Select the figure that completes a matrix using row/column rules - TACHS Ability

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Where Do Figure Matrices Come From?

Have you ever solved a jigsaw puzzle where you had to figure out the missing piece? Figure matrices work the same way. They are grids of shapes that follow a hidden rule, and your job is to find the shape that completes the pattern. Psychologists (scientists who study how the mind works) created these puzzles over 100 years ago to measure how well people spot patterns.

1904 • Spearman's "g" Factor

Charles Spearman proposed that people have a general reasoning ability. He wanted a way to measure it without relying on words or numbers.

1936 • Raven's Progressive Matrices

John C. Raven published a test of figure matrices. Each question showed a grid with a missing piece, and the test-taker chose the best answer from several options.

1960s • Standardized School Testing

Schools began using matrix puzzles on entrance exams like the TACHS to see how well students reason with visual patterns.

Today • TACHS Ability Section

Figure matrix questions appear in the Ability section of the TACHS. They test your pattern-recognition skills without requiring any math formulas.

The big question these puzzles answer is: Can you find a rule that explains how shapes change across a row or down a column? That is exactly the skill you will build in this lesson.

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Core Principles of Figure Matrices

Before you dive into solving, you need to know the building blocks. Every figure matrix follows rules that control how shapes look from one cell to the next. Here are the main ideas to keep in mind.

Row Rule

A pattern that repeats or changes as you read left to right across a single row — just like reading a sentence.

Column Rule

A pattern that repeats or changes as you read top to bottom down a single column — like looking at a list.

Shape Properties

The features you should watch: shape type (circle, square, triangle), size (small, medium, large), shading (empty, striped, filled), and count (how many shapes).

Multiple Rules at Once

A harder matrix may change two or more properties at the same time. For example, shape type changes across the row while shading changes down the column.

Elimination

When in doubt, cross out answer choices that break the rule you found. This narrows your options fast.

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KEY TAKEAWAY

Think of a figure matrix like a playlist with a pattern. If Songs 1, 2, and 3 are rock, pop, rock, you would guess Song 4 is pop. In a matrix, the "songs" are shapes, and the "genres" are their properties — shape, size, shading, and count.

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Seeing the Pattern — A Visual Guide

The diagram below shows a 3 × 3 figure matrix. Each row follows a shape rule (circle → square → triangle), and each column follows a shading rule (empty → striped → filled). The bottom-right cell has a question mark — that is the one you need to solve.

Each row changes shape type (circle → square → triangle). Each column changes shading (empty → striped → filled). The answer to the missing cell must be a filled triangle.

Notice how the diagram has two rules running at the same time. The row rule controls which shape appears (circle, square, triangle). The column rule controls the shading (empty, striped, filled). When you combine both rules, there is only one answer that works.

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How the Rules Work

Figure matrix questions do not require math formulas, but they do follow a logical system. Think of each cell in the matrix as having a set of properties (features). The most common properties are listed below.

Common Properties to Track

Shape type — circle, square, triangle, star, pentagon, etc.
Size — small, medium, large. Sometimes size grows or shrinks step by step.
Shading or fill — empty (outline only), striped (lines inside), or solid (completely filled).
Count — the number of shapes in each cell (1, 2, 3, …).
Rotation — a shape may turn 90° (quarter turn), 180° (half turn), or 270° (three-quarter turn) from cell to cell.
Position inside the cell — a shape might move from left to center to right.

Common Rule Types

Rule Type	What Happens	Example
Cycling	A set of values repeats in order.	Circle → Square → Triangle, then back to Circle.
Progression	A property steadily increases or decreases.	1 dot → 2 dots → 3 dots.
Rotation	A shape turns by the same angle each step.	Arrow at 0° → 90° → 180°.
Addition / Overlay	Elements from column 1 and column 2 combine to form column 3.	A circle plus a triangle stacked together.
Distribution	Each value appears exactly once in every row and column.	Small, medium, and large each show up once per row.

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Pro Tip

Start by checking the row that is most complete (usually the first row). Once you spot a rule there, see if the same rule applies to the other rows. Then do the same for columns.

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Step-by-Step Strategy

When you face a figure matrix on the TACHS, follow this step-by-step approach. The diagram below turns these steps into a flowchart you can picture in your head during the test.

Follow this flowchart from top to bottom every time you see a figure matrix question. Practicing these six steps will make you faster and more accurate.

Let's break down each step. In Step 1, you glance at the entire grid so you get a general feel for the shapes involved. In Step 2, you mentally list the properties that seem to change — shape type? shading? size? In Steps 3 and 4, you test each complete row and column to find the rule. In Step 5, you combine what you learned to predict what the missing cell should look like. Finally, in Step 6, you pick the answer choice that matches your prediction.

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Worked Example

Let's walk through a full example together. Imagine a 3 × 3 matrix where the missing cell is in the bottom-right corner. Here is what each cell contains:

Column 1	Column 2	Column 3
1 small empty circle	2 small empty circles	3 small empty circles
1 medium striped circle	2 medium striped circles	3 medium striped circles
1 large filled circle	2 large filled circles	?

The matrix described in words

Find the Missing Figure

1
Step 1 — Scan the Matrix

All shapes are circles. So shape type does not change — it stays a circle everywhere.
2
Step 2 — List Properties That Change

Three things change: count (1, 2, 3), size (small, medium, large), and shading (empty, striped, filled).
3
Step 3 — Check Rows

Row 1: count goes 1 → 2 → 3. Row 2: count goes 1 → 2 → 3. Row 3: count goes 1 → 2 → ?. The row rule for count is a progression from 1 to 3.

