How to solve matrix reasoning puzzles: the five rule families

Almost every matrix item is built from one of five underlying rules. Learn to recognise them and the puzzles stop feeling like guesswork.

Samir Haddad· Psychometrics writer and test designer· Published June 23, 2026· Updated 2026-08-21· 12 min read

Read the grid before the options

The single most common mistake is jumping to the answer choices. Options are engineered to look reasonable: one matches the shape but not the shading, another the shading but not the count. Reading them first plants exactly the wrong hypotheses.

Instead, describe the transformation out loud, in words, from left to right and then top to bottom. If you can say 'each step adds one dot and rotates the arrow clockwise', you have solved the item and the options become a lookup.

Family 1: progression

Something increases or decreases steadily across the row: the number of elements, the size of a shape, the number of sides, the thickness of a line, the degree of shading.

Watch for progressions that run in both directions simultaneously — dots increasing along rows while shading darkens down columns. Two independent progressions in one item is a standard way to raise difficulty.

Family 2: rotation and reflection

A figure turns by a constant angle each step, or flips across an axis. Rotation is easy to spot in asymmetric shapes and surprisingly hard in near-symmetric ones, which is why test designers love arrows with small tails.

Distinguish rotation from reflection by tracking one identifiable feature — a notch, a dot, a shorter arm. If that feature travels around the figure, it is rotating; if it swaps sides while the rest stays put, it is reflecting.

Family 3: alternation

Features cycle through a fixed sequence: black, white, striped, black, white, striped. Alternation frequently appears alongside another rule, with one attribute cycling while a second progresses.

The tell is a repeated element that reappears at regular intervals. Once you spot a cycle length, check whether it applies along rows, columns, or diagonals.

Family 4: addition and subtraction

The third cell equals the first combined with the second — overlaid, merged, or with shared elements removed. Two variants are common: union, where all marks from both cells appear, and exclusive-or, where marks present in both cancel out.

Test the cancelling version whenever the third cell looks emptier than the first two. It is the rule people miss most often.

Family 5: substitution and distribution

Every row contains the same set of elements in a different arrangement, or a consistent mapping swaps one element for another. Distribution-of-three items are the classic case: three shapes and three fills, each appearing once per row and once per column, like a small Sudoku.

The strategy is inventory rather than transformation: list what each row contains and identify what the incomplete row is missing.

Working under time pressure

Give each item a soft budget. If you have no hypothesis after your first careful pass, mark your best inference and move on: a hard item consumed at the cost of three easy ones is a bad trade in any scoring model.

Guess intelligently when you must. Eliminate options violating the rule you are confident about, then choose among what remains. Partial rule knowledge is real information.

Finally, note that this is trainable — which is exactly why a second attempt at any test inflates your score. Practice teaches you the format, not new reasoning capacity, so compare yourself over time only with fresh item sets.

Frequently asked questions

Does practising matrices raise my real IQ?

It raises your matrix score, mostly between the first and second attempt. Evidence that it transfers to general reasoning is weak.

Is it better to guess or leave an item blank?

On tests without a penalty for wrong answers, always answer. Eliminate what clearly breaks the rule you identified first.

Key sources

  • Carpenter, P., Just, M. & Shell, P. — What one intelligence test measures
  • Raven, J. C. — Progressive Matrices manuals

This article is general education and is not medical, psychological or educational advice. Read our editorial policy.

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