SSC CGL Reasoning · Paper Folding and Cutting

Unfolding a Quarter

A square sheet is folded in half twice, down to one quarter, and cut or punched; opening it mirrors every cut across each fold line in turn, giving a copy in each quarter.

All Paper Folding and Cutting notes

Why this matters

Forty-four questions in the bank, 16 from the 2023 shifts and 28 from the 2024 shifts, all MODERATE but one HARD. Every one is solved by undoing the folds one at a time and mirroring the cuts, and the options differ only in where a copy sits or which way it faces.

Concept 1 of 2: Where the copies land

Folding a sheet in half twice stacks four layers. A cut through all four makes four holes, one in each quarter, placed like reflections of each other in the two fold lines. A cut that sits on a fold line joins its reflection and opens as one larger shape.

Definition

1. Find the folded quarter and its two fold edges: the edges that meet at the centre of the sheet. 2. Undo the last fold first. Each cut appears again on the other side of that fold line, at the same distance from it. 3. Undo the first fold. Every cut now on the paper appears again across that fold line.

How many: a cut away from the folds, through k folds, makes $2^k$ copies: 4 after two folds.
On a fold: half a circle cut on a fold line opens into one full circle. A small square cut at the corner where both folds meet opens into one larger square at the centre.
Near an outer corner: a cut near the quarter's outer corner opens into one near each corner of the sheet.

Copies after k folds

\text{copies of a cut away from every fold} = 2^k

kthe number of folds the cut goes through

Worked example

A square sheet is folded right half onto the left, then top half down, so the folded paper is the bottom-left quarter. One round hole is punched near the quarter's outer bottom-left corner. What does the opened sheet show?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 3 · Q16Moderate

Example 1 · Paper Folding and Cutting · Unfolding a Quarter

A paper is folded and cut as shown below. How will it appear when unfolded?

Counting four copies of a cut on a fold

A cut that touches a fold line joins its reflection, so it opens as one shape, not two. Only a cut clear of every fold gives a separate copy in each quarter.

Placing the copies at the corners when the cut was near the centre

A cut near the corner where the folds meet stays near the centre of the opened sheet. Measure each cut's distance from the fold lines, and keep that distance in every copy.

Undoing the folds in the wrong order

Undo the last fold first, then the one before it. Opening the first fold first puts the copies on the wrong side of the sheet.

Drill 9 more on where the copies land

Concept 2 of 2: Which way each copy faces

Each copy is a reflection, not a slide. Unfolding across a vertical fold reverses a cut left to right; unfolding across a horizontal fold turns it upside down. An arrow, a letter or a pointed triangle shows the difference at once.

Definition

Across a vertical fold: left and right swap. An arrow pointing left becomes one pointing right; an E becomes a reversed E.
Across a horizontal fold: top and bottom swap. A triangle pointing up becomes one pointing down; left and right stay.
After two folds, the quarter diagonally opposite the folded one has its cut both reversed and turned upside down.
A shape that is the same on both sides of a line (a circle, a square, a star) looks the same in every copy, so only its place matters.
Fast route: check the copy in the quarter diagonally opposite the folded one. It is flipped both ways, and most wrong options get it wrong.

Worked example

A sheet is folded bottom half up, then right half over to the left, so the folded paper is the top-left quarter. An arrow pointing left is cut in it. After opening, which way does the arrow in each quarter point?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 1 · Q23Moderate

Example 2 · Paper Folding and Cutting · Unfolding a Quarter

A paper is folded and cut as shown below. How will it appear when unfolded?

Sliding a copy instead of reflecting it

An option whose four arrows all point the same way has slid the cut into each quarter. Unfolding reflects it, so copies on opposite sides of a vertical fold point opposite ways.

Turning the copy upside down across a vertical fold

A vertical fold swaps only left and right. A triangle that points up keeps pointing up in the copy beside it; it turns down only in the copies across the horizontal fold.

Ignoring the diagonally opposite quarter

The copy diagonally across from the folded quarter has been reflected twice, so it is both reversed and upside down. Check it before you choose, because the setter often gets every other quarter right.

Drill 33 more on which way each copy faces

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Where the copies land
Copies after k folds
$\text{copies of a cut away from every fold} = 2^k$

Watch out for (6)

Counting four copies of a cut on a fold
→ Where the copies land
Placing the copies at the corners when the cut was near the centre
→ Where the copies land
Undoing the folds in the wrong order
→ Where the copies land
Sliding a copy instead of reflecting it
→ Which way each copy faces
Turning the copy upside down across a vertical fold
→ Which way each copy faces
Ignoring the diagonally opposite quarter
→ Which way each copy faces

Test yourself on Paper Folding and Cutting

15 past SSC CGL questions from this chapter, timed at 9 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Paper Folding and Cutting chapter test Drill all 44 questions