IQ explained · 8 min read
How to solve matrix reasoning problems
Figure-matrix rule types with worked examples: progression, distribution, two attributes, rotation and counts, plus a method that works before you look at the options.
In short
A matrix reasoning problem is a 3×3 grid of figures with one cell missing. Every such problem follows a small set of rule types: one attribute progresses across a row, three shapes are distributed so that each appears once per row and column, two attributes vary independently by row and column, a shape rotates by a fixed angle, or a count changes. The method is to describe what changes along a row and along a column in words, predict the missing cell, then find it among the options.
Try it on yourself: Abstract reasoning questions with answers · 12 min · free, with the full result.
Practising for a timed test? Abstract reasoning test practice
Untimed and new every time: Untimed puzzles, new every time
New grids with a nudge, the rule, then the working: Matrix reasoning test with hints
The method: describe before you look
- Read the first row left to right. Name what changes: shape, fill, count, rotation, size, position.
- Read the first column top to bottom. Name what changes there.
- Check the second row and column to confirm the rule holds.
- Predict the missing cell in words: for example "the third shape of the row set, filled, count two".
- Only now look at the options and pick the one matching the description. If two options match, you have missed an attribute.
Looking at the options first invites the most common error: choosing the option that looks most like its neighbours rather than the one the rule requires.
Progression
One attribute moves in steps along a row: fill goes from outline to half to solid, or the count goes one, two, three. The rows are independent copies of the same progression, so the missing cell is the next step in its own row.
Answer
Distribution
Three shapes appear so that each row and each column contains every shape exactly once, like a small Latin square. The missing cell is whichever shape is absent from both its row and its column.
Answer
Two attributes, two directions
The row fixes one attribute and the column fixes another: shape by row, fill by column, for instance. The missing cell combines the row's shape with the column's fill. These problems feel harder because both directions matter, but the method is the same: describe the row, describe the column, combine.
Answer
Rotation and count
A figure turns by a fixed angle per column, usually 45 or 90 degrees, while the row controls another attribute such as fill. The trap is the mirror image: a reflection is never a rotation, and option sets often include one. Count rules change how many copies appear, by row or by column, and are easy to miss when the shapes are small.
Practise with the rule named
The abstract-reasoning and matrix tests on this site use these rule types and name the one behind every question in the answer review, so a missed question tells you which rule to practise rather than just that you missed one.
Three grids, worked with hints
Three fixed grids, each solved with the same three hints the practice gives: where to look, the rule, then the worked solution. Open them in order, and before each one, say to yourself what you expect it to tell you.
Learn the rules, try a puzzle and use a hint when you need one.
Your answers are never sent to us. We count how many hints are opened, never which answer you chose; on test pages, masked session replay may record your clicks (privacy section 2.6).
Example 1: Matrix puzzle
Which option completes the grid?
Hint 1: where to look
Look down the first column, then the second. Note how the figures are drawn in each.
Hint 2: the rule
The shading advances by one shade per column, from outlined to half-filled at the start of row 1. Repeat that move for the third tile.
Hint 3: worked solution
- Use the bottom row.
- Tile 1 is 1 outlined diamond, and tile 2 is 1 half-filled diamond.
- The shape stays the same and the shading advances one more step.
- So tile 3 is 1 filled diamond.
Now find it among the options.
Answer: why each option fits or fails
Answer: D. 1 filled diamond.
- AOption A has circle as its shape, taken from the tiles above the gap, not from its own row.
- BOption B repeats outlined for the shading. The bottom row already has it, so the gap needs another.
- COption C has two faults: its shape comes from the tiles above the gap, not its own row, and its shading repeats the bottom row.
- D✓ AnswerThe answer is option D, which is 1 filled diamond.
Example 2: Matrix puzzle
Which option completes the grid?
Hint 1: where to look
Follow the lines along row 1. Compare tile 1 with tile 2, then check what tile 3 kept from them.
Hint 2: the rule
Tile 3 shows the elements found in exactly one of tiles 1 and 2. Row 1 confirms it with frame sides at the bottom and left and corner lines at the top left and top right. Check the bottom row.
Hint 3: worked solution
- Start with tile 1 of the bottom row: the left side and corner lines at the top right and bottom left.
- Now bring in tile 2: frame sides at the top, bottom and left and corner lines at the top left, bottom right and bottom left.
- Check each position and remove any element that appears twice.
- The missing tile has frame sides at the top and bottom and corner lines at the top left, top right and bottom right.
Now find it among the options.
Answer: why each option fits or fails
Answer: C. Frame sides at the top and bottom; corner lines at the top left, top right and bottom right.
- AOption A joins both tiles' frame sides: it keeps the one at the left (in both tiles).
- BOption B has two faults: its frame sides join both tiles (left extra), and its corner lines keep only tile 2's own ones (top right missing).
- C✓ AnswerThe correct one is option C: frame sides at the top and bottom and corner lines at the top left, top right and bottom right.
- DOption D keeps only tile 2's own corner lines: it drops the one at the top right (only in tile 1).
Example 3: Matrix puzzle
Which option completes the grid?
Hint 1: where to look
Look at the shape and number of figures. Which of them stays fixed across a row, and which one varies?
Hint 2: the rule
The number of figures grows by one each column while the shape stays fixed. Row 1: 1, 2, 3. Take the bottom row the same way.
Hint 3: worked solution
- Count the figures in the bottom row's first two tiles: 1 outlined triangle, then 2 outlined triangles.
- Each column adds one more figure, so tile 3 has one more than tile 2.
- That gives 3 outlined triangles.
Now find it among the options.
Answer: why each option fits or fails
Answer: B. 3 outlined triangles.
- AOption A has two faults: its shape comes from the tiles above the gap, not its own row, and its number of figures repeats the bottom row.
- B✓ AnswerThe answer is option B, which is 3 outlined triangles.
- COption C has hexagon as its shape, taken from the tiles above the gap, not from its own row.
- DOption D repeats 1 for the number of figures. The bottom row already has it, so the gap needs another.
When the three make sense, try new grids with the same hints: Matrix reasoning test with hints.
Sources
- The American Psychological Association's topic page on intelligence collects its public material on how intelligence is defined and measured.
- Mensa Norway's own online test page says the test gives an indication of general cognitive abilities and is not a substitute for professional intelligence tests.