Matrix reasoning puzzles (like Raven's Progressive Matrices) present a 3×3 grid (or 4×4 at expert difficulty) where each cell contains shapes with various attributes. One cell is missing, and you must identify the pattern rules to determine what belongs there. Rules can apply to rows, columns, diagonals, or the entire grid. Each puzzle may have multiple simultaneous rules affecting different attributes.

Grid[row][col] = f(position, rules)

Row rule: Same shape repeats across each row
Column rule: Size increases going down each column
Diagonal rule: Corner cells all share the same attribute

How to recognize it

Scan each row for consistent patterns
Scan each column for consistent patterns
Check diagonals and corners for matching elements
Consider shape, size, and fill as independent attributes

Common mistakes

Focusing on only one attribute when multiple rules exist
Missing column patterns while focusing on rows
Not considering diagonal relationships

Step-by-step walkthrough

In a 3x3 matrix, each row contains a circle, square, and triangle. Each column shows small, medium, and large sizes. Cell 9 is missing. What shape and size belongs there?

Identify the row rule: Each row contains all three shapes
Check row 3: cells 7 and 8 show circle and square
Identify the column rule: Sizes increase down each column
Check column 3: cells 3 and 6 show small and medium sizes
Combine both rules to find the answer

Answer: Large triangle

Learn 2

Shape Progression Rules

How shapes change or stay constant across matrix rows and columns.

Shape rules determine which geometric form appears in each cell. The simplest rule is 'constant shape' where all cells contain the same form. Progression rules change the shape systematically—either the same order repeats in each row, each column has its own shape, or each row/column contains three different shapes. Understanding shape rules helps isolate this attribute from size and fill.

shape(cell) = f(row, col) or constant

Constant: All cells contain circles
Row pattern: Each row shows circle → square → triangle
All different: Each row has 3 unique shapes (order may vary)

How to recognize it

Check if all shapes are identical (constant rule)
Compare shapes across each row—do they follow a pattern?
Compare shapes down each column—do they follow a pattern?
Note: shape is independent of size and fill

Common mistakes

Confusing shape order with size or fill patterns
Assuming shapes must change when they might be constant
Missing that different rows can have different shape sets

Step-by-step walkthrough

In a 3x3 matrix, each row shows the same shape progression. Row 1: circle, square, triangle. Row 2: circle, square, triangle. Row 3: circle, square, ?. What comes next?

Examine row 1 to find the shape sequence
Confirm the pattern repeats in row 2
Apply the pattern to row 3
The third position in the sequence is always triangle

Answer: Triangle

Learn 3

Size Progression Rules

How size changes systematically across matrix rows or columns.

Size rules govern whether shapes are small, medium, or large. Common patterns include constant size (all same), increasing size (small→medium→large across rows or down columns), and decreasing size (large→medium→small). Size progression is often the most visually obvious pattern but can be subtle when combined with other rules.

size(cell) = small | medium | large, following position rule

Constant: All cells have medium-sized shapes
Row increase: Each row goes small → medium → large
Column decrease: Each column goes large → medium → small

How to recognize it

Compare sizes within each row—are they equal or changing?
Compare sizes within each column—are they equal or changing?
Note direction: increasing (growth) or decreasing (shrinking)
Size is independent of shape and fill

Common mistakes

Confusing size direction (increasing vs decreasing)
Missing size patterns when distracted by shape changes
Assuming size must vary when it might be constant

Step-by-step walkthrough

In a 3x3 matrix, shapes increase in size from left to right in each row. Row 3 shows: small circle, medium circle, ?. What size belongs in the missing cell?

Identify the size pattern across rows
Verify the pattern in rows 1 and 2
Apply the pattern to row 3
The third position follows the increasing sequence

Answer: Large circle

Learn 4

Fill Pattern Rules

How shape fills (empty, half-filled, filled) change across the matrix.

