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.
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.
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
Size Change
Shading / Fill Change
Rotation
Number of Parts
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.
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.
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.
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.
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.
| 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 |
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?
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. |
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.
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.
Lesson 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.