Select the unfolded pattern that matches the folding and...

SECTION 1

Where Did Paper Folding Puzzles Come From?

People have been folding paper for hundreds of years. Paper folding puzzles test your ability to picture what happens inside your mind. These puzzles show up on tests like the TACHS because they measure how well you can think about shapes and space. Let's look at how this idea developed over time.

~600 AD

Origins of Origami

Paper folding begins in Japan as an art form called origami. People learn that folding creates symmetry (mirror images) in the paper.

1900s

Spatial Reasoning Tests Emerge

Psychologists begin using paper folding tasks to measure how well people can picture objects in their minds. These tasks become part of IQ and ability tests.

1940s

Fold-and-Punch Problems Appear

Standardized tests start including fold-and-punch questions. A piece of paper is shown being folded, then a hole is punched through it. Students must predict the hole pattern when the paper is unfolded.

Today

TACHS and Other Entrance Exams

Paper folding questions are a key part of the TACHS Ability section. They test spatial reasoning, which is your brain's power to rotate and flip images mentally.

So here's the big question these puzzles ask: If you fold a piece of paper, punch a hole through it, and then unfold it, where will all the holes be? Answering this correctly takes practice, but once you learn the rules, it becomes much easier.

SECTION 2

Core Principles of Paper Folding and Punching

Before you try any problems, you need to understand a few key ideas. These rules work every single time, no matter how the paper is folded.

Symmetry Across the Fold

When you fold paper and punch a hole, the hole appears on every layer. When you unfold, each hole is a mirror image across the fold line. Think of the fold line like a mirror.

Layers Multiply Holes

Each fold doubles the number of layers. One fold = 2 layers = 2 holes per punch. Two folds = 4 layers = up to 4 holes per punch. The more folds, the more holes.

Distance from the Fold Matters

A hole punched close to the fold line will create holes close together. A hole punched far from the fold will create holes far apart. The distance is always equal on both sides.

Track the Fold Direction

Pay close attention to which direction the paper folds: left-to-right, top-to-bottom, or corner-to-corner. The fold direction tells you where the mirror line is.

✦ KEY TAKEAWAY

Think of folding paper like closing a book. If you poke a hole through a closed book, the hole goes through every page. When you open the book back up, every page has a hole in the same spot. The fold line is the spine of the book, and holes appear as mirror images on each side of it.

SECTION 3

Seeing It Step by Step

The diagram below shows a simple example. A square piece of paper is folded once from right to left. Then a hole is punched in the folded paper. Finally, the paper is unfolded to reveal two holes.

This diagram walks through the four steps: start with a full square, fold it in half from right to left, punch a hole, then unfold. Notice how the two holes are mirror images across the fold line.

The key idea here is the fold line. It acts like a mirror. Whatever appears on one side of that line also appears on the other side, at the same distance away. In the diagram, the pink hole on the left is the same distance from the purple dashed fold line as the pink hole on the right.

SECTION 4

How It Works — Counting Holes and Finding Positions

You don't need fancy math for these problems, but there is a simple counting rule that always works.

HOLE COUNT RULE

Number of holes = 2 raised to the power of (number of folds)

With 1 fold you get 2 holes per punch. With 2 folds you get 4 holes per punch. With 3 folds you get 8 holes per punch. Each fold doubles the number of holes.

Finding the Position of Each Hole

To find where every hole ends up, work backward through the folds. Start with the punched hole on the folded paper. Then reverse the last fold and reflect the hole across that fold line. Keep reversing folds, one at a time, until the paper is fully open. Each time you reverse a fold, you double the number of holes.

💡 Pro Tip

Always undo the folds in reverse order. The last fold you made is the first fold you undo. Think of it like unstacking pancakes — the one on top comes off first.

What About Holes on the Fold Line?

If a hole is punched exactly on the fold line, something special happens. When you unfold, that hole does not create two separate holes. Instead, it stays as one hole right on the fold line. The two halves of the hole just open up into one larger opening. This is a tricky detail that test writers love to include!

SECTION 5

Different Types of Folds

On the TACHS, you'll see several different fold directions. The diagram below shows the four most common types. Each one creates a different mirror line.

The top row shows three common single-fold directions. The bottom row shows a double fold — first folded left to right, then top to bottom — creating 4 layers and up to 4 holes per punch.

Common fold types and the number of holes each creates per single punch

Fold Type	Fold Line Direction	Holes per Punch
Left to Right (or Right to Left)	Vertical line down the center	2
Top to Bottom (or Bottom to Top)	Horizontal line across the center	2
Diagonal (corner to corner)	Diagonal line from one corner to the opposite	2
Two folds (any combination)	Two fold lines	Up to 4

SECTION 6

Worked Example — Two Folds and One Punch

Let's work through a problem like the ones you'll see on the TACHS. Imagine a square piece of paper. First it is folded in half from right to left. Then it is folded in half again from bottom to top. Finally, a hole is punched in the upper-left area of the folded paper. What does the paper look like when it is completely unfolded?

Two Folds, One Punch

Step 1 — Identify the FoldsThe paper is folded twice. Fold 1 goes from right to left (creating a vertical fold line on the right side). Fold 2 goes from bottom to top (creating a horizontal fold line on the bottom). Two folds means we could have up to 4 holes when we unfold.