Count in the missing cell = 3
4
Step 4 — Check Columns

Column 1: size goes small → medium → large. Column 1 shading goes empty → striped → filled. Column 3 should follow the same pattern. Row 3 already has large and filled in column 1, so column 3, row 3 should also be large and filled.

Size = large, Shading = filled
5
Step 5 — Predict and Match

Combine all the properties: 3 large filled circles. Look at the answer choices and pick the one that shows exactly that.

Answer: 3 large filled circles

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Common Tips and Traps

Even when you know the strategy, certain traps can trip you up on the TACHS. The table below compares good habits with common mistakes.

Good Habit ✅	Common Mistake ❌
Check BOTH rows and columns before choosing.	Only check rows and ignore columns (or vice versa).
Track each property separately (shape, size, shading, count).	Look at the whole picture at once and guess based on a feeling.
Eliminate wrong answer choices one at a time.	Pick the first answer that looks "close enough."
Verify your answer fits EVERY row and EVERY column.	Check one row and assume the answer is right.
Stay calm and follow the 6-step strategy.	Rush through the question because it looks hard.

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KEY TAKEAWAY

Think of a figure matrix like a Sudoku puzzle. In Sudoku, every row and column must contain each number exactly once. In a figure matrix, every row and column must follow a pattern. If your answer breaks the pattern in any direction, it is wrong.

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Connecting to Harder Puzzles

Once you master 3 × 3 matrices, you are ready for trickier variations. The table below shows how basic matrices compare to more advanced versions you might encounter on harder tests or in future grades.

Feature	Basic Matrix (TACHS Level)	Advanced Matrix
Grid size	2 × 2 or 3 × 3	4 × 4 or larger
Number of rules	1 or 2 properties change	3 or more properties change at once
Rule types	Cycling and progression	Overlay, subtraction, XOR logic
Answer choices	4 choices, usually only 1 is close	6–8 choices, several look similar

The good news is that the strategy you learned here — scan, list properties, check rows, check columns, predict, match — works for all levels of figure matrices. If you build this habit now, you'll be ahead of the game on any future reasoning test.

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Practice Problems

Try these five problems. Each one gets a little harder. After you think of your answer, check the solution below.

Question 1: CONCEPTUAL

A 3 × 3 matrix has circles in row 1, squares in row 2, and triangles in row 3. All shapes in column 1 are small, all in column 2 are medium, and all in column 3 are large. The cell at row 3, column 3 is missing. What should it be?

Show answer

The answer is a large triangle. The row rule tells us row 3 uses triangles. The column rule tells us column 3 uses large shapes. Combining both, the missing figure must be a large triangle. Always check both directions to be sure.

Question 2: BASIC CALCULATION

In a 3 × 3 matrix, the count of dots across each row is 1, 2, 3. Down each column the shading is empty, striped, filled. The cell at row 2, column 3 is missing. How many dots does it have, and what is its shading?

Show answer

The answer is 3 striped dots. Row 2 tells us the count goes 1 → 2 → 3, so column 3 gets 3 dots. Column 3 shading goes empty (row 1) → striped (row 2) → filled (row 3), so row 2 is striped. Putting it together: 3 striped dots.

Question 3: INTERMEDIATE

A 3 × 3 matrix has two rules. Across each row the shape rotates 90° clockwise each step. Down each column the shape changes from an arrow to a cross to a star. Row 1 starts with an arrow pointing up. What does the cell at row 2, column 2 look like?

Show answer

Row 2 uses crosses (column rule). Column 2 means the shape has been rotated 90° once from its starting position (row rule). So the answer is a cross rotated 90° clockwise from its starting position. When tracking rotation, mark the orientation of each shape carefully in the completed cells to confirm the rotation amount is consistent.

Question 4: APPLIED

You see a 3 × 3 matrix on the TACHS. Each cell has a shape inside a frame. In row 1 the shapes are a small empty circle, a small empty square, and a small empty triangle. In row 2 the shapes are a medium striped circle, a medium striped square, and a medium striped triangle. Row 3 shows a large filled circle and a large filled square, but column 3 is missing. Four answer choices are: (A) large filled triangle, (B) large empty triangle, (C) medium filled triangle, (D) small striped triangle. Which do you choose?

Show answer

The answer is (A) large filled triangle. The row rule tells us row 3 should have shapes that are large and filled. The column rule tells us column 3 uses triangles. Choice (B) has the wrong shading (empty instead of filled). Choice (C) has the wrong size (medium instead of large). Choice (D) has the wrong size and wrong shading. Only (A) matches every rule.

Question 5: CRITICAL THINKING

A 3 × 3 matrix uses an overlay rule: the figure in column 3 is formed by combining the shapes from columns 1 and 2. Row 1 has a circle in column 1, a square in column 2, and a circle with a square inside it in column 3. Row 2 has a triangle in column 1, a diamond in column 2, and a triangle with a diamond inside it in column 3. Row 3 has a pentagon in column 1, a star in column 2, and the missing cell in column 3. What should be in the missing cell, and how did you figure it out?

Show answer

The answer is a pentagon with a star inside it. The rule here is overlay (combination): column 3 always places the column 2 shape inside the column 1 shape. You can confirm this by checking rows 1 and 2, where this rule holds perfectly. For row 3, place the star inside the pentagon. This kind of rule is trickier because you have to notice that two shapes merge into one rather than just changing a single property.

SUMMARY

Summary

A figure matrix is a grid of shapes that follows hidden row rules and column rules. To find the missing figure, track shape type, size, shading, count, and rotation across each row and down each column. Common rule types include cycling, progression, rotation, overlay, and distribution.

Use the 6-step strategy: scan the matrix, list properties, check rows, check columns, predict the answer, then match it to the choices. Always eliminate wrong answers by testing them against every row and column. With practice, these puzzles become fast and fun!