What rules do matrix puzzles use?

Carpenter, Just and Shell (1990) found five rules behind Raven's Advanced Progressive Matrices: constant in a row, quantitative progression, figure addition or subtraction, distribution of three values, and distribution of two values.

What's Your IQ

What is a matrix reasoning test?

A matrix reasoning test shows a grid of figures, usually 3×3, with one cell missing. You work out the rules that change the figures across the rows and down the columns, then pick the option that completes the grid.

Because the puzzles are purely visual, they measure fluid reasoning: solving a problem you've never seen, without relying on learned facts or language. Raven's Progressive Matrices made the format famous, and a matrix subtest now appears in most major IQ batteries.

1938Raven's Progressive Matrices first published, by John C. Raven

3×3the standard grid: eight figures and one gap

5rule types behind Raven's advanced items (Carpenter et al., 1990)

0words or numbers needed to solve one

THE FIVE RULES

Almost every matrix uses one of five rules

In 1990 Carpenter, Just and Shell analysed Raven's Advanced Progressive Matrices and found that the items are built from just five kinds of rule, often stacked together. Pick one to see it in action.

Constant in a row

One attribute stays the same across a row but changes from row to row.

IN THIS EXAMPLEEach row keeps its own shape: circles, then squares, then triangles.

How to spot it:Read along a row: if something never changes, it's fixed for that row.

THE TEST

15 matrices, 15 minutes

Before you start

1Each matrix has 8 figures and a gap in the bottom-right corner. Choose the option that fits.

2Look along the rows first, then down the columns. Several rules can apply at once.

3Matrices get harder as you go: 5 warm-up, 5 core and 5 hard items.

4You have 15 minutes in total. You can skip and return using the numbered squares.

Answers are saved only on this device.

WARM-UP · ONE OR TWO RULES · MATRIX 1

Which option completes the matrix?

YOUR RESULT0 / 15Just startingThis practice test isn't normed against a population, so it doesn't give an IQ score. It shows how many rule combinations you can hold and test at once; the hard items need three or four.

Warm-up0 / 5

Core0 / 5

Hard0 / 5

29854925:45total time

1990328:23per matrix

Every answer, explained

YOURS

ANSWER

Matrix 1 · SkippedConstant in a row: each row keeps one shape. Progression: the count rises 1, 2, 3 along every row, so three triangles.

YOURS

ANSWER

Matrix 2 · SkippedThe fill depends only on the column: outline, grey, solid. The last column is solid.

YOURS

ANSWER

Matrix 3 · SkippedDistribution of three: circle, triangle and square each appear once in every row and column. The missing one in the last row is the triangle.

YOURS

ANSWER

Matrix 4 · SkippedProgression: the triangle turns 90° clockwise each step along a row, so it ends pointing down.

YOURS

ANSWER

Matrix 5 · SkippedShape follows the column, count follows the row: three hexagons.

YOURS

ANSWER

Matrix 6 · SkippedTwo distributions of three running in opposite directions: shapes shift one way, counts shift the other.

YOURS

ANSWER

Matrix 7 · SkippedThe count rises by one along rows and down columns (1 to 5), and each row has its own fill. Five solid circles.

YOURS

ANSWER

Matrix 8 · SkippedOuter shape is constant in each row; the small inner shape is a distribution of three (circle, triangle, square).

YOURS

ANSWER

Matrix 9 · SkippedThree rules at once: count by row, fill by column, and the triangles flip upside down and back along each row.

YOURS

ANSWER

Matrix 10 · SkippedShapes and fills are separate distributions of three running in opposite directions; the count is constant in each row.

YOURS

ANSWER

Matrix 11 · SkippedOuter and inner shapes are two distributions running opposite ways, and each row has its own fill.

YOURS

ANSWER

Matrix 12 · SkippedShape and fill move together as one distribution, while the count runs the other way. Track the count separately.

YOURS

ANSWER

Matrix 13 · SkippedShape constant in each row; count and fill are distributions of three in opposite directions.

YOURS

ANSWER

Matrix 14 · SkippedFour rules: outer and inner shapes in opposite distributions, count set by column (1, 2, 1), fill following the inner shape's diagonal.

YOURS

ANSWER

Matrix 15 · SkippedCount and rotation travel together along one diagonal, while fill runs the other. The answer combines three counts, a 180° turn and the right fill.

METHOD

How strong solvers work through a matrix

Eye-tracking studies show two styles: constructive matching (work out the answer first, then look for it) and response elimination (compare each option with the grid). Constructive matching is linked to higher scores.

01Scan one rowCompare the three figures in a complete row and name everything that changes: shape, count, fill, angle, inner part.

02Check the columnsDoes the same change happen down the columns? Rules often run both ways, or one each way.

03Build the answerBefore looking at the options, describe the missing figure in words. This is constructive matching.

04Then eliminateFind the option that matches your description. If none does, one rule is wrong: go back to step one.

How the matrix became the IQ item

1936Penrose and Raven.Lionel Penrose and John C. Raven describe a new matrix-based test of reasoning.

1938Raven's Progressive Matrices.The Standard Progressive Matrices are published, designed to measure reasoning without language.

1990Carpenter, Just and Shell.Their analysis of Raven's advanced items identifies the five rule types and links scores to working memory.

1997Matrices join the Wechsler.The WAIS-III adds a Matrix Reasoning subtest, now part of the Wechsler adult and child scales.

2014Open matrices.The International Cognitive Ability Resource (ICAR) releases free matrix items for research.

Keep going

TRAININGPattern recognitionGames for spotting rules in shapes and sequences.

GAMEMatrix Reasoning gameEndless adaptive matrices, scored as you go.

TESTAbstract reasoning testSeries, odd-one-out and matrices mixed.

TESTInductive reasoning testFind the rule from examples.

Nine cells, one missing. Find the rule that fills it.

What is a matrix reasoning test?

Almost every matrix uses one of five rules

15 matrices, 15 minutes

How strong solvers work through a matrix

How the matrix became the IQ item

Keep going

Matrices are one part of IQ. See the whole picture.