Track folds of a square piece of paper through a sequence of folds - TACHS Ability

SECTION 1

Where Did Paper-Folding Questions Come From?

People have been folding paper for hundreds of years. In Japan, the art of origami (decorative paper folding) dates back to at least the 1600s. Over time, teachers realized that folding paper is a great way to test how well someone can picture shapes in their mind. That skill is called spatial reasoning — the ability to imagine objects moving, flipping, or changing in space.

1600s • Origami Blossoms in Japan

Japanese artists develop complex paper-folding patterns. These designs require tracking each fold mentally.

1900s • IQ Tests Include Spatial Tasks

Psychologists begin adding paper-folding items to intelligence and aptitude tests to measure spatial reasoning.

1950s • Standardized Tests Adopt the Format

Exams like the TACHS start using paper-folding questions to evaluate students' ability to visualize transformations.

Today • Paper Folding on the TACHS

The TACHS Ability section still tests paper folding. It checks whether you can track a sequence of folds and predict the final result.

So here is the big question these problems ask: if you fold a square piece of paper one, two, or three times — and maybe punch a hole — what does it look like when you unfold it? Let's learn how to answer that!

SECTION 2

Core Principles of Paper Folding

Before you dive into practice, you need a few big ideas. These are the rules that every paper-folding question follows.

The Fold Line Is a Mirror

When you fold paper, the fold acts like a mirror. Whatever is on one side of the fold lands directly on top of the matching spot on the other side.

Layers Stack Up

Each fold doubles the number of layers. One fold gives 2 layers, two folds give 4 layers, and three folds give 8 layers.

Holes Go Through All Layers

If a hole is punched through the folded paper, it goes through every layer at once. When you unfold, each layer shows a hole.

Unfold in Reverse Order

To figure out the final result, undo each fold one at a time — starting with the last fold. Think of it like rewinding a video.

Symmetry Is Your Friend

Each fold creates a line of symmetry. Holes and marks always appear as mirror images across each fold line.

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KEY TAKEAWAY

Think of each fold like closing a book. When you close it, the left page lands on top of the right page. If you poke a hole through the closed book, both pages get a hole. When you open the book again, the holes are in mirror-image spots — one on the left page and one on the right page.

SECTION 3

Seeing a Single Fold in Action

The diagram below shows the simplest case: one fold and one hole punch. Study each step carefully.

A single vertical fold creates two layers. One hole punch produces two mirror-image holes when the paper is unfolded.

Notice how the two pink holes in Step 3 are the same distance from the fold line but on opposite sides. That mirror-image pattern is the key to every paper-folding question you will see on the TACHS.

SECTION 4

How Folding Works Step by Step

Counting Layers

Every fold doubles the layers. You can use a simple rule:

LAYER RULE

Layers = 2 raised to the number of folds

1 fold → 2 layers. 2 folds → 4 layers. 3 folds → 8 layers.

Counting Holes

When you punch a hole, it goes through all the layers at once. So the number of holes on the unfolded paper equals the number of layers — as long as the hole is not on a fold line.

HOLE RULE (hole NOT on a fold line)

Number of holes = number of layers = 2^(number of folds)

If a hole is punched right on a fold line, some holes overlap when you unfold. This means fewer holes appear.

Direction of Folds Matters

A fold can go in different directions. A vertical fold brings the right side onto the left (or vice versa). A horizontal fold brings the bottom up to the top (or vice versa). A diagonal fold goes corner to corner. Always pay attention to the arrow or description that tells you which way to fold.

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Test Tip

On the TACHS, the question usually shows a series of pictures. Read them left to right. The shaded or darker area shows the part that moved on top. Follow the shading to figure out which direction the paper was folded.

SECTION 5

Types of Folds and Their Effects

The diagram below shows the three main fold directions you will see on the test. Each one creates a different line of symmetry.

Three fold types with a single hole punch. Vertical folds mirror left-right, horizontal folds mirror top-bottom, and diagonal folds mirror across the corner-to-corner line.

Fold Direction	Fold Line Runs	Holes Mirror
Vertical	Top to bottom (up-down)	Left ↔ Right
Horizontal	Left to right (side to side)	Top ↔ Bottom
Diagonal	Corner to corner	Across the diagonal line

How each fold direction affects hole placement

SECTION 6

Worked Example: Two Folds and a Hole Punch

Let's walk through a problem with two folds. This is the kind of question you will see most often on the TACHS.

Two-Fold Problem

1
Step 1 — Start with a Square

Picture a flat square piece of paper lying on a table. Label the corners: top-left (A), top-right (B), bottom-right (C), bottom-left (D).
2
Step 2 — First Fold (Vertical)

Fold the right half onto the left half along a vertical center line. The paper is now a tall rectangle with 2 layers. Corner B sits on top of corner A, and corner C sits on top of corner D.

Shape: tall rectangle, 2 layers.
3
Step 3 — Second Fold (Horizontal)

Now fold the bottom half up onto the top half along a horizontal center line. The paper is now a small square with 4 layers.

Shape: small square, 4 layers.
4
Step 4 — Punch a Hole

Punch a hole in the center of this small square. The hole goes through all 4 layers.
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Step 5 — Unfold the Second Fold

Undo the last fold (the horizontal one) first. The hole reflects across the horizontal fold line. Now there are 2 holes — one in the top half and one in the bottom half of the tall rectangle.

2 holes, mirrored top-to-bottom.
6
Step 6 — Unfold the First Fold

Now undo the first fold (the vertical one). Each of the 2 holes reflects across the vertical fold line. That gives 4 holes total — one in each quarter of the original square.