Fill rules determine whether shapes are empty (outline only), half-filled (partial shading), or fully filled (solid). Like other attributes, fill can be constant across the grid or progress systematically across rows or columns. Fill patterns often create visual rhythm that helps identify the correct answer.

fill(cell) = empty | half | filled, following position rule

Row pattern: Each row shows empty → half → filled
Column pattern: Each column shows empty → half → filled
All different per column: Each column has one of each fill

How to recognize it

Examine fill states across each row
Examine fill states down each column
Look for the empty→half→filled progression
Fill is independent of shape and size

Common mistakes

Confusing 'half' with 'empty' or 'filled'
Missing fill patterns while focused on shape
Not checking both row and column directions

Step-by-step walkthrough

In a 3x3 matrix, fill progresses across each row as empty, half-filled, filled. Row 3 shows: empty square, half-filled square, ?. What fill belongs in the missing cell?

Identify the fill pattern across rows
Verify the pattern in rows 1 and 2
Apply the pattern to row 3
The third position completes the fill sequence

Answer: Filled square

Learn 5

Diagonal and Corner Patterns

Rules that apply to diagonal lines or corner cells of the matrix.

Beyond rows and columns, matrices can have diagonal rules. The main diagonal runs from top-left to bottom-right; the anti-diagonal runs from top-right to bottom-left. Corner rules specifically match the four corner cells. These patterns are typically found in harder puzzles and require looking beyond the obvious row/column structure.

Diagonal: cells[0,0], cells[1,1], cells[2,2] share attribute. Corners: cells at (0,0), (0,2), (2,0), (2,2) share attribute.

Diagonal shape: Top-left, center, bottom-right all have same shape
Anti-diagonal fill: Top-right, center, bottom-left all have same fill
Corners same: All four corner cells have matching shapes

How to recognize it

After checking rows and columns, examine the diagonals
Compare the four corner cells for matching attributes
Diagonal rules often combine with row/column rules
These patterns appear in harder difficulty levels

Common mistakes

Only checking rows and columns, missing diagonal patterns
Confusing main diagonal with anti-diagonal
Forgetting that diagonal rules can apply to any attribute

Step-by-step walkthrough

In a 3x3 matrix, all cells on the main diagonal share the same shape. Cells 1 and 5 contain circles. Cell 9 is missing. What shape belongs there?

Identify the main diagonal cells
Check the diagonal pattern: cell 1 has a circle
Confirm the pattern: cell 5 also has a circle
Apply the diagonal rule to cell 9

Answer: Circle

Learn 6

Multi-Rule Solving

Strategies for puzzles with multiple simultaneous pattern rules.

Harder matrix puzzles combine multiple rules—for example, shapes might follow a row pattern while sizes follow a column pattern and fill follows a diagonal. Solving these requires systematically checking each attribute independently, then combining the constraints to find the unique answer that satisfies all rules simultaneously.

Answer = intersection of all rule constraints

2 rules: Row shapes (circle, square, triangle) + Column sizes (small, medium, large)
3 rules: Constant shape + Row fill + Column size
4 rules: Diagonal shape + Anti-diagonal fill + Row size + All-different shapes per row

How to recognize it

Analyze each attribute (shape, size, fill) separately
Write down what each attribute should be based on its rule
Combine constraints: the answer must satisfy ALL rules
Eliminate options that violate any single rule

Common mistakes

Finding one rule and stopping the search
Not checking all three attributes independently
Choosing an answer that matches some but not all rules

Step-by-step walkthrough

In a 3x3 matrix: shapes follow circle, square, triangle across rows; sizes increase down columns (small, medium, large); fill is constant (all filled). What belongs in cell 9?

Analyze the shape rule: each row has circle, square, triangle
Analyze the size rule: column 3 goes small, medium, large
Analyze the fill rule: all shapes are filled
Combine all three constraints

Answer: Large filled triangle

Learn 7

Rotation Patterns

Shapes rotate by fixed angles across rows or columns.

Rotation patterns involve shapes turning by consistent angles (90°, 180°, 270°) as you move across rows or down columns. The shape type remains the same, but its orientation changes predictably.

