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Paper Folding and Cutting

Reasoning · Non-Verbal Reasoning

7 Part1 worked examples4 practice questions

Fold a square coloured paper in the middle and punch one hole with a paper punch. Now open it — you find not one but two holes. When folded, two layers lay on top of each other, and the punch went through both at once.

In the exam you see a picture of the folding steps and the final hole, and you are asked how the paper will look when opened. You have neither paper nor punch — you must unfold it in your head. The good news is that there is one firm rule for it, which you learn in the first part of this chapter.

Every figure in this chapter was made by a computer simulation: the paper was really folded, punched and opened, so each answer is certain. The wrong options are the ones students actually make by mistake — such as forgetting to open one fold or mirroring in the wrong line.

Map of this chapter

PartWhat you learn
1. FoundationThe mirror-twin rule; kinds of folds; how to read the picture
2. UnfoldingOne fold, two folds, slanting folds
3. Cuts and countingSquare and triangle cuts, edge notches, the total number of holes
4. BackwardsThe opened paper is given — where was the hole made?
RememberIn every question open the last fold first. This one habit stops half of the mistakes.
1

Foundation — mirror twins and folds

Every mark has a twin, across the fold line

The mirror-twin rule, the kinds of folds and how to read a folding picture.

The mirror-twin rule

The fold line works like a mirror

How to find the new place of every mark on opening, in plain words.

When you fold a paper along a line, the part on one side comes to sit on the part on the other side — just as a mirror makes your reflection. That is why the fold line = a mirror.

The rule for finding the twin
1. Draw a straight perpendicular from the mark to the fold line2. Go the same distance further in the same direction3. The twin is there — its shape is also flipped like a mirror image
Orange marks are the originals, teal ones their twins. The distance from the dotted line is equal, and the triangle tip points the same way relative to the line

Notice three things: (1) the twin appears straight across, not slantwise; (2) a mark close to the fold line has its twin close to it too; (3) the direction of the mark is also reversed like a mirror image — do not treat it as a “copy” (sliding across).

CheckA simple test: if you fold the opened paper again along the fold line, every mark must fall exactly on its twin. If it does not, the twin is wrong.

In grid language: say the paper has 8 columns. When opening an upright fold in the middle, the twin of column c is column 9 − c at the same height; for a flat fold the height flips but the left–right place stays; for a slanting fold the place is swapped: a mark in row r and column c goes to row c and column r (for the main diagonal).

?ABCD
Top: the folding steps and the final punch. Below: four possible opened papers
Example 1 · Basic
A square paper is folded as shown and one hole is punched in the last fold. How will the paper look when it is opened out?
Solution, step by step
  1. 1Remember: when unfolding, every mark gets a mirror twin on the other side of the fold line, at the same distance from it.
  2. 2Open fold 1 (folded along a flat line: the lower part goes up onto the upper part) — 1 hole becomes 2 holes.
  3. 3The opened paper has its 2 marks in exactly these places — that is option D.
Answer D
In the solved figure the orange rim is the real hole and the teal rim is its twin.
Solution: opening from the last fold — orange rims are the original marks, teal rims are the new mirror twins
?ABCD
Top: the folding steps and the final punch. Below: four possible opened papers
BasicTry it yourself first
A square paper is folded as shown and one hole is punched in the last fold. How will the paper look when it is opened out?
?ABCD
Top: the folding steps and the final punch. Below: four possible opened papers
BasicTry it yourself first
A square paper is folded as shown and one hole is punched in the last fold. How will the paper look when it is opened out?

Kinds of folds and reading the picture

Dotted line = fold, arrow = which part goes where

Upright, flat and slanting folds, the layers doubling and the shape of the folded paper.

A folding picture has three things: the shape of the paper, the dotted line (where it will fold) and the arrow (which part flips over to which side). After each fold the shape changes — the square becomes half, a rectangle or a triangle.

FoldLineShape after folding
Upright foldThe middle upright lineA half-width rectangle
Flat foldThe middle flat lineA half-height rectangle
Slanting foldCorner to cornerA right-angled triangle
Two folds: square → rectangle → small square; then the hole and the opened paper (orange real, teal and purple twins from the two folds)

The layers double. After one fold there are 2 layers, after two folds 4 layers. The punch goes through all layers at once, so one hole becomes 4 holes after two folds (if it does not lie on a fold line).

TrapThe line for the next fold is often drawn on the already folded paper. Do not confuse it with a line on the first sheet. At each step look at the present shape of the paper.
?ABCD
Top: the folding steps and the final punch. Below: four possible opened papers
BasicTry it yourself first
A square paper is folded as shown and one hole is punched in the last fold. How will the paper look when it is opened out?
?ABCD
Top: the folding steps and the final punch. Below: four possible opened papers
BasicTry it yourself first
A square paper is folded as shown and one hole is punched in the last fold. How will the paper look when it is opened out?
2

Unfolding — fold by fold

One fold, two folds, slanting folds

How to open the paper starting from the last fold, for three kinds of folds.

One fold

One mirror, one twin

Questions with one fold: the twin of the hole across the fold line.

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Two folds

Open the last fold first, then the first

The order of opening with two folds and how four holes appear.

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Slanting folds

The mirror runs corner to corner

The triangular fold made along a diagonal and how it opens.

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3

Cuts, edge notches and counting

Square, triangle and edge cuts besides round holes

The direction of cut shapes, notches on an edge and counting the total holes.

Square and triangle cuts

A round hole worries only about place; a triangle also about direction

The mirror image of a cut shape and a check of its direction.

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Notches on an edge

A half shape cut on the fold opens into a whole shape

Triangles cut from an edge and the diamond-like shape formed on a fold line.

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Counting the total holes

Holes × 2 for every fold

The formula for the number of holes after opening, based on layers.

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4

Thinking backwards

The opened paper is given — where was the hole made?

Finding the hole in the folded paper from the opened paper, and the tricks of the question.

From the opened paper to the folded hole

Open every option and test it

The method for reverse questions: opening the options and removing the mismatches.

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5

Strategy guide

The firm order of opening and the slips to avoid

The order for every question, clues for cutting options and common mistakes.

The firm order of solving

First the count, then the place, then the direction

A table for cutting options quickly.

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Common traps

The mistakes that keep happening

The commonest reasons for losing marks in paper folding questions.

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6

Mixed practice

Opening, counting and backwards — all together

Questions of every kind in three levels.

Basic practice

One fold, one hole

One upright, flat or slanting fold and plain counting.

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Standard practice

Two folds, cuts and counting

Two folds, triangle cuts, edge notches and reverse questions.

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Advanced practice

Two folds, two marks, slanting folds and direction

Two holes, double slanting folds, triangle cuts and edge notches.

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7

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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