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

SECTION 1

Why Do We Use Figure Matrices?

Have you ever tried to figure out the next picture in a pattern? That's exactly what figure matrices ask you to do. A figure matrix is a grid of pictures, usually organized in rows and columns. One picture is missing, and your job is to find the rule that connects the figures in each row. Then you use that rule to pick the correct missing piece.

People have been studying pattern recognition (the ability to find rules in visual information) for over a hundred years. Psychologists realized that spotting patterns is one of the best ways to measure reasoning skills. Here's a quick look at how this idea developed.

1904 • Spearman's "g Factor"

Charles Spearman proposed that a single general intelligence factor connects all mental abilities. Pattern puzzles became a key way to measure it.

1938 • Raven's Progressive Matrices

John C. Raven created a famous test using figure matrices. Test-takers had to find rules across rows and columns to pick the missing figure.

1960s • Standardized School Tests

Figure matrix questions started appearing on school aptitude exams because they test reasoning without relying on reading or math skills.

Today • TACHS and Other Admissions Tests

The TACHS (Test for Admission into Catholic High Schools) uses figure matrices in its Ability section to measure how well students recognize visual patterns.

The big question these puzzles answer is simple: can you figure out a rule just by looking at pictures? Let's learn exactly how to do that.

SECTION 2

Core Principles of Row Rules

Every figure matrix follows at least one rule. A rule is a pattern that tells you how the figures change from left to right in each row. Once you name the rule, you can predict what the missing figure should look like. Here are the main types of rules you'll see.

Shape Change

The shape itself changes across the row. For example, circle → triangle → square. Look at the number of sides or the overall form.

Size Change

The figure gets bigger or smaller as you move across the row. It might grow steadily, or it might alternate between large and small.

Shading or Fill

The fill pattern changes: empty → striped → solid, or white → gray → black. Sometimes shading rotates through a cycle.

Rotation or Flip

The figure turns (rotates) a certain number of degrees or flips over an invisible line. Common rotations are 90° and 180°.

Number of Elements

The count of objects inside the figure changes. You might see one dot, then two dots, then three dots across a row.

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KEY TAKEAWAY

Think of a figure matrix like a recipe. Each row follows the same "recipe" of changes. If you can figure out the recipe for the complete rows, you can cook up the missing figure yourself. It's just like noticing that every sandwich at lunch has bread, meat, then cheese — if the cheese is missing, you know what goes there!

SECTION 3

Seeing the Pattern: A Visual Guide

The diagram below shows a 3 × 3 figure matrix. Study it carefully. Each row follows the same two rules: the shape changes (circle → triangle → square) and the shading changes (empty → striped → solid). The bottom-right cell is missing — that's the one you need to figure out.

Each row follows the same two rules: shape changes from circle to triangle to square, and shading changes from empty to striped to solid. The missing figure must be a solid square.

Notice how each row repeats the same pattern. Row 1 and Row 2 are complete, so you can check your rule against both of them. If the rule works for both complete rows, you can trust it for Row 3.

SECTION 4

How to Find the Rule — Step by Step

Finding the rule is like being a detective. You gather clues from the complete rows and use them to solve the mystery of the missing figure. Here is a simple system you can follow every time.

The SCAN Method

S — Shape

Look at the shape in each cell. Does it stay the same, or does it change? Count the sides if you're not sure.

C — Color / Shading

Check the fill. Is the figure empty, shaded, striped, or solid? Does the color or darkness change across the row?

A — Angle / Position

See if the figure rotates, flips, or moves to a different spot inside the cell. Track the direction it points.

N — Number

Count any elements inside or around the figure — dots, lines, small shapes. Does the count go up, down, or stay the same?

Run through SCAN for each complete row. Write down what stays the same and what changes. Most TACHS questions involve one or two of these features changing at the same time. The features that change are the rule.

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Pro Tip

Always check your rule against every complete row, not just one. If it works for all complete rows, you've found the real rule. If it only works for one row, keep looking!

SECTION 5

Common Rule Types in Detail

Let's look at the most common rules you'll see on the TACHS, with a visual example for each one. The diagram below shows five mini-rows. Each mini-row demonstrates a different rule type.

Five mini-rows showing the most common figure matrix rules: shape change, size change, shading change, rotation, and number change.

Rule Type	What to Look For	Example Clue
Shape Change	Number of sides or overall form	3 sides → 4 sides → 5 sides
Size Change	Figure gets larger or smaller	Small → Medium → Large
Shading/Fill	Darkness or pattern of fill	White → Gray → Black
Rotation	Direction or angle the figure points	Pointing up → right → down
Number of Elements	Count of dots, lines, or small shapes	1 dot → 2 dots → 3 dots

Quick reference for the five most common rule types

SECTION 6

Worked Example: Solving a Figure Matrix

Let's walk through a complete problem. Imagine a 3 × 3 matrix where each row has an arrow. In Row 1 the arrow points up, then right, then down. In Row 2 the arrow points up, then right, then down. In Row 3 the arrow points up, then right, then the third cell is missing. What goes in the missing cell?

Finding the Missing Arrow

1
Step 1 — Look at Row 1

The arrow starts pointing up, then turns to point right, then turns again to point down. Each step, the arrow rotates 90° clockwise.

Rule guess: rotate 90° clockwise each step
2
Step 2 — Check Row 2

The arrow in Row 2 also goes: up → right → down. That's the same 90° clockwise rotation. Our rule holds for both complete rows.

