Use transformations (addition/subtraction of elements, rotation, shading) - TACHS Ability

SECTION 1

Why Do We Study Pattern Puzzles?

People have been studying patterns for thousands of years. Figure matrices (grids of shapes that follow a rule) grew out of the idea that you can measure how well someone reasons without using words or numbers. Scientists wanted a fair way to test thinking skills no matter what language you speak.

1936 • Raven's Progressive Matrices

John C. Raven created a test using grids of shapes. Each grid followed a hidden rule, and the test-taker had to pick the missing piece. This test became one of the most popular reasoning tests in the world.

1950s • School Entrance Exams Adopt Matrices

Schools began using figure-matrix questions on entrance exams. These puzzles helped measure a student's ability to think logically without relying on reading or math skills.

1970s • Transformation Rules Are Classified

Researchers grouped the rules in figure matrices into categories: adding or removing shapes, rotating shapes, and changing shading. This made it easier to teach students how to solve them.

Today • TACHS and Modern Ability Tests

The TACHS (Test for Admission into Catholic High Schools) uses figure matrices to measure your visual reasoning. Knowing how transformations work gives you a real advantage on test day.

So here is the big question these puzzles answer: Can you figure out the hidden rule that connects a set of shapes, and then use that rule to predict what comes next? Let's learn exactly how to do that.

SECTION 2

Core Principles of Transformations

A transformation is any change that happens to a shape as you move across a row or down a column in a figure matrix. There are three main types you need to know for the TACHS.

Addition / Subtraction of Elements

Shapes or parts are added to or removed from the figure as you move across a row or down a column. For example, a circle might gain an extra line in each cell.

Rotation

A shape turns by a set amount (like 45°, 90°, or 180°) from one cell to the next. The shape stays the same size and form — it just spins.

Shading Changes

The fill of a shape changes in a pattern. It might go from empty → striped → solid, or the shaded area might move around the figure.

Combined Transformations

Many TACHS questions use two or more rules at once. A shape might rotate AND gain a new element at the same time. Always check for multiple changes.

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KEY TAKEAWAY

Think of a figure matrix like a recipe with steps. Each cell is a step in the recipe. If Step 1 is "put dough on a pan" and Step 2 is "add sauce," then Step 3 must be "add cheese" — because each step adds one new ingredient. In a matrix, each cell adds, removes, rotates, or re-shades something following the same kind of steady rule.

SECTION 3

Seeing Transformations in Action

The diagram below shows a 3 × 3 figure matrix. Look at each row from left to right. In Row 1, a new line is added inside the circle each time. In Row 2, the arrow rotates 90° clockwise. In Row 3, the square's shading changes from empty to half-filled to fully filled. The missing piece is in Row 3, Column 3.

Row 1 shows addition (one new line per cell). Row 2 shows rotation (the triangle turns 90° each time). Row 3 shows shading (empty → half-filled → ?). The missing piece must be a fully filled square.

When you look at a figure matrix on the TACHS, always scan across each row first, then down each column. Ask yourself: "What changed between Cell 1 and Cell 2?" Then check if the same change happens between Cell 2 and Cell 3. If it does, you have found the rule!

SECTION 4

How Each Transformation Works

Addition and Subtraction of Elements

In an addition pattern, a new piece appears in each cell. It could be a dot, a line, a small shape, or an entire new figure layered on top. Count the number of elements in Cell 1, then in Cell 2. If Cell 2 has one more element, you can predict that Cell 3 will have one more still.

In a subtraction pattern, a piece disappears in each cell. The figure gets simpler as you move across the row. Sometimes one row uses addition while another row uses subtraction — so check every row independently.

Rotation

When a shape rotates, it spins around a center point. The most common rotation amounts on the TACHS are 45°, 90°, and 180°. A 90° turn is like turning a book so the top faces to the right. A 180° turn flips it upside down. Pay attention to whether the turn is clockwise (like the hands on a clock) or counterclockwise (opposite direction).

Shading Changes

Shading transformations change the fill of a shape. Common shading patterns include: empty → half-shaded → fully shaded, or a cycle where the shading moves to a different section of the shape. Sometimes shading alternates: shaded → empty → shaded. Think of it like traffic lights cycling through colors in a set order.

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Pro Tip

On the TACHS, many questions combine two transformations at once. A shape might rotate AND change its shading at the same time. If you only notice one change, look again — there may be a second rule hiding in the same row.

