Predict how punched holes or marks appear when the paper is unfolded - TACHS Ability

SECTION 1

Why Paper Folding Matters

People have been folding paper for thousands of years. The art of origami (paper folding) started in Japan and China long ago. Over time, scientists and teachers realized that folding paper is a great way to test how well someone can think in pictures — a skill called spatial reasoning (the ability to imagine how shapes look when they move or change).

~600 • Early Paper Folding

Paper folding begins in China and Japan as an art form and a way to wrap gifts.

1800s • Paper Folding in Schools

German educator Friedrich Froebel uses paper folding to teach geometry to young children in kindergarten.

1900s • Ability Tests Emerge

Psychologists begin using paper-folding puzzles on aptitude tests to measure spatial reasoning skills.

Today • TACHS and Other Exams

Paper-folding and hole-punch questions appear on the TACHS Ability section as a key way to measure visual thinking.

On the TACHS, you will see a picture of a piece of paper being folded. Then a hole is punched through the folded paper. Your job is to figure out: What does the paper look like when you unfold it? This lesson will teach you exactly how to solve these problems step by step.

SECTION 2

Core Principles of Paper Folding

Before you practice any problems, you need to understand a few big ideas. These ideas are the building blocks for every paper-folding question you will see.

The Fold Line Is a Mirror

When you fold paper, the fold acts like a mirror line. Any hole punched on one side will appear in a matching spot on the other side when you unfold.

Holes Multiply with Folds

Each fold doubles the layers. One fold means 2 layers. Two folds means 4 layers. More layers = more holes when you unfold.

Unfold in Reverse Order

Always undo the last fold first. If the paper was folded right then down, unfold up first, then unfold left.

Distance Is Preserved

A hole's distance from the fold line stays the same on both sides. If the hole is 1 inch from the fold, its mirror image is also 1 inch from the fold — on the opposite side.

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KEY TAKEAWAY

Think of a fold line like the center of a butterfly's wings. If you paint a dot on one wing, the same dot appears in the exact matching spot on the other wing. The fold line is the butterfly's body — the line of symmetry.

SECTION 3

Seeing It: One Fold, One Hole

Let's start with the simplest case. Imagine a square piece of paper. You fold it in half from right to left. Then you punch one hole near the top-right corner of the folded paper. What happens when you unfold it?

This diagram shows a single fold from right to left. The pink hole is punched through both layers. When unfolded, two holes appear, each the same distance from the fold line (shown in purple).

Notice how the two holes in the final picture are mirror images of each other across the fold line. The fold line acts like a mirror. Whatever is on one side is copied to the other side in a flipped position. This is the single most important idea for paper-folding problems.

SECTION 4

How Folds and Holes Work Together

Counting Holes

Here is a simple rule to remember. Each fold doubles the number of layers. When you punch one hole through all the layers, you get one hole per layer.

HOLE COUNT RULE

Number of holes = 2 raised to the number of folds

With 1 fold: 2¹ = 2 holes. With 2 folds: 2² = 4 holes. With 3 folds: 2³ = 8 holes. (This only applies when punching one single hole through all layers.)

Direction of the Fold Matters

A fold to the left or right creates a vertical fold line (up and down). Holes will be mirrored left and right. A fold up or down creates a horizontal fold line (side to side). Holes will be mirrored above and below.

Corner and Diagonal Folds

Sometimes a corner of the paper is folded over to create a diagonal fold line. In this case, the mirror reflection flips the hole across the diagonal. These problems are trickier, but the same mirror rule still applies. Just reflect the hole across the slanted line.

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Pro Tip

On the TACHS, always look at the arrows in the picture. They tell you which direction the paper was folded. Follow the arrows carefully before you think about the holes.

SECTION 5

Two Folds: Predicting Four Holes

Many TACHS questions involve two folds. When the paper is folded twice, you get four layers. A single hole punch goes through all four layers, so the unfolded paper will show four holes. The key is unfolding in reverse order — undo the second fold first, then undo the first fold.

