrow pointing up : an arrow pointing right :: a flag pointing left : ? Bridge: A rotates 90° clockwise to become B. Apply to the flag: pointing left, rotated 90° clockwise, swings to pointing up. Check with the clock-hand image: a hand at 9 o'clock moved a quarter-turn clockwise lands at 12 o'clock.
The Elimination Strategy
When the rule does not reveal itself, work backward from the options:
- Pick the most distinctive feature in the item — usually shading or element count.
- Decide what that feature must look like in the correct answer.
- Cross out every option that fails that single test; often two of the four fall immediately.
- Repeat with a second feature until exactly one option survives.
Elimination converts a confusing item into at most two quick checks, and it works even when you never fully understand the pattern — a legitimate edge under exam time pressure with roughly 150 items to finish.
Common Traps
- Mirror versus rotation. A flipped (mirrored) figure is not the same as a rotated one. If A becomes B by rotation, the answer must rotate C — an option showing a mirror image of C is a classic distractor.
- Changing two features when the bridge changed one. If A : B altered only shading, the correct answer alters only the shading of C. Extra changes are always wrong, even when they look attractive.
- Miscounting sides or dots. Count deliberately, pointing at each side or dot. A hexagon mistaken for a pentagon breaks the entire chain of reasoning.
Exam Tips
- Whisper one short sentence per frame — dot moves one corner clockwise, or figure gains one line. Naming the change makes it concrete and keeps you from re-checking features you already solved.
- Budget under a minute per figure item. If the checklist and elimination both stall, shade your best surviving option, mark the item, and move on — every item carries the same weight, so never sacrifice three easy points chasing one hard one.
A square shows a single dot in its top-left corner. In the next frame the dot sits in the top-right corner, and in the third frame it sits in the bottom-right corner. Where should the dot appear in the fourth frame?
Complete the analogy: a small shaded circle is to a large shaded circle as a small unshaded square is to ___.