SECTION 5

A Closer Look at Each Transformation Type

The diagram below breaks down the three main transformation types side by side. Each column shows how one cell leads to the next. Study the arrows between the figures to see exactly what changes.

Three columns show how each transformation type progresses. Addition adds one dot per step. Rotation turns the arrow 90° clockwise per step. Shading increases the fill by half per step.

Transformation	What to Count or Track	Common Patterns
Addition	Number of lines, dots, or shapes in each cell	+1 per cell, +2 per cell, or elements appear in new positions
Subtraction	Number of elements that disappear each step	−1 per cell, outermost layer removed, smallest shape removed
Rotation	Direction a shape or line is pointing	45° CW, 90° CW, 90° CCW, 180° flip
Shading	Fill level or shaded region of each shape	Empty → striped → solid, shading moves to next quadrant

Quick-reference guide for spotting each transformation type

SECTION 6

Worked Example: Solving a Figure Matrix

Imagine a 2 × 2 figure matrix (with the bottom-right cell missing). Cell 1 (top-left) has a pentagon with one shaded section and a small dot inside. Cell 2 (top-right) has the same pentagon rotated 90° clockwise, with the shaded section in a new position and two dots inside. Cell 3 (bottom-left) has a hexagon with one shaded section and one dot inside. What does Cell 4 (bottom-right) look like?

Finding the Missing Figure

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Step 1 — Identify the row rule

Compare Cell 1 to Cell 2 (top row). The pentagon rotated 90° clockwise, the shaded section moved with the rotation, and the number of dots increased from 1 to 2. So the row rule has two parts: rotate 90° CW and add one dot.

Row rule: rotate 90° CW + add 1 dot
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Step 2 — Check the column rule

Compare Cell 1 to Cell 3 (left column). The shape changed from a pentagon to a hexagon, but the number of dots stayed the same (1 dot) and the shading stayed in the same relative position. So the column rule is: change the shape (add one side) while keeping everything else the same.

Column rule: add one side to the shape
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Step 3 — Apply both rules to find Cell 4

Starting from Cell 3 (hexagon, 1 dot, original position), apply the row rule: rotate the hexagon 90° clockwise and add 1 dot. Or, starting from Cell 2 (pentagon rotated, 2 dots), apply the column rule: change the pentagon to a hexagon (add one side).

Cell 4 = hexagon, rotated 90° CW, with 2 dots inside and the shaded section in the rotated position.
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Step 4 — Verify with answer choices

Look at the answer choices. Eliminate any option that shows a pentagon (wrong shape), any option with only 1 dot (wrong count), and any option where the hexagon is not rotated. The correct answer is the one that matches all three changes: hexagon + rotated + 2 dots.

Always double-check by reading both the row AND the column.

SECTION 7

Strategies, Strengths, and Common Mistakes

Strategy	Why It Helps	Watch Out For
Scan rows first, then columns	Most matrices set the main rule across the row. Checking rows first gives you the fastest path to the answer.	Some matrices have a rule that only shows up in the columns. If the row doesn't show a clear pattern, switch to columns.
Count elements in each cell	Counting reveals addition or subtraction patterns instantly.	Don't just count total elements — also note their positions. Two dots on the left side is different from two dots on the right.
Look for a 'pointer' on the shape	A small notch, arrow, or asymmetric feature tells you when a shape has been rotated.	If a shape is symmetric (like a plain circle), rotation alone won't produce a visible change. The rule may be shading instead.
Use elimination	On the TACHS, you only have 4 answer choices. Ruling out even one wrong option makes guessing easier.	Be careful — some wrong answers are 'close' on purpose. They might match one rule but break another.

Helpful strategies and common pitfalls for figure-matrix questions

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KEY TAKEAWAY

Think of solving a figure matrix like solving a Sudoku puzzle. You look at what's already there, figure out the rules, and use those rules to fill in the blank. The more you practice spotting patterns, the faster you'll get — and speed matters on the TACHS!

SECTION 8

From Simple Matrices to Complex Reasoning

The transformations you learn for the TACHS are the building blocks of more advanced visual reasoning. As questions get harder, you'll see bigger matrices (3 × 3 instead of 2 × 2) and rules that combine three or more transformations at once.

