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Paper Folding & Cutting

Outline & Learning Goals

By the end of this chapter, you will be able to:

• Visualise what happens when a paper is folded once, twice, or thrice.
• Predict the pattern of holes or cuts after the paper is unfolded.
• Use the "mirror rule" to solve questions in seconds.
• Spot and avoid the examiner's favourite traps.

The Hook: One Punch, Many Holes

Imagine this. You fold a sheet of paper just two times, punch one single hole through it, and then open it up. How many holes do you see?

Not one. Four.

That is the magic — and the logic — of this chapter. Every fold doubles the layers of paper. One punch through 4 layers = 4 holes. One punch through 8 layers = 8 holes. Once you truly *feel* this idea, most questions in this chapter become straightforward.

"But I Can't Visualise 3D Shapes!" — You're Not Alone

Most students look at a Paper Folding question and their mind goes blank. The figures look like a puzzle from another planet. Here is the truth: toppers don't have special brains — they have a method. They never try to "see" the whole answer at once. They unfold the paper one fold at a time, in reverse order. That is exactly what you will learn now.

Chunk 1: The Golden Rule — Fold Forward, Unfold Backward

Every question shows you a sequence:

Step 1 → Step 2 → Step 3 → ?

The paper is folded in steps 1 and 2, and a cut/punch is made in step 3. You must find how it looks when fully opened.

The method:

1. Look at the last fold made.
2. Mentally unfold it — the cut gets reflected (mirrored) across that fold line.
3. Now unfold the previous fold — all the cuts you now have get mirrored again.
4. Repeat until the paper is fully open.

> Key line to memorise: *"Last fold, first unfold."* Always work in reverse order.

🧠 Active Recall: If a paper is folded twice and then punched once, in which order do you unfold — first fold first, or last fold first? (Answer: last fold first — always reverse order.)

Chunk 2: The Mirror Rule (Your Best Friend)

When you unfold a fold, the hole or cut does not just "copy" — it reflects, like an image in a mirror placed along the fold line.

• Fold was vertical → the copy appears on the other side, left-right flipped.
• Fold was horizontal → the copy appears above/below, top-bottom flipped.
• Fold was diagonal → the copy flips across the diagonal line.

Mnemonic — "MIRROR":
> Mirror Image Reflects, Reverse Order Rules.

Every cut is a mirror image across the fold line, and you always unfold in reverse order.

🧠 Active Recall: A paper is folded upward along a horizontal line and a triangle is cut near the fold. When unfolded, where does the second triangle appear, and in what orientation? (Answer: on the other side of the horizontal fold, flipped top-to-bottom.)

Chunk 3: Counting Holes — The Doubling Logic

Each fold doubles the paper layers at the punched spot:

But caution! This doubling works only if the punch lies in the region covered by all the folds. If a punch falls on a part covered by only one fold, it produces only 2 holes, not 4. This is exactly where the examiner traps you.

🧠 Active Recall: Paper folded twice, one punch gives 4 holes — true always, or only sometimes? (Answer: only when the punch point is covered by both folds.)

Chunk 4: Cuts on the Fold Line and on the Edges

Two special cases that exams love:

1. Cut exactly ON the fold line: When opened, the two mirrored halves join into one symmetric shape. A half-circle cut on the fold becomes a full circle; a half-diamond becomes a full diamond.
2. Cut on the outer edge (not on a fold): It produces separate identical cuts on the corresponding edges — they do not merge.

> Quick check: Cut on fold = shapes merge. Cut away from fold = shapes multiply.

🧠 Active Recall: A semicircle is cut on a fold line. After opening, do you get one circle or two semicircles? (Answer: one full circle — the halves merge.)

Chunk 5: A Step-by-Step Solved Example

Question type: A square paper is folded (i) left half onto the right half, then (ii) bottom half upward. A small circle is punched at the centre of the folded paper. What appears on opening?

Solution (reverse order):

1. Unfold fold (ii): the circle mirrors across the horizontal fold → now 2 circles, one above the other.
2. Unfold fold (i): both circles mirror across the vertical fold → now 4 circles, arranged in a 2×2 symmetric pattern.

Answer: 4 circles in a square arrangement. Notice — we never "imagined" the final figure. We just mirrored, step by step.

Summary Table

📌 Review this chapter after 2 days. Redraw one fold sequence on rough paper, punch it with a pen tip, and check your prediction — 2 minutes of real folding fixes this chapter in memory forever.

⚠️ Caution: Examiner's Trap

Trap 1 — Unclear figures on screen/paper.
In the exam, fold arrows and dotted lines can look faint or confusing. Students misread a diagonal fold as a horizontal fold and lose the mark. Fix: before solving, spend 3 seconds confirming the direction of every arrow — which side is folding onto which.

Trap 2 — Similar-looking options with hidden differences.
The four answer figures often differ only in the position of one hole or the orientation of one cut. The examiner hides the difference in a corner. Fix: don't scan the whole figure — check one hole at a time, starting from the hole nearest to the last fold.

Trap 3 — Slight change in rotation/reflection.
A cut shape (like an L or a triangle) is shown rotated in the option instead of mirrored. Rotation and reflection look similar but are not the same — unfolding always produces a mirror image, never a rotation. Fix: ask yourself for every option — "Is this a mirror flip across the fold line, or just a turned-around copy?" Eliminate the rotated ones.