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Visual Pattern Puzzles and the Four Rules They Use

Figural pattern puzzles run on translation, rotation, reflection and glide reflection, plus a counted attribute. How to name what changes, and check it.

By Published Updated 6 min read

Reign Creative LLC makes IQ Test & Brain Training Games, which this article links to, so we have a commercial interest in it. How we write and check articles.

Key takeaways

  • Four transformations account for most of what moves in a figural puzzle: translation, rotation, reflection and glide reflection, which is a translation and a reflection combined.
  • Anything that changes without moving — count, shading, size, colour — is a separate layer, and hard items usually run one transformation and one counted attribute at the same time.
  • Rotation and reflection are told apart by handedness: rotating a figure never turns it into its mirror image, so one asymmetric feature settles the question.
  • Two terms are never enough to fix a rule. A third term is what rules out the competing explanation, and a fourth is what confirms it.
  • Perception research finds that patterns containing more symmetry types are detected more readily than the simpler ones nested inside them, which is why some patterns seem to jump out and others do not.
  • Our own app, IQ Test & Brain Training Games, has pattern and rule-discovery logic games and an IQ-style test covering patterns and spatial reasoning, but its listing does not say whether any of them use the rotation, reflection or translation rules described here.

Nearly every figural pattern puzzle you will meet is built from four transformations — translation, rotation, reflection and glide reflection — applied to a figure, plus one or more attributes that change in place, such as count, shading or size. The skill being tested is not seeing harder; it is describing. A solver who can say "the arrow turns 90 degrees clockwise while the number of dots goes up by one" has already finished, and a solver staring at the figures waiting for something to click usually has not started.

One statement first, because this hub is attached to an app: IQ Test & Brain Training Games, which is our own app, has pattern-based logic games and an IQ-style test covering patterns and spatial reasoning, but its Google Play listing does not describe mirror-image or rotation tasks, so this article is about the puzzle format, not about a feature.

The four transformations

A transformation is a rule for moving a figure without changing its size or shape. In two dimensions there are exactly four, and research on pattern perception treats them as the basic vocabulary.

  • Translation — the figure slides. Nothing turns or flips; only position changes.
  • Rotation — the figure turns about a point, by a consistent angle each step.
  • Reflection — the figure flips across a line, producing a mirror image.
  • Glide reflection — a reflection combined with a shift along the mirror line. Footprints are the everyday case: each print is the mirror of the one before, and a step further on.

A 2021 paper by Peter Kohler and Alasdair Clarke in Proceedings of the Royal Society B notes that these four operations combine into exactly seventeen distinct regular textures — the wallpaper groups — which is the complete catalogue of repeating two-dimensional patterns. That number is worth knowing because it puts a ceiling on the subject: there are not endlessly many kinds of regularity, only a short and enumerable list.

Their experimental finding is the more useful one for a puzzle solver. Patterns belonging to richer groups, containing more symmetry types, were detected more readily than the simpler patterns nested inside them, needing shorter presentation times to be reliably seen. In other words, more structure is easier to notice, not harder. A pattern that you are struggling to see is often one carrying a single, thin regularity rather than a complicated one.

The attributes that change without moving

Transformations are only half of it. The other half is the set of attributes that can change while the figure stays put:

  • Count — how many elements are present.
  • Shading or colour — filled, half-filled, empty.
  • Size — a figure growing or shrinking in steps.
  • Composition — elements being added to or removed from a figure between steps.

The IMak item generator, documented by Diego Blum and Heinz Holling in Frontiers in Psychology, is a good illustration of how few rules are needed to produce a large bank of figural items: it manipulates a handful of rules such as rotating the main shape, reflecting it across an axis, removing line segments and moving a dot along an edge. Items differ mainly in how many of those rules operate at once.

That is the practical point. Difficulty in figural puzzles comes far more from simultaneity than from exotic rules. Two easy rules running together are harder than one difficult rule running alone.

Telling a rotation from a reflection

This is the one genuinely tricky discrimination, and it has a mechanical answer: handedness. Rotating a figure can never turn it into its mirror image, and reflecting it always does. So find one asymmetric feature and follow it.

Write a capital F. Turn the page through any angle you like and the letter remains a recognisable F viewed from an odd direction. Hold it to a mirror and you get a shape no rotation of an F will ever produce. If the figures in a puzzle are symmetrical enough that this test fails — a plain square, a circle with a centred dot — then rotation and reflection are genuinely indistinguishable on that item, and the setter has either marked one arbitrarily or used a figure with a mark on it precisely to avoid the ambiguity.