Final answer: 4 holes arranged in a 2 × 2 pattern in the center of the square.

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REMEMBER

Always unfold in reverse order! It is like taking off your shoes and socks — you put socks on first, but you take shoes off first. The last fold you made is the first fold you undo.

SECTION 7

Common Mistakes and How to Avoid Them

Many students lose points not because they don't understand folding, but because they rush and make small errors. Here are the most common mistakes and how to dodge them.

Mistake	Why It Happens	How to Fix It
Unfolding in the wrong order	Students undo the first fold first instead of the last.	Number each fold (1, 2, 3) and undo them 3, 2, 1.
Reflecting the hole the wrong way	Confusing which side of the fold line the hole mirrors to.	Ask: 'Which direction did the paper move?' The hole mirrors that same direction.
Forgetting holes on a fold line overlap	Expecting more holes than actually appear.	If a hole sits right on a fold edge, it stays as one hole — it does not split into two.
Losing track of which corner is which	After two or three folds, it is hard to keep track.	Mentally label corners (A, B, C, D) or use scratch paper.

Common paper-folding errors

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PRO TIP

If you are allowed scratch paper on the test, actually fold a small piece of paper! Poke it with your pencil tip and unfold it. Seeing the real result helps you build the mental picture you need.

SECTION 8

Going Further: Three Folds and Multiple Holes

Some harder questions may give you three folds or more than one hole punch. The same rules still apply — you just repeat the process more times.

Number of Folds	Layers	Holes per Punch (max)	Difficulty
1	2	2	Easy
2	4	4	Medium
3	8	8	Hard

How folds multiply holes

If a question has two separate hole punches on a doubly-folded paper, you track each hole independently. First unfold the last fold and mirror both holes. Then unfold the second fold and mirror all the holes again. At the end you might have up to 8 holes from just 2 punches on a 3-fold paper!

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Looking Ahead

In high school geometry and beyond, this same skill — imagining reflections across lines — shows up in topics like transformations, coordinate reflections, and even symmetry in art and design. Mastering paper folding now gives you a head start!

SECTION 9

Practice Problems

Try these five problems. Start with the easier ones and work your way up. For each problem, picture the paper in your mind — or use a real piece of paper to check your answer.

Question 1: CONCEPTUAL

You fold a square piece of paper in half using a single vertical fold (right side onto left side). You punch one hole in the center of the folded paper. How many holes will you see when you unfold the paper?

Show answer

You will see 2 holes. One fold creates 2 layers, so the punch goes through both layers. When you unfold, each layer has one hole, and they are mirror images of each other across the vertical fold line. The two holes appear side by side in the center of the paper.

Question 2: BASIC CALCULATION

A square piece of paper is folded twice — first a vertical fold (right onto left), then a horizontal fold (bottom onto top). How many layers does the paper have after both folds?

Show answer

The paper has 4 layers. Each fold doubles the layers. After the first fold: 2 layers. After the second fold: 2 × 2 = 4 layers. You can also use the rule: layers = 2 raised to the number of folds, so 2² = 4.

Question 3: INTERMEDIATE

A square piece of paper is folded in half vertically (right onto left), then folded in half horizontally (bottom onto top). A hole is punched in the top-left corner of the small folded square. Describe where the holes appear when the paper is fully unfolded.

Show answer

There will be 4 holes, one in each corner of the original square. First, undo the horizontal fold: the top-left hole mirrors to the bottom-left, giving 2 holes on the left side of the tall rectangle. Then undo the vertical fold: each of those 2 holes mirrors to the right side. The result is 4 holes — one at each corner of the original square (top-left, top-right, bottom-left, bottom-right).

Question 4: APPLIED

A square piece of paper is folded in half vertically (right onto left). A hole is punched right on the fold line (the left edge of the folded rectangle). When you unfold the paper, how many holes do you see, and where are they?

Show answer

You see only 1 hole, located on the center vertical line of the original square. Because the hole was punched on the fold line itself, both layers share the same hole. When you unfold, the hole does not split into two separate holes — it stays as a single hole right on the fold line. This is why holes on the fold edge are tricky: they do not double.

Question 5: CRITICAL THINKING

A square piece of paper is folded three times: first vertically (right onto left), then horizontally (bottom onto top), and finally diagonally (bottom-left corner to top-right corner of the small square). One hole is punched through the paper. What is the maximum number of holes you could see when the paper is fully unfolded? Explain your reasoning.

Show answer

The maximum number of holes is 8. Three folds create 2³ = 8 layers. If the hole is not on any fold line, it passes through all 8 layers, and each layer produces a separate hole when unfolded. You undo the diagonal fold first (mirroring across the diagonal), then the horizontal fold (mirroring top-to-bottom), and finally the vertical fold (mirroring left-to-right). Each unfolding step doubles the holes: 1 → 2 → 4 → 8. If the hole happened to be on one of the fold lines, some holes would overlap and the total would be less than 8.

SUMMARY

Paper Folding — Quick Review

Summary

To solve paper-folding problems, remember three big ideas. First, each fold line acts like a mirror, reflecting everything to the other side. Second, every fold doubles the number of layers, so 1 fold = 2 layers, 2 folds = 4 layers, and 3 folds = 8 layers. Third, always unfold in reverse order — undo the last fold first, then the one before it.

Pay attention to the direction of each fold (vertical, horizontal, or diagonal) because it determines which way holes mirror. Watch out for holes on a fold line — they do not split into two when you unfold. Practice with real paper to build your mental picture, and you will be ready for any paper-folding question the TACHS throws at you!