13.1 2D Sequences, Series Completion & Progression Rules

Non-Verbal Reasoning (NVR) evaluates a candidate's ability to decipher visual information, identify underlying geometric rules, and apply inductive logic without relying on language or advanced mathematical skills. On the ISEB Common Pre-Test, Non-Verbal Reasoning forms a distinct 30-minute test component alongside Spatial Reasoning. Administered in a computer-adaptive format, the examination dynamically adjusts item difficulty based on candidate performance. Because candidates cannot return to previous questions or skip ahead, possessing a systematic, structured methodology for dissecting complex visual sequences is vital.

In series completion items, candidates are presented with a sequence of frames—typically a progression of five boxes—and must determine which figure logically completes the series as the sixth box. Candidates who attempt to absorb every diagram as a unified whole often experience cognitive overload. High-scoring candidates treat each figure as a composite system of distinct geometric variables, isolating and evaluating one element at a time.

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1. The "SHAPES" Feature-Isolation Framework

When confronting any 2D sequence question, apply the SHAPES mnemonic checklist to decompose the visual elements systematically:

┌────────────────────────────────────────────────────────────────────────┐
│                     THE "SHAPES" MNEMONIC CHECKLIST                    │
├─────────────┬──────────────────────────────────────────────────────────┤
│ S — Shape   │ Form, side count (3, 4, 5, 6, 8), straight vs curved     │
│ H — Heading │ Orientation, pointing direction, rotation angle (45°, 90°)│
│ A — Arrange │ Position (corners, centre, edges), nesting, layering     │
│ P — Proport │ Relative size (enlarging, shrinking, scale swaps)        │
│ E — Element │ Count of discrete items (dots, crosses, lines, vertices) │
│ S — Shading │ Fill type (solid black, hollow white, hatching, stripes) │
└─────────────┴──────────────────────────────────────────────────────────┘

S — Shape (Geometric Form & Side Count)

Examine the primary figures and any secondary symbols enclosed within them:

• Side and Vertex Count: Count the number of edges. A progression might cycle through polygons in ascending order: triangle (3) → quadrilateral (4) → pentagon (5) → hexagon (6) → heptagon (7) → octagon (8).
• Straight vs Curved: Observe whether figures consist entirely of straight lines, circular arcs, or a combination.
• Regularity vs Irregularity: Note whether polygons are regular (equal sides and angles) or irregular.
• Closure: Check whether outlines are fully closed polygons or open line paths.

H — Heading & Orientation (Directional Bearing)

Observe which way an asymmetric element or directional pointer is oriented:

• Rotational Increments: Pointers, arrows, or asymmetric shapes frequently rotate in uniform angular steps: 45° (one-eighth turn), 90° (quarter turn), 135° (three-eighths turn), or 180° (half turn).
• Direction of Turn: Distinguish strictly between clockwise (CW) and anticlockwise (ACW) rotational motion.
• Reflective Heading: Determine whether an asymmetric element is rotating in 2D space or being flipped across an invisible vertical or horizontal mirror line.

A — Arrangement & Position (Spatial Coordinates)

Track the physical location of shapes within the bounding box:

• Perimeter Navigation: Elements may step systematically between the four corners (top-left, top-right, bottom-right, bottom-left) or edge midpoints.
• Clock Positions: Circular paths often map to standard clock face coordinates: 12:00, 3:00, 6:00, and 9:00 (or intermediate steps like 1:30, 4:30, 7:30, 10:30).
• Concentric and Nested Layering: Elements may move from the exterior of a container to the interior, or shift between foreground and background.

P — Proportion & Relative Size

Monitor scale variations across consecutive frames:

• Monotonic Scaling: A shape may grow progressively larger (small → medium → large → maximum) or shrink symmetrically.
• Relative Scale Inversion: Two nested shapes might swap proportions: an outer large square enclosing a tiny circle may evolve into a large circle enclosing a tiny square.
• Aspect Ratio: A rectangle or ellipse might oscillate between tall/narrow (vertical aspect) and wide/flat (horizontal aspect).

