13.2 2D Analogies, Similarities & Odd-One-Out

Non-verbal reasoning assessments frequently examine visual problem solving through relational comparisons rather than linear sequences. On the ISEB Common Pre-Test, these take three primary formats: 2D Shape Analogies ($A : B :: C : ?$), Odd-One-Out (identifying the single figure that violates a common rule among five options), and Grouping / Shape Classification (sorting shapes into categories based on shared geometric properties). Each format tests a candidate's ability to abstract structural rules from geometric diagrams and apply those rules rigorously without being deceived by superficial visual distractors.

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1. 2D Shape Analogies ($A : B :: C : D$)

Visual analogies operate on proportional reasoning: Figure A transforms into Figure B according to a specific set of operations. You must apply the identical operations to Figure C to produce Figure D.

A clear practice analogy uses visible arrow directions:

The stated rule is rotate 90° clockwise. Up becomes right; applying the same rule to left gives up. A single pair can fit more than one imaginable rule, so use the instruction and any extra examples rather than assuming a unique transformation from minimal evidence.

The Analogy Transformation Grammar

Every analogy transformation is constructed from a defined vocabulary of geometric operations:

1. Reflective Flipping (Mirror Inversion):
   • Reflection across a horizontal mirror line: Figure flips upside down ($y \to -y$). Top features move to the bottom.
   • Reflection across a vertical mirror line: Figure flips left-to-right ($x \to -x$). Features on the left swap to the right.
2. Rotational Turning:
   • The entire figure (or an internal component) rotates by a fixed angle: 45°, 90°, 180°, or 270°, either clockwise or anticlockwise.
   • Critical Distinction: Do not confuse a 180° rotation with a reflection. For symmetric shapes they may appear identical, but for asymmetric shapes (such as the letter 'F' or an asymmetric flag), a 180° rotation preserves chirality (handedness), whereas a reflection reverses it.
3. Structural Nesting Inversion (Inside-Out Swap):
   • The outer container shape shrinks to become the inner element, while the inner element expands to become the outer container.
4. Shading & Fill Transposition:
   • Colours or fills invert: solid black becomes hollow white, or striped patterns transfer from the outer container to the inner shape.
5. Arithmetic Mutation (Side & Element Adjustments):
   • The number of sides on a polygon increases or decreases by a fixed integer (e.g., triangle with 3 sides becomes square with 4 sides, so a pentagon with 5 sides must become a hexagon with 6 sides: $n \to n+1$).
   • Elements split, duplicate, or halve in count.

The Two-Step Analogy Protocol

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• Step 1: Verbalize the Exact Rule Sentence: State the transition from A to B aloud or in your mind with complete precision: "The outer hexagon loses one side to become a pentagon, rotates 90° clockwise, and its fill changes from white to black."
• Step 2: Apply the Template to Figure C: Take Figure C (e.g., a square) and apply the exact rules: "The square must lose one side to become a triangle (4 − 1 = 3), rotate 90° clockwise, and its fill must change from white to black."

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2. Odd-One-Out (Similarities and Differences)

In Odd-One-Out questions, five figures are displayed. Four of the figures share a common, definable geometric principle, while exactly one figure violates it. The task is to discover the unifying law and identify the outlier.

The Hierarchy of Common Odd-One-Out Rules

When examining five figures, test candidate rules in order of geometric rigor:

┌────────────────────────────────────────────────────────────────────────┐
│                  ODD-ONE-OUT GEOMETRIC RULE TAXONOMY                   │
├─────────────────────────┬──────────────────────────────────────────────┤
│ 1. Symmetry             │ Lines of reflective symmetry (0, 1, 2, 4) or │
│                         │ rotational symmetry order (Order 2, 4)       │
│ 2. Chirality / Rotation │ 4 shapes are identical rotations; 1 is a     │
│                         │ reflected mirror image (reversed handedness) │
│ 3. Side & Vertex Parity │ 4 shapes have even sides; 1 has odd sides    │
│ 4. Parallel Lines       │ 4 shapes contain parallel pairs; 1 has none  │
│ 5. Angle Class          │ 4 shapes are convex; 1 is concave (reflex)   │
│ 6. Topology & Closure   │ 4 shapes are closed loops; 1 is open         │
│ 7. Shading Ratios       │ 4 shapes are exactly 50% shaded; 1 is not    │
└─────────────────────────┴──────────────────────────────────────────────┘

1. Symmetry (Reflective & Rotational)

• Lines of Symmetry: Count the number of straight lines across which the shape can be folded into two congruent, overlapping halves. A classic pattern presents four shapes with exactly two lines of symmetry (such as a rectangle, rhombus, or oval) and one shape with only one line of symmetry (such as an isosceles trapezoid or kite).
• Rotational Symmetry: Rotate the shape 360° around its centre. How many times does it match its starting outline? An equilateral triangle has rotational symmetry of order 3; a regular hexagon has order 6.