Two folds create four layers. One hole punch produces four symmetric holes when fully unfolded. Notice how we unfold in reverse: undo fold 2 first (cyan line), then undo fold 1 (purple line).

The four holes form a rectangle pattern. The top-left pair mirrors across the vertical fold line to make the top-right pair. The bottom pair mirrors from the top pair across the horizontal fold line. Every hole is the same distance from its fold line as the original hole.

SECTION 6

Worked Example: Step by Step

Let's walk through a complete problem just like the ones you will see on the TACHS.

Paper Folded Twice with One Hole

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Step 1 — Read the Folds

A square piece of paper is folded in half from left to right (creating a vertical fold line on the left side). Then it is folded from top to bottom (creating a horizontal fold line on the top). The paper is now a small square — one quarter of the original size.

2 folds → 4 layers → expect up to 4 holes per punch
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Step 2 — Locate the Hole

A single hole is punched in the bottom-right corner of the folded paper. This is far from both fold lines, so the hole is not on a fold.

Hole position: bottom-right of folded paper
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Step 3 — Undo the Last Fold (top to bottom → flip up)

The last fold went top to bottom, so we flip the paper back up. The hole in the bottom-right creates a mirror hole in the top-right. Now we have a rectangle (half of the original) with two holes on the right side — one near the top-right corner and one near the bottom-right corner.

After first unfold: 2 holes (top-right and bottom-right)
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Step 4 — Undo the First Fold (left to right → flip left)

The first fold went left to right, so we flip the paper back to the left. Each hole on the right side creates a mirror hole on the left side. The top-right hole mirrors to top-left. The bottom-right hole mirrors to bottom-left.

After second unfold: 4 holes — one in each corner of the paper
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Step 5 — Check Your Answer

The final paper has four holes, one near each corner. They form a symmetric pattern: top-left, top-right, bottom-left, and bottom-right are all mirror images across both fold lines. The answer choice that shows four corner holes is correct.

Answer: Four holes, one near each corner

SECTION 7

Common Mistakes and How to Avoid Them

Common Mistake	Why It Happens	How to Fix It
Wrong number of holes	Forgetting that each fold doubles the layers and therefore doubles the holes.	Use the rule: holes = 2 raised to the number of folds. 1 fold = 2 holes, 2 folds = 4 holes.
Holes in wrong positions	Unfolding in the wrong order or reflecting across the wrong line.	Always unfold the LAST fold first. Reverse the folding order like rewinding a video.
Forgetting holes on the fold line	If a hole is punched right on a fold line, it does not double — it stays as one hole on the edge.	Check if the hole touches the fold line. If yes, that hole does NOT create a mirror copy.
Mixing up fold direction	Confusing "fold left" with a fold line on the left vs. the right.	The fold LINE is on the side the paper stays. If you fold right to left, the fold line is on the LEFT.

Avoid these common paper-folding errors

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KEY TAKEAWAY

Think of unfolding like pressing "undo" on a computer. You always undo the most recent action first. If you typed three words, pressing undo removes the third word first, then the second, then the first. Unfolding works the same way — reverse the order of the folds.

SECTION 8

Special Cases and Advanced Patterns

Some TACHS questions try to trick you with special situations. Let's look at a few of them so you will be prepared.

Situation	What Happens
Hole on the fold line	The hole sits right on the crease. When unfolded, it becomes one elongated hole (or stays as one hole), not two separate holes.
Hole at the corner where two folds meet	If the hole is at the intersection of two fold lines, it only appears once because it is on both fold lines.
Diagonal fold	Reflect the hole across the diagonal line. The mirror image flips both left-right and up-down at an angle.
Multiple holes punched	Each hole is treated separately. Apply the mirror rule to every hole individually, then combine results.
Three folds	Three folds = 8 layers. One hole punch = up to 8 holes. Unfold three times in reverse order.