The ambiguity is not hypothetical. Consider a single dot on the rim of a circle, appearing at the twelve o'clock position, then three o'clock, then six o'clock. A quarter-turn clockwise explains it. So does reflecting across a diagonal each time, and so does a translation of the dot around the rim. With three terms the rotation is the simplest surviving rule, and the fourth position would be nine o'clock — but that conclusion rests on a convention about simplicity, not on proof. NRICH, the mathematics education project at the University of Cambridge, collects problems on reflection, rotation and enlargement that make this concrete, including several that ask what happens when two transformations are applied one after the other.

A procedure that works on hard items

Four steps, in order:

  1. List the attributes. Before looking for any rule, name everything that could vary: shape, count, size, shading, orientation, position. Writing the list down is not fussiness; it is what stops you from finding one rule and stopping.
  2. Take one attribute at a time. Follow count across the whole sequence, ignoring everything else. Then follow shading. Mixing attributes is how people talk themselves into a rule that fits nothing.
  3. Test in both directions where a grid allows it. A rule that holds across the rows and fails down the columns was a coincidence. The structure of a grid item is covered in more depth in our article on how a matrix puzzle is built.
  4. Predict before you look at the options. Distractors are constructed by breaking one rule at a time, so an option that looks right is often the one built from the rule you missed.

Here is a counted example to work on. A sequence of dot arrangements contains 1 dot, then 3, then 6, then 10. The differences are 2, 3 and 4, so the next difference is 5, giving 15. Those are the triangular numbers, and the visual version — dots stacked into a triangle with one more row each time — makes the rule obvious in a way the bare list does not. Notice that the count rule needed four terms before it was safe; with only 1 and 3 in front of you, doubling-and-adding-one fits just as well.

What transfers, and what this app actually contains

None of the above requires shapes. The habit that makes it work — name the attributes, isolate one, insist on a third and fourth term before committing — applies identically to a run of numbers, and to the pattern and rule-discovery games in our app. IQ Test & Brain Training Games ships a Number Pattern Challenge mode that asks for the rule behind a numerical run, at three difficulty settings, and records accuracy and response time per session. It is a one-dimensional analogue of the same discipline, and it is practice rather than assessment; the app's separate IQ-style test, which covers patterns and spatial reasoning, produces results its listing calls estimates based on app activities, not a professional assessment. What the app tracks is set out on the IQ Test & Brain Training Games app page.

Two related habits are worth a look. Reading a pattern that is moving rather than static is a different problem, and our guide to reading boss patterns treats it directly. Holding several partial hypotheses while testing another is a genuine capacity limit rather than a character flaw, and working memory and training explains what that limit is. For the full taxonomy of rule-finding problems, start with number sequence puzzles, browse the logic and pattern puzzles hub, or see everything in the education and brain training section.

Good to know

Frequently asked questions

What is the difference between a rotation and a reflection?
A rotation turns a figure about a point and preserves its handedness; a reflection flips it across a line and reverses handedness. The practical test is to pick an asymmetric feature and follow it. A capital F rotated by any angle is still an ordinary F seen from an odd direction, but a reflected F is a mirror F that no amount of turning will reproduce.
What is a glide reflection?
A glide reflection is a reflection across a line combined with a shift along that same line. Footprints in sand are the standard example: each print is the mirror image of the one before it, and also a step further forward. It is counted as one of the four fundamental symmetry operations in two dimensions, alongside translation, rotation and reflection.
How many terms do I need before I can be sure of a pattern?
More than two, and usually more than three. Two terms are consistent with a large number of rules, so any answer based on them is a guess. A third term eliminates most competitors and a fourth confirms the survivor. If a puzzle gives you only two terms and asks for the next, it is relying on a convention about which rule is simplest rather than on evidence.
Can I practise visual pattern puzzles in IQ Test & Brain Training Games?
Its Google Play listing does not say directly. It describes logic games built on patterns and rule discovery, memory games that use symbols, colours and grids, and an IQ-style test covering patterns and spatial reasoning, without saying whether any item uses the rotation, reflection or translation rules this article describes. The listing also describes the test's results as estimates based on app activities, not a professional assessment.

Sources

  1. The human visual system preserves the hierarchy of two-dimensional pattern regularity (opens in a new tab)Proceedings of the Royal Society B, via PubMed Central — accessed
  2. Transformations: Age 11-16 (opens in a new tab)NRICH, University of Cambridge — accessed
  3. Automatic Generation of Figural Analogies With the IMak Package (opens in a new tab)Frontiers in Psychology, via PubMed Central — accessed

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