Each cell's rotation = base_rotation + (position × rotation_step)

Triangle pointing up → right → down → left across a row
Square rotated 45° in column 1, 90° in column 2, 135° in column 3
All shapes rotated by the same constant angle

How to recognize it

Same shape appears in different orientations
Rotation follows a regular pattern (90° increments)
Check both rows and columns for rotation progressions

Common mistakes

Confusing rotation with different shapes
Missing subtle rotation differences
Not considering all four rotation states

Step-by-step walkthrough

In a 3x3 matrix, triangles rotate 90 degrees clockwise across each row. Row 3 shows: triangle pointing up, triangle pointing right, ?. What orientation comes next?

Identify the rotation pattern
Track the sequence: up (0 degrees), right (90 degrees)
Apply the next 90-degree rotation
The triangle rotates to point downward

Answer: Triangle pointing down

Learn 8

Count Patterns

The number of shapes in each cell increases or decreases systematically.

Count patterns involve the quantity of shapes changing across rows or down columns. A cell might contain 1, 2, or 3 identical shapes, with the count following a predictable progression.

Cell count = base_count + (position × count_step)

1 circle → 2 circles → 3 circles across a row
Single shape in row 1, pairs in row 2, triplets in row 3
Count increases from left to right in each row

How to recognize it

Count the shapes in each cell
Look for consistent increases (1→2→3)
Check if the pattern applies to rows or columns

Common mistakes

Focusing only on shape type, ignoring quantity
Assuming all cells have one shape
Missing the count pattern when shapes overlap

Step-by-step walkthrough

In a 3x3 matrix, the number of shapes increases across each row. Row 3 shows: 1 circle, 2 circles, ?. How many circles belong in the missing cell?

Count the shapes in row 3, cell 1
Count the shapes in row 3, cell 2
Identify the counting pattern
Apply the pattern to find the next count

Answer: 3 circles

Learn 9

Symmetry Patterns

The grid exhibits horizontal, vertical, or point symmetry.

Symmetry patterns require the grid to mirror itself along an axis or around a central point. Horizontal symmetry means left and right columns are reflections. Vertical symmetry means top and bottom rows mirror each other. Point symmetry (180° rotational) means opposite corners match.

Horizontal: cell[r][0] = cell[r][2], Vertical: cell[0][c] = cell[2][c], Point: cell[r][c] = cell[2-r][2-c]

Left column shapes identical to right column shapes
Top row mirrored in bottom row
Corners with matching shapes (top-left = bottom-right)

How to recognize it

Compare opposite sides of the grid
Check if folding the grid would align shapes
Look for matching pairs across the center

Common mistakes

Confusing horizontal and vertical symmetry
Not checking all symmetric pairs
Missing point symmetry around the center

Step-by-step walkthrough

In a 3x3 matrix with horizontal symmetry, column 1 mirrors column 3. Column 1 shows: triangle, square, circle (top to bottom). Column 3, row 3 is missing. What shape belongs there?

Identify the symmetry type
Find the corresponding cell in column 1
Apply the mirror rule
The missing cell must match its mirror

Answer: Circle

Learn 10

Color Patterns

Shape colors change systematically across rows or columns.

Color patterns involve the hue of shapes changing as you move through the grid. Unlike fill (empty/half/filled), color refers to the actual hue (primary, secondary, tertiary). Colors may progress across rows, down columns, or share the same color along diagonals.

Color follows sequence: primary → secondary → tertiary

Purple → Blue → Green across each row
Same color in all diagonal cells
Colors change down columns but stay consistent in rows

How to recognize it

Look for hue changes (not just brightness/fill)
Check if colors follow a sequence
Compare diagonal cells for matching colors

Common mistakes

Confusing color with fill pattern
Not noticing subtle color differences
Assuming color is constant when it varies

Step-by-step walkthrough

In a 3x3 matrix, colors progress across each row as red, blue, green. Row 3 shows: red circle, blue circle, ?. What color belongs in the missing cell?