2. Chirality and Rotational Equivalence (Handedness)

In a rotation-versus-reflection practice puzzle, you may be given five asymmetric figures (for example, an asymmetric polygon with an attached flag or hook). Four of the figures are physically identical shapes merely rotated to different angles on the screen. The fifth figure is a reflected (mirror) version which can never be rotated in 2D space to match the others. Mentally rotating each shape to a common upright orientation immediately reveals the reflected outlier.

3. Side, Line, and Vertex Parity

• Count the total line segments forming the outer boundary. In many items, four figures have an even number of edges (e.g., 4, 6, 8, 10), while the odd-one-out has an odd number (e.g., 5 or 7).
• Alternatively, tally interior components: four shapes contain an even number of dots, while one contains an odd number.

4. Parallel and Perpendicular Edges

• Inspect pairs of lines. Four shapes may be trapezoids, parallelograms, or regular hexagons containing at least one pair of parallel lines, while the outlier has no parallel sides (e.g., an irregular scalene quadrilateral).
• Look for right angles (90°): four figures might possess at least one right angle, while the fifth possesses only acute and obtuse angles.

5. Topological Properties (Closedness & Intersections)

• A closed curve partitions the 2D plane into an inside and an outside. Four figures might be closed polygons, while one figure has an imperceptible open gap.
• Count the number of line intersections or regions formed by overlapping loops.

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3. Grouping & Two-Group Classification

In shape classification items, candidates are presented with two distinct sets of figures: Group 1 (containing 2 or 3 exemplars sharing Rule A) and Group 2 (containing 2 or 3 exemplars sharing Rule B). Candidates are then given a target probe shape and must determine whether it belongs to Group 1, Group 2, or neither.

  GROUP 1: Shapes with an EVEN number of sides and a SOLID BLACK dot
  ┌───────┐   ┌───────┐   ┌───────┐
  │   ●   │   │   ●   │   │   ●   │
  │Square │   │Hexagon│   │Octagon│
  └───────┘   └───────┘   └───────┘

  GROUP 2: Shapes with an ODD number of sides and a HOLLOW WHITE dot
  ┌───────┐   ┌───────┐   ┌───────┐
  │   ○   │   │   ○   │   │   ○   │
  │Triangl│   │Pentag │   │Heptag │
  └───────┘   └───────┘   └───────┘

  TARGET SHAPE: Pentagon with a solid black dot.
  Deduction: Has an odd number of sides (5) but a black dot. Matches neither group!

The Classification Strategy:

1. Isolate Group 1's Rule: Compare all exemplars within Group 1. What single geometric property (or combination of two properties) do they all hold without exception?
2. Isolate Group 2's Rule: Compare all exemplars within Group 2. Formulate their unifying law. Ensure that Group 1's rule and Group 2's rule are mutually distinct.
3. Test the Target Shape: Evaluate the probe shape against Group 1's rule. If it satisfies every condition, assign it to Group 1. If not, test Group 2. If it violates both, choose the appropriate "neither" option.

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4. Detailed Worked Examples

Worked Example 1: two changes together

Here △ is an outline triangle pointing up, ▼ is a filled triangle pointing down, and ◁ is an outline triangle pointing left. The stated rule is to turn the triangle 180° and change outline to filled. Applying it to C gives a filled triangle pointing right (▶). A filled triangle pointing left has changed only the fill; an outline triangle pointing right has changed only the direction. Both are incomplete answers.

Worked Example 2: rotations and a reflected outlier

Each # is one occupied square; o marks the end of the shorter arm. Dots are empty background. Compare the whole figure, including the marker.

     A           B           C           D           E     
# . . . .   # # # # #   o # # . .   . . . . o   . . # . .  
# . . . .   # . . . .   . . # . .   . . . . #   . . # . .  
# . . . .   o . . . .   . . # . .   # # # # #   . . # . .  
# . . . .   . . . . .   . . # . .   . . . . .   . . # . .  
# # o . .   . . . . .   . . # . .   . . . . .   o # # . .  

A is the reference. B, C and D are its 90°, 180° and 270° clockwise rotations. E is its left-to-right reflection. To check E, start at the unmarked tip of the long arm, travel to the bend and then to o. Rotation preserves the turn you make at the bend; E reverses it. Because the two arms have different lengths and the short end is marked, swapping the ends cannot disguise the difference.

This is a visual check on similarities and differences. It also shows why you must track a distinctive feature, rather than comparing the bounding boxes alone.