Special cases you may encounter on the TACHS

As you move into high school, these spatial reasoning skills connect to bigger ideas in geometry. Reflections across fold lines are the same as line reflections in coordinate geometry. When you study transformations (slides, flips, and turns), you will see that paper folding is really just practicing reflections — one of the key geometric transformations.

SECTION 9

Practice Problems

Try these five problems. Start with the easier ones and work your way up. For each problem, picture the folds in your mind and unfold step by step.

Question 1: CONCEPTUAL

A square piece of paper is folded once from right to left. One hole is punched in the center of the folded paper. How many holes appear when the paper is unfolded?

Show answer

There will be 2 holes. One fold creates 2 layers. The hole goes through both layers. When you unfold, the original hole stays, and a mirror image appears on the other side of the fold line. The two holes will be side by side, each the same distance from the center crease.

Question 2: BASIC CALCULATION

A square piece of paper is folded in half from top to bottom. A hole is punched near the bottom-left corner of the folded paper. When unfolded, where do the holes appear?

Show answer

There will be 2 holes. The fold line is horizontal (across the middle). The hole near the bottom-left is below the fold line. When you unfold upward, its mirror image appears near the top-left corner. The two holes are stacked vertically — one near the top-left and one near the bottom-left — each the same distance from the horizontal fold line.

Question 3: INTERMEDIATE

A square piece of paper is folded in half from left to right, then folded again from top to bottom. A hole is punched near the top-left corner of the small folded square. How many holes appear, and where are they, when the paper is fully unfolded?

Show answer

There will be 4 holes (2 folds → 2² = 4). First, undo the last fold (top to bottom → flip up). The top-left hole mirrors to the bottom-left. Now you have 2 holes on the left side. Next, undo the first fold (left to right → flip left). Each left-side hole mirrors to the right side. The final result is 4 holes arranged in a rectangle — one near each corner of the top and bottom edges, equally spaced from the fold lines.

Question 4: APPLIED

A square piece of paper is folded from bottom to top. Then a hole is punched right on the fold line (the bottom edge of the folded paper), near the right side. How many holes appear when the paper is unfolded, and why?

Show answer

Only 1 hole appears, even though the paper was folded once. When a hole is punched directly on the fold line, both layers share the same hole location. Unfolding reveals a single hole (which may be slightly elongated into a slit shape) right on the center crease near the right side. The key rule is: a hole ON the fold does not create a mirror copy because the original position and the mirror position are the same point.

Question 5: CRITICAL THINKING

A square piece of paper is folded in half from right to left, then folded again from bottom to top, and then folded a third time by bringing the top-right corner down to the bottom-left corner (a diagonal fold). One hole is punched near the center of the resulting shape, away from all fold lines. How many holes will appear when the paper is fully unfolded?

Show answer

There will be 8 holes. Three folds create 2³ = 8 layers. Since the hole is not on any fold line, it punches through all 8 layers. When you unfold in reverse — first undo the diagonal fold (reflecting across the diagonal), then undo the bottom-to-top fold (reflecting across the horizontal), and finally undo the right-to-left fold (reflecting across the vertical) — you get 8 holes. They form a symmetric pattern with mirror symmetry across all three fold lines. This is the most complex type of paper-folding problem you might see.

SUMMARY

Lesson Summary

Summary

Paper-folding problems test your spatial reasoning — your ability to picture shapes changing in your mind. Every fold line acts as a mirror. When you punch a hole, it goes through all the layers. Each fold doubles the layers, so one fold gives 2 holes, two folds give 4, and three folds give 8. The distance from the hole to the fold line is always the same on both sides.

To solve these problems, always unfold in reverse order — undo the last fold first. Watch out for special cases: a hole punched on the fold line does NOT create an extra hole. Practice picturing the folds in your mind, and you will be ready for every paper-folding question on the TACHS.