2 folds → up to 4 holes

Step 2 — Locate the PunchThe hole is punched in the upper-left area of the folded piece. This is important. We need to remember exactly where this hole sits relative to both fold lines.

Hole is in the upper-left corner of the folded paper

Step 3 — Undo the Last Fold FirstThe last fold was bottom to top. So we undo this fold by flipping the paper back down. The fold line is horizontal. The hole in the upper-left gets a mirror copy in the lower-left. Now we have 2 holes: one in the upper-left and one in the lower-left, equal distances from the horizontal fold line.

2 holes — upper-left and lower-left

Step 4 — Undo the First FoldThe first fold was right to left. So we undo it by flipping the paper back to the right. The fold line is vertical. Each of the 2 holes gets a mirror copy on the right side. The upper-left hole creates a copy in the upper-right. The lower-left hole creates a copy in the lower-right.

4 holes — upper-left, upper-right, lower-left, lower-right

Step 5 — Select the Matching AnswerThe correct answer choice is the one showing four holes arranged symmetrically. There is one hole in each corner area of the paper. All four holes are the same distance from the center of the paper. Look for the answer that shows this symmetric pattern.

Answer: 4 holes, one in each corner area, equally spaced from center

SECTION 7

Strategies and Common Mistakes

Knowing the rules is only part of the battle. You also need good strategies and you need to avoid the traps that test writers set for you.

Strategies vs. common mistakes for paper folding problems

Strategy (Do This ✓)	Common Mistake (Avoid This ✗)
Undo folds in reverse order — last fold first.	Undoing the first fold first, which puts holes in wrong spots.
Count the folds to predict how many holes there will be.	Guessing the number of holes without counting layers.
Use the fold line as a mirror to find reflected positions.	Placing holes at random spots instead of reflecting them.
Check if any holes land on a fold line (these don't double).	Always doubling every hole, even when it sits on the fold.
Use scratch paper to draw it out if you're stuck.	Trying to do everything in your head on hard problems.

✦ KEY TAKEAWAY

Think of each fold like taking a selfie with a mirror. The hole and its reflection are always the same distance from the mirror. If you use two mirrors at right angles (like two folds), you see four copies of yourself — just like four holes from two folds!

SECTION 8

Beyond the Basics — Multiple Punches and Complex Folds

Some harder TACHS questions will throw a curveball at you. Here's what to look out for as problems get trickier.

How basic problems compare to advanced problems

Basic Version	Advanced Version
1 fold, 1 punch	2 folds, 1 punch (up to 4 holes)
Fold goes left-right or top-bottom	Diagonal fold (corner to corner)
1 punch in the middle of the paper	Multiple punches in different spots
Hole is away from fold lines	Hole is on the fold line (doesn't double)
Square paper	Rectangular paper (changes proportions)

When you see multiple punches, just treat each punch separately. Unfold each punch one at a time using the same mirror-reflection rules. Then combine all the holes together on one picture. The same rules always apply, no matter how complicated the problem looks.

🔭 Looking Ahead

Paper folding is related to a math topic called symmetry and transformations. In high school geometry, you'll learn about reflections, rotations, and translations. The fold-and-punch skill you're building now is the foundation for understanding all of these geometric transformations.

SECTION 9

Practice Problems

Try these five problems. They go from easier to harder. For each one, picture the folding and punching in your mind, then figure out the unfolded pattern.

PROBLEM 1 — CONCEPTUAL

A square piece of paper is folded once from left to right. A single hole is punched through the folded paper. When the paper is unfolded, how many holes will there be?

PROBLEM 2 — BASIC CALCULATION

A square piece of paper is folded from top to bottom. A hole is punched in the lower-left corner of the folded paper. Describe where the holes appear when unfolded.

PROBLEM 3 — INTERMEDIATE

A square piece of paper is folded from right to left, then folded again from bottom to top. A single hole is punched in the center of the folded paper (not on any fold line). How many holes will appear when the paper is fully unfolded, and describe their arrangement.

PROBLEM 4 — APPLIED

A square piece of paper is folded from left to right. Then a hole is punched exactly on the fold line (the right edge of the folded paper). How many holes appear when unfolded? Now imagine a second hole is also punched near the upper-left corner (not on the fold line). How many total holes appear?

PROBLEM 5 — CRITICAL THINKING

A square piece of paper is folded diagonally from the bottom-left corner to the top-right corner, creating a triangle. A hole is punched near the bottom-right area of the triangle (close to but not on the fold line). Describe where the holes appear when unfolded, and explain why diagonal folds can be confusing.

SUMMARY

Summary — Paper Folding and Punching

Paper folding and punching problems test your spatial reasoning — your ability to picture shapes in your mind. Every fold creates a fold line that acts like a mirror. Holes reflect across that fold line when the paper is unfolded. Each fold doubles the number of holes per punch, so one fold gives 2 holes and two folds give up to 4 holes.

To solve these problems, always undo folds in reverse order (last fold first). Use each fold line as a mirror to reflect holes to their matching positions. Watch out for holes on the fold line — these do not double. With practice, you can master these problems and boost your TACHS Ability score!