Identify a rule describing how figures in each column are related - TACHS Ability

SECTION 1

Where Do Figure Matrices Come From?

Have you ever solved a puzzle where you had to figure out what comes next? People have been using pattern puzzles to measure thinking skills for over a hundred years. Figure matrices (grids of shapes that follow a hidden rule) are one of the most popular types. They show up on tests like the TACHS because they measure how well you can spot patterns without needing any special knowledge.

1936 • Raven's Progressive Matrices

Psychologist John Raven created the first widely used figure matrix test. It used grids of shapes with one missing piece.

1960s • Standardized School Tests

Schools started adding figure matrix questions to entrance exams. These puzzles tested reasoning, not memorized facts.

1990s • TACHS Exam Developed

The Test for Admission into Catholic High Schools (TACHS) included figure matrices in its Ability section to measure logical thinking.

Today • Modern Test Prep

Students now practice figure matrices with interactive tools and visual guides. The core skill — finding the rule — has not changed.

The big question figure matrices ask is simple: What rule connects these shapes? In this lesson, you will learn how to find that rule by looking down each column of a grid. Once you can do that, you can pick the correct missing figure every time.

SECTION 2

Core Principles of Column Rules

A figure matrix is a grid — usually 3 rows by 3 columns — filled with shapes. One cell is empty, and your job is to pick the shape that belongs there. You can find the answer by studying columns (up-and-down) or rows (left-to-right). In this lesson we focus on column rules — the pattern that connects the top figure to the middle figure to the bottom figure in the same column.

Shape Change

The shape itself may change as you go down a column. For example, a circle may become a square, then a triangle.

Size Change

The figure may get bigger or smaller from row to row. Watch for a shape that grows or shrinks step by step.

Shading / Fill Change

A shape might go from empty (white) to striped to solid (black) as you move down the column.

Rotation

An arrow or pointer may turn 90° clockwise each row. Rotation rules are common on the TACHS.

Number of Parts

The count of dots, lines, or small shapes inside a figure may increase or decrease by the same amount each row.

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KEY TAKEAWAY

Think of a column like an elevator ride. Each floor (row) changes one thing about the figure — maybe the shape, the shading, or the direction it faces. Your job is to figure out what changes at every floor so you can predict what you will see when the doors open on the last floor.

SECTION 3

Seeing the Column Rule

The diagram below shows a sample 3 × 3 figure matrix. Look at each column separately. In Column 1, the shape changes from a circle to a square to a triangle. In Column 2, the same pattern repeats. Once you see the rule, you can figure out the missing figure in Column 3.

Each column follows the same rule: circle → square → triangle. The missing figure in Column 3 must be a triangle.

Notice how the rule is the same in every column. Column 1 goes circle, square, triangle. Column 2 does the same. So Column 3 must follow the same pattern. That is the key idea: find the rule in columns you can see, then apply it to the column with the missing piece.

SECTION 4

How Column Rules Work Step by Step

Figure matrix problems do not use formulas like algebra. Instead, they rely on logical steps. Here is a simple method you can follow every time.

The 4-Step Column Method

Step 1 — Isolate one complete column. Pick a column that has all three figures showing (no question mark). Cover the rest of the grid with your hand or eyes.
Step 2 — Ask: What changes from row to row? Check shape, size, shading, rotation, and number of parts.
Step 3 — Confirm with another column. Does the same rule work in a second complete column? If yes, you found the rule.
Step 4 — Apply the rule to the missing cell. Use the pattern to predict the missing figure, then match it to the answer choices.

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Pro Tip

Sometimes the column rule involves more than one change at the same time. A shape might rotate AND change shading. Check every feature separately before deciding on your answer.

On the TACHS, you usually have four answer choices. After you predict what the missing figure should look like, scan the choices. Often two or three choices will be wrong in an obvious way — maybe the wrong shape or wrong shading. Eliminate those first, then compare the remaining options carefully.

SECTION 5

Common Column Rule Types

Not every matrix uses the same kind of rule. Below is a diagram showing five common rule types you will see on the TACHS. Study each column carefully.

Five rule types: shape change, size change, shading change, rotation, and count change.

Rule Type	What to Look For	Example Pattern
Shape	The outline of the figure changes.	Circle → Square → Triangle
Size	The figure gets bigger or smaller.	Small → Medium → Large
Shading	The fill inside changes.	Empty → Striped → Solid
Rotation	The figure turns around a center point.	Pointing up → Right → Down
Count	The number of small parts changes.	1 dot → 2 dots → 3 dots

Summary of the five most common column rules on the TACHS.

SECTION 6

Worked Example: Solving a Column Rule Problem

Let's walk through a full problem together. Imagine a 3 × 3 matrix. In Column 1, you see a small white circle, then a medium gray circle, then a large black circle. Column 2 shows a small white square, then a medium gray square, then a large black square. Column 3 shows a small white triangle, then a medium gray triangle, then a question mark. What is the missing figure?

Finding Two Rules at Once

1
Step 1 — Isolate Column 1

Look at Column 1 from top to bottom. The shape stays a circle. But two things change: the size grows (small → medium → large) and the shading darkens (white → gray → black).
2
Step 2 — Confirm with Column 2

Column 2 uses squares instead of circles. But the same two rules hold: size grows from small to large, and shading goes from white to gray to black. The rules match!
3
Step 3 — Apply to Column 3

Column 3 has triangles. Row 1 is a small white triangle. Row 2 is a medium gray triangle. Following the pattern, Row 3 should be a large black triangle.

