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Free guide · Abstract reasoning · 8 min read

How to solve matrix reasoning puzzles

The five transformation rules

Nearly every 3×3 matrix puzzle is built from a small set of transformations applied across rows and columns. Learn to name them, and the hard puzzles become bookkeeping.

Updated

Matrix puzzles look infinite. They are not. Research that reverse-engineered the classic matrix tests (Carpenter, Just & Shell, 1990) found that almost every item is generated by five rules applied to simple attributes — shape, count, size, shading, corner dots. Your job is never to 'see the answer'; it is to name which rules are running.

Rule 1 — Constant in a row

An attribute is identical across each row and changes only between rows. If every cell in row one holds a circle and every cell in row two holds a square, the shape channel is 'constant in a row' — read it and move on.

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The shape holds constant along each row while the shading changes — the shape channel is 'constant in a row'.

Rule 2 — Progression

Something changes by a constant step along the row: the figure grows a size step, or a dot count climbs 1 → 2 → 3. Ask: what is being added each step?

The dot count progresses +1 per cell — so the next cell must hold four.

Rule 3 — Distribution of three

Three values of an attribute — say circle, square, triangle — each appear exactly once per row and once per column, like a tiny Sudoku. Spot two values in the bottom row and the third is forced.

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Each shape appears exactly once per row and column — a tiny Sudoku.

Rules 4 & 5 — Addition, subtraction, and XOR

The third cell combines the first two: their elements added together (union), the second removed from the first (subtraction), or — the hardest rule on any matrix test — XOR: an element appears in the third cell only when it appears in exactly one of the first two. When a matrix looks lawless, test XOR.

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XOR: a corner dot survives only where it appears in exactly one of the first two cells.

The strategy

  1. Rows first. State the rule for row 1, confirm it on row 2, apply it to row 3.
  2. One attribute at a time. Shape, then count, then size, then shading, then the corner dots — never all at once.
  3. Predict before peeking. Decide what the answer must look like, then scan the options. The wrong options are engineered to tempt scanners.
  4. 30-second rule. Stuck? Flag it and move on — points come from the whole test, not one puzzle.

Common traps

  • Options that copy a neighboring cell — the 'repetition' distractor. Your predicted answer almost never equals a visible cell.
  • Options that satisfy one rule but break another. Check every attribute channel before committing.
  • Diagonal reading. Rules run along rows (and columns) — diagonals are almost never the key on 3×3 items.

Speed · ~45s/item: Target 25–60 seconds per matrix depending on difficulty. The rows-first scan is fast because it is ordered — undirected staring is what burns time.

Worked example

Made by the same generator and answer checker as the practice questions, at middling difficulty, but never served in a test. Commit to an option before you reveal — the pick is yours alone and is never recorded.

Which piece completes the pattern?

?
Reveal the answer

The answer is A.

Show why — the worked reasoning
Work rows first, one attribute at a time (see the matrix lesson): - Distribution of three — the three shapes each appear exactly once in every row and every column (a Latin square). - Constant in a row — the shading is identical across each row. Apply each rule to the bottom row and the missing cell must be a large hatched circle.

Try it yourself: 4 practice questions

Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.

Question 1 · Easy

Which piece completes the pattern?

?
Reveal the answer

The answer is F.

Show why — the worked reasoning
Work rows first, one attribute at a time (see the matrix lesson): - Constant in a row — every cell in a row uses the same shape; the shape changes only between rows. - Progression — the figure grows by one size step at each cell, left to right (small → medium → large). Apply each rule to the bottom row and the missing cell must be a large outlined square.
Question 2 · Easy

Which piece completes the pattern?

?
Reveal the answer

The answer is E.

Show why — the worked reasoning
Work rows first, one attribute at a time (see the matrix lesson): - Constant in a row — every cell in a row uses the same shape; the shape changes only between rows. - Progression — the figure grows by one size step at each cell, left to right (small → medium → large). Apply each rule to the bottom row and the missing cell must be a large outlined pentagon.
Question 3 · Easy

Which piece completes the pattern?

?
Reveal the answer

The answer is F.

Show why — the worked reasoning
Work rows first, one attribute at a time (see the matrix lesson): - Constant in a row — every cell in a row uses the same shape; the shape changes only between rows. - Progression — the dot count along the top increases by one at each cell (1 → 2 → 3). Apply each rule to the bottom row and the missing cell must be a large outlined hexagon and 3 dot(s) along the top.
Question 4 · Easy

Which piece completes the pattern?

?
Reveal the answer

The answer is C.

Show why — the worked reasoning
Work rows first, one attribute at a time (see the matrix lesson): - Constant in a row — every cell in a row uses the same shape; the shape changes only between rows. - Progression — the dot count along the top increases by one at each cell (1 → 2 → 3). Apply each rule to the bottom row and the missing cell must be a large outlined pentagon and 3 dot(s) along the top.

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Questions people ask

What is a matrix reasoning question?

A 3×3 grid of figures with the last cell missing. You choose the figure that completes the rules running along the rows and columns. It is the format of Raven's Progressive Matrices and of the Matrix Reasoning subtest in the Wechsler scales.

How long should a matrix question take?

About 25 to 60 seconds, depending on difficulty. If you have no rule after about 30 seconds, flag the question and come back to it.

Can practising matrices raise your IQ?

Practice makes you faster and more accurate on this question type, which can lift your score on a test that uses it. Research on practice effects finds that such gains stay with the trained kind of task and do not carry over to unfamiliar reasoning problems, so it is test skill rather than higher intelligence.

The evidence: the studies behind this answer

Will matrix reasoning questions be on the free test?

Usually. The free test gives abstract reasoning 6 of its 20 questions and spreads them across the 6 abstract reasoning question types, so each type normally appears once.

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