Figure Analogies — Explained
Detailed Explanation
What makes a figure analogy a reasoning problem?
A figure analogy presents a transformation between two diagrams and asks for the corresponding result when that transformation is applied to another diagram. Its structure is A becomes B, so C becomes what?
The essential task is to identify the relationship between the first pair, not to choose an option that merely resembles B. Shapes may rotate, reflect, move between positions, change shading or gain and lose elements.
The descriptions and numerical examples here are original exercises. They illustrate reasoning within the broad analytical and mental-ability areas of the 2026 Preliminary Paper II syllabus, without claiming to reproduce past questions.
Describe features before guessing a rule
Represent each diagram using a small set of observable features: number of elements, shape, position, orientation and shading. These features answer different questions. A triangle can remain a triangle while its position changes; an arrow can stay in the same place while its direction changes.
An unfilled circle can become filled without moving. If several features change together, record each change separately. This prevents the common mistake of noticing one conspicuous change and ignoring another that distinguishes the correct answer from a close alternative.
Suppose the first pair contains an outlined triangle pointing upward, followed by a filled triangle pointing right. A possible stated rule is a quarter-turn clockwise plus a switch from outline to fill.
Applying that rule to an outlined left-pointing arrow gives a filled upward-pointing arrow. A filled right-pointing arrow copies the appearance of the example’s endpoint instead of transferring its transformation.
An outlined upward arrow applies the rotation but misses the shading change. A filled downward arrow applies a quarter-turn in the wrong direction. Each is related to the task, but fails a different part of the rule.
Separate rotation from reflection
Rotation turns a figure around a specified centre. A clockwise quarter-turn sends up to right, right to down, down to left and left to up. Reflection instead places points at equal perpendicular distances on opposite sides of a mirror line.
A vertical reflection exchanges left and right positions while retaining height. A horizontal reflection exchanges upper and lower positions while retaining horizontal position. These operations can look identical for a highly symmetric figure, so an asymmetric marker is useful when the question needs to distinguish them.
Use coordinates to check a rotation when a visual sketch is uncertain. With the centre at the origin and east and north positive, a clockwise quarter-turn maps (x, y) to (y, −x). Thus a dot at (2, 1) moves to (1, −2).
Reflection in the vertical axis instead gives (−2, 1). A half-turn gives (−2, −1). These explicit coordinates distinguish three transformations that a hurried reading might confuse. The centre matters: a rule about rotation around a diagram’s centre cannot be applied around a different corner without changing the result.
Track position and orientation independently
A diagram can move an object to the next compartment without rotating the object itself. Imagine a four-cell ring labelled top, right, bottom and left. If a right-pointing arrow moves from the top cell to the right cell while continuing to point right, the transformation changes location but preserves orientation.
Applying the same rule to an upward-pointing arrow in the left cell places it in the top cell, still pointing upward. Rotating the arrow as well would introduce a second change that the example does not establish.
Conversely, the example may rotate both the arrangement and each object as part of a rigid whole. Then orientation changes along with position. Read whether the transformation concerns independent symbols or the entire diagram. A small feature ledger helps: position, orientation, shading and count each get their own before-and-after entry. The correct option must satisfy the complete ledger. This is more reliable than treating any clockwise movement as proof that every feature should rotate.
Use counts and shading as separate constraints
Some analogies change the number of elements according to a stated relationship, such as replacing each outlined mark with two filled marks. If three outlined marks become six filled marks, applying that explicit rule to four outlined marks gives eight filled marks.
Adding three because the first pair increased by three would match the example numerically but implement a different rule. Without a stated operation or another example, both rules could fit one pair.
Good practice material identifies that ambiguity instead of pretending that one observed input-output pair establishes a unique mathematical function.
Shading can also encode a swap rather than a one-way conversion. If the rule explicitly exchanges filled and unfilled cells, filled cells become unfilled as well as the reverse. A candidate that fills the formerly empty cells but keeps the original filled cells unchanged has applied accumulation rather than inversion. Check the total number of positions as well as the number shaded. A shading change does not create extra cells unless a separate rule says that it does.
Resolve ambiguity with the supplied evidence
An unmarked square rotated by a quarter-turn looks unchanged. Its appearance alone cannot distinguish that rotation from doing nothing. A dot near one corner, a directional arrow or a second asymmetric example can supply the missing information.
The response to an ambiguous practice question should be to identify the competing transformations and specify what extra observation would distinguish them. Avoid inventing a hidden convention merely because an answer key selects one option.
In a well-specified exercise, the stem or the example set should make the intended transformation defensible.
A repeatable solving sequence
First read the requested analogy and mark the reference centre or axes. Second, compare the first pair feature by feature. Third, test the proposed transformation against each supplied example. Fourth, apply it to the new figure without copying irrelevant details from the example.
Finally, compare every feature of the result with the options. Verify the closest distractor by naming its precise error: reversed rotation, wrong mirror, missed shading, changed count or orientation altered during a position-only move.
This sequence builds speed through a stable method while keeping uncertainty visible when the evidence cannot support a unique transformation.