Identify the color sequence in rows 1 and 2
Verify the pattern applies to row 3
Determine the next color in the sequence
Apply the color to the missing cell

Answer: Green circle

Learn 11

Position Patterns

Shapes shift position within their cells across rows or columns.

Position patterns involve shapes moving within their cell boundaries. A shape might be positioned left, center, or right within its cell, with the position changing systematically across rows or down columns.

Position sequence: left → center → right (or top → center → bottom)

Shape in left of cell → center → right across row
Shapes shift from top to bottom down each column
All cells in a row have shapes at the same position

How to recognize it

Notice where shapes sit within their cells
Look for consistent position shifts
Check if position changes row-wise or column-wise

Common mistakes

Ignoring position and focusing only on shape type
Confusing size changes with position shifts
Not noticing subtle position differences

Step-by-step walkthrough

In a 3x3 matrix, shapes shift position within their cells across each row. Row 3 shows: shape at left, shape at center, ?. What position belongs in the missing cell?

Identify the position sequence
Check row 3: first cell has shape at left
Check row 3: second cell has shape at center
Apply the pattern to find the third position

Answer: Shape positioned at right

Learn 12

Border Patterns

Shape outline styles change across rows or columns.

Border patterns involve the outline style of shapes changing systematically. Borders can be none (invisible outline), dashed, or solid. This attribute changes across rows or down columns following a predictable pattern.

Border sequence: none → dashed → solid

No border → dashed border → solid border across row
Column 1 has no borders, column 2 has dashed, column 3 has solid
All shapes in a row share the same border style

How to recognize it

Examine the outline of each shape
Look for dashed vs solid vs invisible borders
Check if border style follows row or column pattern

Common mistakes

Confusing border style with fill pattern
Not noticing subtle border differences
Assuming all shapes have the same border

Step-by-step walkthrough

In a 3x3 matrix, border styles progress across each row as none, dashed, solid. Row 3 shows: circle with no border, circle with dashed border, ?. What border style belongs in the missing cell?

Identify the border sequence in rows 1 and 2
Verify the pattern in row 3
Determine the next border style
Apply the border style to the missing cell

Answer: Circle with solid border

Learn 13

Shape Sequence Patterns

Shapes follow a defined ordered sequence across rows or columns.

Shape sequence patterns involve shapes appearing in a specific order. Unlike random shape distribution, these follow predictable progressions like circle→square→triangle. The sequence may also appear in reverse order.

Shapes follow ordered progression: shape[n] → shape[n+1] → shape[n+2]

Circle → Square → Triangle in each row
Same three-shape sequence repeating down columns
Reverse sequence: Triangle → Square → Circle across row

How to recognize it

Identify the specific shape sequence
Check if sequence repeats in every row/column
Look for reversed sequences in harder puzzles

Common mistakes

Missing the ordered nature of the sequence
Confusing sequence with random shape variation
Not recognizing reverse sequences

Step-by-step walkthrough

In a 3x3 matrix, each row follows the sequence circle, square, triangle. Row 3 shows: circle, square, ?. What shape comes next?

Identify the shape sequence in row 1
Verify the same sequence in row 2
Apply the sequence to row 3
The third shape in the sequence is triangle

Answer: Triangle

Learn 14

Geometric Analogy

A is to B as C is to ? — apply the same transformation to predict D.

Geometric analogy puzzles show a 2×2 grid: top row has cells A → B (where some single-attribute transformation is applied), bottom row has C → ?. The transformation that maps A onto B must be the same one that maps C onto ?. The user picks D from four candidates. The format dates to early-20th-century intelligence testing — Spearman cited analogy tests as a primary measure of g (general intelligence) around 1923, and similar items appear throughout WAIS, WISC, and many subsequent batteries.