Answer: Large black triangle
4
Step 4 — Check the Answer Choices

Scan the options for a large black triangle. If two choices look similar, compare the exact size and shade to the rest of the grid.

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Watch Out!

A common mistake is to spot only one rule and ignore a second one. In this example, if you noticed the size change but missed the shading change, you might pick a large white triangle — which would be wrong. Always check all features.

SECTION 7

Strategies, Strengths, and Pitfalls

Knowing the column method is powerful, but different strategies work better in different situations. Here is a quick comparison.

Strategy	Strength	Limitation
Column-first	Works great when changes happen from top to bottom. Catches size, shading, and count rules quickly.	May miss rules that only appear across rows.
Row-first	Good for patterns that change from left to right, like adding a shape each cell.	Can overlook vertical shading or rotation rules.
Both together	The most reliable method. Checks rows AND columns for consistency.	Takes a bit more time, but it is worth it on tough problems.

Comparing three common approaches to figure matrix problems.

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BEST PRACTICE

Think of it like checking your answers on a math test. You solved the problem one way (columns). Now double-check with the rows. If your answer fits both the column rule and the row rule, you can be very confident it is correct.

SECTION 8

From Simple Rules to Combined Rules

On easier TACHS questions, only one thing changes in each column. On harder questions, two or even three things change at the same time. The table below shows how difficulty increases.

Difficulty	Number of Changes	Example
Easy	1 change	Shape stays the same; only shading changes.
Medium	2 changes	Shape changes AND size changes.
Hard	3 changes	Shape changes, shading changes, AND rotation changes.

Don't panic when you see a complex matrix. Just use the same 4-step method. Check each feature one at a time: shape, then size, then shading, then rotation, then count. Write down what you notice on scratch paper if it helps. Breaking a hard problem into small checks makes it manageable.

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Looking Ahead

Some advanced reasoning tests use 4 × 4 or even 5 × 5 matrices. The same column-rule strategy applies — there are just more cells to examine. Mastering 3 × 3 matrices now prepares you for any size grid in the future.

SECTION 9

Practice Problems

Try these five problems. Start with the easier ones and work your way up. For each one, identify the column rule before picking your answer.

Question 1: CONCEPTUAL

In a 3 × 3 matrix, every column shows the same three shapes in the same order: pentagon in Row 1, hexagon in Row 2, and octagon in Row 3. The bottom cell of Column 3 is missing. What shape belongs there?

Show answer

The missing shape is an octagon. The column rule is that the number of sides increases: 5 sides (pentagon), 6 sides (hexagon), 8 sides (octagon). Since this rule repeats in every column, Column 3 must end with an octagon.

Question 2: BASIC CALCULATION

A column in a figure matrix shows 1 dot in Row 1, 3 dots in Row 2, and a question mark in Row 3. The dots increase by the same number each row. How many dots should appear in Row 3?

Show answer

There should be 5 dots. From Row 1 to Row 2, the count increased by 2 (1 + 2 = 3). Applying the same increase, Row 3 should have 3 + 2 = 5 dots. The column rule is: add 2 dots each row.

Question 3: INTERMEDIATE

In a matrix, Column 1 shows: a small white circle, a medium gray circle, a large black circle. Column 2 shows: a small white triangle, a medium gray triangle, and a question mark. Column 3 shows: a small white square, a question mark is NOT here but all three are present. What figure belongs in the missing cell of Column 2?

Show answer

The missing figure is a large black triangle. Two rules work at the same time in each column: size increases (small → medium → large) and shading darkens (white → gray → black). Column 2 follows triangles, so the bottom cell must be a large black triangle. Always check both rules before choosing.

Question 4: APPLIED

You see a matrix where each column has an arrow. In Column 1, the arrow points up in Row 1, right in Row 2, and down in Row 3. In Column 2, the arrow points up in Row 1 and right in Row 2. The Row 3 cell in Column 2 is missing. Also, each arrow in Column 2 is thicker than the one above it. What does the missing figure look like?

Show answer

The missing figure is a thick arrow pointing down. The rotation rule is: the arrow turns 90° clockwise each row (up → right → down). The thickness rule is: the arrow gets thicker with each row. Combining both rules, Row 3 in Column 2 must be the thickest arrow and it must point downward. Remember to check every feature — rotation and thickness — before selecting your answer.

Question 5: CRITICAL THINKING

In a tricky matrix, Column 1 shows a white star with 3 inner dots, then a gray star with 2 inner dots, then a black star with 1 inner dot. Column 2 shows a white diamond with 3 inner dots, then a gray diamond with 2 inner dots, then a question mark. A classmate says the answer is a black diamond with 3 inner dots. Is your classmate correct? Explain why or why not.

Show answer

Your classmate is incorrect. The answer should be a black diamond with 1 inner dot. There are two column rules: shading darkens each row (white → gray → black) and the number of inner dots decreases by 1 each row (3 → 2 → 1). Your classmate correctly identified the shading rule but missed the count rule. The dot count should continue to decrease to 1, not reset to 3. This problem shows why you must check every feature — shape, shading, and count — before choosing your answer.

SUMMARY

Lesson Summary

Summary

A figure matrix is a grid of shapes where one cell is missing. To find the missing figure, look for a column rule — a pattern that repeats from top to bottom. Common rules involve changes in shape, size, shading, rotation, or count of parts.

Use the 4-step column method: isolate a complete column, identify what changes, confirm the rule in another column, and apply it to the missing cell. On harder questions, watch for multiple rules working at the same time. Always check every feature — and double-check your answer against both columns and rows for full confidence.