Rule confirmed: 90° clockwise rotation
3
Step 3 — Apply to Row 3

Row 3 starts with an arrow pointing up, then an arrow pointing right. One more 90° clockwise turn from "right" gives us "down."

Answer: the missing figure is an arrow pointing down
4
Step 4 — Pick the Answer Choice

Look at the answer choices and find the arrow pointing down. Eliminate any arrows that point up, left, or right — those don't match the rule.

Select the down arrow ↓

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KEY TAKEAWAY

Think of this process like following a dance routine. First you watch someone do the full routine (complete rows). Then you copy the moves yourself (apply the rule to the incomplete row). If the dance always goes "step-spin-clap," you know the next move after "step-spin" must be "clap."

SECTION 7

Strategies, Strengths, and Pitfalls

Some strategies work better than others when you're under time pressure on a test. The table below compares helpful approaches with common mistakes.

Good Strategy ✓	Common Mistake ✗
Check the rule against every complete row before choosing.	Only looking at one row and assuming it works for all.
Use SCAN to check Shape, Color, Angle, and Number systematically.	Focusing on only one feature and missing a second rule.
Eliminate answer choices that break the rule.	Picking the first answer that looks right without checking.
Look at columns too — they often follow their own pattern.	Ignoring columns completely and missing helpful clues.
Stay calm and move on if stuck — come back later.	Spending too long on one question and running out of time.

Comparison of effective strategies vs. common pitfalls

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DON'T FORGET

Many TACHS figure matrix questions involve TWO rules at the same time (for example, shape changes AND shading changes). Always run through the full SCAN checklist so you don't miss a second rule hiding in plain sight.

SECTION 8

Connecting Row Rules to Column and Diagonal Rules

So far we've focused on rules that run across each row. But figure matrices can also follow rules down each column, or even along the diagonal. On the TACHS, row rules are the most common, but knowing about columns and diagonals can help you double-check your answer.

Direction	How It Works	When to Use It
Row Rule	Figures change left to right within each row.	Start here first — most common on TACHS.
Column Rule	Figures change top to bottom within each column.	Use to double-check your answer or if row rules are unclear.
Diagonal Rule	Figures on the diagonal share a feature (e.g., all solid).	Rare on the TACHS, but good to know for tricky problems.

As you move into high school and take more advanced reasoning tests, you'll see matrices with more cells, more rules, and trickier patterns. But the basic skill — finding a rule by comparing what changes and what stays the same — stays exactly the same. Master it now, and you'll be ready for anything.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In a figure matrix, what does it mean to "identify the rule" for a row?

Show answer

Identifying the rule means describing exactly how the figures change from one cell to the next within a row. You look at features like shape, size, shading, rotation, and number of elements, then state which features change and how they change. For example, "the shape gains one more side each step" is a rule. A good rule works for every complete row in the matrix, not just one.

Question 2: BASIC CALCULATION

A row shows: a small circle, then a medium circle, then a missing figure. In the other complete rows, every row follows the same rule: the circle gets bigger each step. What is the missing figure?

Show answer

The missing figure is a large circle. The rule is a size change — the circle grows from small to medium to large. Since the other complete rows follow the same pattern (small → medium → large), you can be confident the answer is a large circle. The shape stays the same; only the size increases.

Question 3: INTERMEDIATE

A row contains: a white triangle pointing up, a gray triangle pointing right, and a missing figure. The other rows follow the same pair of rules. What two rules are at work, and what is the missing figure?

Show answer

Two rules are working at the same time. First, the triangle rotates 90° clockwise each step (up → right → down). Second, the shading gets darker each step (white → gray → black). The missing figure must be a black triangle pointing down. Always check both rules against the other complete rows to make sure they match before choosing your answer.

Question 4: APPLIED

You see a 3 × 3 matrix. Row 1: square with 1 dot inside, square with 2 dots inside, square with 3 dots inside. Row 2: circle with 1 dot, circle with 2 dots, circle with 3 dots. Row 3: triangle with 1 dot, triangle with 2 dots, and a missing figure. What is the missing figure, and how do you know?

Show answer

The missing figure is a triangle with 3 dots inside. The row rule is that the number of dots increases by one each step (1 → 2 → 3). The shape stays the same across each row. You can also check the columns: Column 1 is always the shape with 1 dot, Column 2 always has 2 dots, and Column 3 always has 3 dots. The column rule confirms the answer because Column 3, Row 3 should have 3 dots inside a triangle.

Question 5: CRITICAL THINKING

A student looks at a matrix and says, "The rule is that every figure in the row is different." Is this a useful rule? Why or why not? What makes a rule strong enough to predict the missing figure?

Show answer

Saying "every figure is different" is too vague to be a useful rule. A good rule must be specific enough to predict exactly what the missing figure looks like. It should describe HOW the figures are different — for example, "the shape adds one side each step" or "the shading goes from light to dark." A vague rule could match many answer choices, so it won't help you pick the right one. A strong rule is testable: you can apply it to every complete row and get a match, and it narrows the missing figure down to one specific answer.

SUMMARY

Putting It All Together

Summary

A figure matrix is a grid of pictures where one picture is missing. To find the missing piece, you identify the rule that describes how figures in each row are related. Use the SCAN method to check four features: Shape, Color/shading, Angle/position, and Number of elements. The five most common rules involve shape change, size change, shading change, rotation, and counting changes.

Always test your rule against every complete row before picking an answer. Watch out for problems with two rules at the same time — run through the full SCAN checklist so you don't miss anything. You can also check column patterns to double-check your answer. With practice, these steps become fast and automatic!