T(A) = B ∧ T(C) = D where T is a single-attribute transformation

size: small circle → large circle :: small triangle → large triangle
fill: empty pentagon → half pentagon :: empty diamond → half diamond
rotation: triangle 0° → triangle 90° :: pentagon 0° → pentagon 90°
color: primary square → secondary square :: primary hexagon → secondary hexagon

How to recognize it

Compare A and B — what single attribute changed?
That attribute is the transformation; everything else stays the same
Apply the same change to C to find D

Common mistakes

Picking C unchanged (forgot to apply the transformation)
Picking D with A's shape instead of C's (mixed up which cell is the starting point)
Picking the right axis of change but the wrong destination value

Step-by-step walkthrough

A: empty pentagon. B: half pentagon. C: empty triangle. D: ?

Compare A and B: only the fill changed (empty → half)
Apply the same change to C
Pick the answer that matches: half triangle (everything else identical to C)

Answer: Half-filled triangle

Learn 15

Odd-One-Out

A 3×3 grid where eight cells follow a rule and one breaks it — click the violator.

Odd-one-out puzzles show a full 3×3 grid in which a single row-rule or column-rule (e.g., 'each column has three different shapes', 'fill increases across each row') governs eight of the nine cells. Exactly one cell — the violator — has a different value for the rule's controlled attribute and breaks the pattern. The user clicks the violator directly. Cattell's Culture Fair Intelligence Test (1949) popularized this format because it requires rule discovery without relying on language; it is widely used in cross-cultural and culture-fair assessments.

∀ i ∈ cells \ {v} : rule(i) holds ∧ ¬rule(v)

Rule borderRow: every row has none/dashed/solid borders. Violator has 'solid' where 'dashed' is expected.
Rule shapeCol: every column has three different shapes. Violator has a duplicate shape.
Rule fillRow: every row has empty/half/filled fills. Violator has 'half' where 'filled' is expected.

How to recognize it

Scan rows for a consistent pattern; then columns; then diagonals
Once you find the rule, walk every cell and test conformance
The single cell that fails is the answer

Common mistakes

Assuming the violator is visually unusual (it may differ only in border or fill)
Spotting the rule but forgetting to test every cell
Confirming the first 'odd-looking' cell without verifying the rule

Step-by-step walkthrough

A 3×3 grid where every row should have borders [none, dashed, solid] in column order. One cell breaks this. Find it.

Read row 0: borders are none, dashed, solid — conforms
Read row 1: borders are none, none, solid — middle cell should be dashed
Click the violator

Answer: Row 1, column 1 (border 'none' where 'dashed' is required)

Learn 16

Visual Sequence Continuation

A horizontal strip shows the first three frames of a value progression — predict the fourth.

Visual sequence continuation puzzles show a 1×4 horizontal strip with three cells visible and a fourth marked '?'. A single attribute (size, fill, border, count, or rotation) progresses through a fixed value set across the cells; predict the fourth value to complete the sequence. The mechanic is the visual analog of numeric sequence completion (the literature on number-series tests dates to early 20th-century psychometrics).

cell[i] = values[i % N] for i ∈ {0, 1, 2, 3}

size: small → medium → large → ? (predict xlarge)
fill: empty → quarter → half → ? (predict filled)
border: none → dotted → dashed → ? (predict solid)
count: 1 → 2 → 3 → ? (predict 4)
rotation: 0° → 90° → 180° → ? (predict 270°)

How to recognize it

Identify which single attribute changes from frame to frame
Confirm the change is monotonic (always growing/turning in one direction)
Pick the answer that continues the progression by the same step

Common mistakes

Picking a frame that is identical to one already shown
Predicting in the wrong direction (e.g. shrinking when sequence is growing)
Confusing the rule attribute with another attribute that happens to be uniform

Step-by-step walkthrough

A 1×4 strip shows three small circles, then medium circles, then large circles. The fourth cell is '?'. What goes there?

Identify the changing attribute: size
Confirm the progression direction: ascending
Apply the next step in the progression: xlarge

Answer: Extra-large circle

Worked examples

The worked examples below preserve the original practice set at every difficulty. Open an example to compare the prompt, answer choices, correct answer, and explanation.

Example

Explanation

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