Feature	Basic TACHS Matrix	Advanced Matrix
Grid size	2 × 2 or 3 × 3	3 × 3 or larger
Number of rules	1–2 per row	2–3 per row, sometimes different rules per column
Transformation types	Add/subtract, rotate, shade	Add/subtract, rotate, shade, reflection, overlapping, XOR logic
Distractor answers	Clearly wrong or match only 1 rule	Match 2 out of 3 rules — very tricky

How matrices increase in complexity

Don't worry about the advanced level right now. If you master the three core transformations — addition/subtraction, rotation, and shading — you'll have the foundation for all future visual reasoning tasks.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A row of three figures shows a triangle with 1 star inside, then a triangle with 2 stars inside, then a missing figure. What transformation is being used, and what should the missing figure look like?

Show answer

The transformation is addition. One star is added inside the triangle in each step. Cell 1 has 1 star, Cell 2 has 2 stars, so Cell 3 must have 3 stars inside the same triangle. The triangle itself doesn't change — only the number of stars increases.

Question 2: BASIC CALCULATION

In a figure matrix, an arrow points UP in Cell 1 and points to the RIGHT in Cell 2. If the same rotation rule continues, which direction does the arrow point in Cell 3? Is the rotation clockwise or counterclockwise?

Show answer

The arrow moved from UP to RIGHT, which is a 90° clockwise rotation. Continuing the same rule, the arrow in Cell 3 rotates another 90° clockwise from RIGHT, so it now points DOWN. The rotation is clockwise because UP → RIGHT follows the same direction as clock hands.

Question 3: INTERMEDIATE

A 3 × 3 matrix has this pattern in Row 1: a square with a black top-left quarter, then a square with a black top-right quarter, then a square with a black bottom-right quarter. What is the shading rule? If Row 2 starts with a circle that has a black left half, what would the next two circles look like?

Show answer

The shading rule is that the shaded area rotates 90° clockwise around the shape. In the square: top-left → top-right → bottom-right. Applying the same 90° clockwise shading rule to the circle: the black left half moves to become a black top half in Cell 2, then a black right half in Cell 3. Even though the shapes are different, the shading rule stays the same across rows.

Question 4: APPLIED

In a 2 × 2 matrix: Cell 1 (top-left) shows a white diamond with 2 horizontal lines inside. Cell 2 (top-right) shows a gray diamond with 3 horizontal lines inside. Cell 3 (bottom-left) shows a white hexagon with 2 horizontal lines inside. What does Cell 4 (bottom-right) look like? Identify every transformation at work.

Show answer

There are three transformations. First, shading: the shape changes from white to gray across the row. Second, addition: one horizontal line is added (2 → 3). Third, the shape changes down the column from diamond to hexagon. Applying all rules: Cell 4 is a gray hexagon with 3 horizontal lines inside.

Question 5: CRITICAL THINKING

A 3 × 3 matrix has this pattern: across each row, a shape rotates 90° clockwise and one element is removed. Down each column, the shape's shading cycles through empty → striped → solid. If Cell 7 (Row 3, Column 1) is a solid right-pointing arrow with 4 dots, and Cell 8 (Row 3, Column 2) follows the row rules, describe Cell 8 completely. Then explain how you would verify your answer using the column rule.

Show answer

Applying the row rule to Cell 7: the right-pointing arrow rotates 90° clockwise to become a downward-pointing arrow, and one dot is subtracted, leaving 3 dots. The shading stays solid because the row rule doesn't change shading — shading is the column rule. So Cell 8 is a solid downward arrow with 3 dots. To verify, check Column 2. Cell 2 should be empty, Cell 5 should be striped, and Cell 8 should be solid. If the shading matches the column cycle, your answer is correct. This cross-checking method helps catch mistakes.

SUMMARY

Putting It All Together

Summary

Figure matrices test your ability to spot transformations — the rules that change shapes from one cell to the next. The three main types are addition or subtraction of elements (counting dots, lines, or shapes), rotation (turning a shape by a fixed angle like 45°, 90°, or 180°), and shading changes (empty → half-filled → solid, or shading that moves around the shape).

To solve a TACHS matrix, scan rows first to find the pattern, then check columns to confirm it. Watch for combined transformations — many questions use two rules at once. Use elimination to rule out answer choices that break any of the rules you found. Practice makes these patterns easier to spot quickly on test day!