E — Elements & Quantitative Count

Count every discrete item, symbol, or subdivision:

• Arithmetic Sequences: The number of dots, lines, stars, or tick marks may change by a constant difference: +1, +2, −1, −2.
• Compound Counts: Total line segments across all shapes in a box might remain constant (e.g., always summing to 12) or follow a formula (e.g., number of dots equals number of outer sides minus 2).
• Vertex-to-Dot Relationships: Look for direct correspondences between the quantity of interior elements and the exterior polygon's geometric properties.

S — Shading & Fill Patterns

Track changes in internal fill, texture, and colour:

• Fill States: Common states include solid black, clear white (hollow), vertical hatching, horizontal striping, diagonal lines, and stippling (dots).
• Cyclic Shading Sequences: Shading often cycles predictably through a recurring sequence: White → Hatched → Black → White.
• Fill Inversion: In alternating series, two components may swap shading states from box to box (e.g., Box 1 has a black star in a white circle; Box 2 has a white star in a black circle).

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2. Core Progression Archetypes

Visual sequences in non-verbal reasoning typically follow one of four structural archetypes. Identifying which archetype governs the sequence determines how to track its components.

Archetype 1: Continuous Linear Progression

In a linear progression, every successive box applies the same incremental operation to each visual variable. Multiple elements operate concurrently, each following its own independent rule:

Here the triangle moves one corner clockwise each step. The dot moves along the bottom edge and bounces at either end. If those stated rules continue, Frame 6 has a triangle at top-right and a dot at bottom-middle. Track both features: the first rule alone does not locate the dot.

Archetype 2: Alternating (Interleaved) Series

When a sequence does not appear to follow a smooth, continuous transition from Box 1 to Box 2, it is frequently an alternating series. An alternating sequence contains two separate sub-sequences interleaved into a single progression:

   Odd Sub-Sequence:    Box 1 ──────► Box 3 ──────► Box 5 ──────► [Box 7]
   Even Sub-Sequence:          Box 2 ──────► Box 4 ──────► [Box 6]

• Diagnostic Indicator: If an element toggles between two contrasting states (such as pointing up, then down, then up, then down), or if Box 1 and Box 3 share obvious structural similarities that are absent in Box 2, immediately test an alternating hypothesis.
• Target Identification: If you are asked to determine Box 6, your primary focus should be the rule connecting Box 2 and Box 4, because Box 6 is the third term of the even sub-sequence.

Archetype 3: Cyclic Perimeter and Coordinate Motion

Elements frequently travel along a defined geometric perimeter. Understanding boundary behaviors prevents common missteps:

• Wrap-Around Motion: An element reaching the rightmost edge wraps around to reappear on the leftmost edge in the following box.
• Reflective Bounce: An element travelling rightward hits the bounding wall and reverses direction, travelling leftward in subsequent steps.
• Corner Navigation: In a square bounding box, a clockwise movement of 90° jumps from one corner to the adjacent corner; a movement of 45° steps from a corner to an edge midpoint.

Archetype 4: Line Addition, Subtraction & XOR Overlap Logic

In line-based sequences, components do not merely shift position; they combine, dissolve, or interact through geometric arithmetic:

     Box 1              Box 2              Box 3
  ┌─────────┐        ┌─────────┐        ┌─────────┐
  │    │    │        │         │        │    │    │
  │ ───┼─── │   +    │ ─────── │   =    │    │    │   (XOR: Overlapping
  │    │    │        │         │        │    │    │    horizontal line cancels!)
  └─────────┘        └─────────┘        └─────────┘

• Cumulative Addition: Each step adds a fixed number of line segments (e.g., drawing the edges of a cube or star one stroke at a time).
• Sequential Eradication (Subtraction): Edges vanish one by one, typically in a clockwise or outward-inward order.
• Superposition & XOR Logic (Exclusive OR): When two figures overlap, features present in only one figure are preserved, but features present in both figures cancel out and disappear entirely. In competitive entrance assessments, XOR logic is a hallmark of higher-difficulty items.

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3. The Step-by-Step Strategic Elimination Protocol

Under adaptive exam conditions, pacing is paramount. Use a steady pace; there is no published per-item time allowance. Never spend time analyzing features that have already been resolved.

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1. Step 1: Scan and Select: Inspect the five sequence boxes. Identify the most prominent, unambiguous variable (for instance, the number of sides on the main exterior polygon, or the position of a solid black dot).
2. Step 2: Formulate the Rule: Track that single variable across all five boxes. State the rule explicitly: "The outer shape gains one side each step: 3 sides, 4 sides, 5 sides, 6 sides, 7 sides. Box 6 must have 8 sides (an octagon)."
3. Step 3: First Wave Elimination: Scan the five answer options (A, B, C, D, E). Immediately eliminate any option that does not feature an octagon. In many items, this single step eliminates three of the four distractors.
4. Step 4: Resolve Remaining Ambiguities: If two options remain (e.g., both are octagons), identify how they differ. Perhaps Option A contains a shaded inner diamond while Option D contains an unshaded diamond. Now check only the shading rule across the five sequence boxes to make the final determination.

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4. Detailed Worked Examples

Worked Example 1: Multi-Feature Linear Sequence

Problem Statement: Examine the five-box sequence below and determine which figure correctly completes the pattern as Box 6.

Visual Breakdown:

• Element 1 (Central Arrow):
  • Box 1: Points UP (12:00).
  • Box 2: Points RIGHT (3:00) — rotated 90° clockwise.
  • Box 3: Points DOWN (6:00) — rotated 90° clockwise.
  • Box 4: Points LEFT (9:00) — rotated 90° clockwise.
  • Box 5: Points UP (12:00) — rotated 90° clockwise.
  • Deduction for Box 6: Must point RIGHT (3:00).
• Element 2 (Dot Count & Position):
  • Box 1: 1 dot positioned below the arrow.
  • Box 2: 2 dots positioned below the arrow.
  • Box 3: 3 dots positioned below the arrow.
  • Box 4: 4 dots positioned below the arrow.
  • Box 5: 5 dots positioned below the arrow.
  • Deduction for Box 6: Must have 6 dots positioned below the arrow.
• Answer Verification: Look for the option with a central arrow pointing horizontally to the right accompanied by exactly 6 dots beneath it.

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Worked Example 2: Alternating (Interleaved) Sequence

Problem Statement: Determine which figure should appear as Box 6 in the sequence below:

Box 1         Box 2         Box 3         Box 4         Box 5
┌─────────┐   ┌─────────┐   ┌─────────┐   ┌─────────┐   ┌─────────┐
│  ■      │   │       ○ │   │      ■  │   │         │   │         │
│         │   │         │   │         │   │         │   │         │
│         │   │         │   │         │   │       ○ │   │      ■  │
│         │   │         │   │         │   │         │   │         │
└─────────┘   └─────────┘   └─────────┘   └─────────┘   └─────────┘
(Black Sq)    (White Cir)   (Black Sq)    (White Cir)   (Black Sq)

Visual Breakdown:

• Analyze Sequence Structure: Observe that the shape alternates between a solid black square (Boxes 1, 3, 5) and a hollow white circle (Boxes 2, 4). Because we are solving for Box 6, Box 6 belongs to the even sub-sequence (Box 2 → Box 4 → Box 6).
• Track the Even Sub-Sequence:
  • In Box 2, a hollow white circle is located in the top-right corner.
  • In Box 4, the hollow white circle has moved to the bottom-right corner (moved down along the right vertical edge).
  • Deduction for Box 6: Continuing the path along the perimeter in a clockwise direction, the circle must move to the bottom-left corner.
  • The shape must be a hollow white circle.
• Answer Verification: Eliminate any option containing a black square. Select the option featuring a single hollow white circle in the bottom-left corner.

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Worked Example 3: cancelling repeated strokes

Use a horizontal midline and a vertical midline as two whole features. The rule for this exercise is: overlay the previous two frames; keep a stroke present in exactly one, and remove a stroke present in both. This is exclusive OR (XOR) applied to whole strokes.

Frame 1 has horizontal and vertical strokes. Frame 2 has horizontal only. The shared horizontal stroke cancels, leaving the vertical stroke in Frame 3. Next, horizontal-only XOR vertical-only gives both strokes in Frame 4. Frame 3 XOR Frame 4 cancels vertical, leaving horizontal in Frame 5. Finally Frame 4 XOR Frame 5 cancels horizontal, leaving vertical only in Frame 6.

Do not confuse coincident strokes with lines merely crossing at a point. A horizontal stroke and a vertical stroke remain two different features; crossing them does not create a diagonal slash. When a problem uses regions or pixels rather than whole strokes, apply the unit of comparison specified in its instructions.