Paper Folding, Cutting and Unfolding Short Notes for GATE CS
Paper Folding, Cutting and Unfolding short notes for GATE CS: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi
paper folding cutting and unfolding short notes
Summary: Folding and Dimensions Checklist
Quick Revision Checklist
Identify the Fold Axis: Determine which dimension is perpendicular to the fold line.
Halve Correctly: Only the perpendicular dimension is divided by 2. The parallel dimension is untouched.
Update State: After each fold, re-evaluate which dimension is the "long edge" for the next step.
Final Calculation:
Perimeter = 2×(final length+final width) Do not confuse with Area, which is conserved across layers.
Sanity Check: Ensure the final dimensions make physical sense (e.g., no dimension can be larger than the original starting dimensions).
Exam Readiness: Cut Pattern Checklist
Final Revision Checklist
Work Backwards: Always unfold in the reverse order of the folds.
Mirror Strictly: Treat every fold line as a rigid axis of reflection.
Identify the Center: The corner where all folds converge becomes the exact center of the unfolded paper.
Edge Check:
Folded edge cut = Merged central shape.
Open edge cut = Boundary notches or separate holes.
Layer Count Verification: Total isolated interior holes must equal the number of layers pierced.
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Question 1
Level 1: Warm-up
A rectangular paper of dimensions 16 cm×4 cm is folded twice. In each fold, the paper is folded along a line of symmetry perpendicular to one of its edges. What is the minimum possible perimeter (in cm) of the final folded sheet?
Question 2
Level 1: Warm-up
A square paper is folded in half vertically, and then in half horizontally. A small circular hole is punched at the corner where all four layers meet. Which of the following statements is TRUE about the paper when it is completely unfolded?
Question 3
Level 1: Warm-up
A square paper is folded in half vertically (right over left), then in half horizontally (top over bottom). A small square hole is punched at the top-right corner of the folded square. Which of the following statements is TRUE about the unfolded paper?
Question 4
Level 1: Warm-up
A rectangular paper of dimensions 30 cm×12 cm is to be folded exactly once along a line of symmetry. What is the maximum possible perimeter (in cm) of the folded sheet?
Question 5
Level 1: Warm-up
Consider the following two statements about paper folding and cutting:
Assertion (A): A rectangular paper is folded in half. A semicircular cut is made on the folded edge. Upon unfolding, the paper will have two separate semicircular notches on opposite edges.
Reason (R): A cut made on the folded edge of a paper acts as a single unit that merges into one complete symmetrical shape when unfolded.
Which of the following options is correct?
Question 6
Level 1: Warm-up
A square paper is folded in half twice, resulting in 4 layers. A single circular cut is made. Depending on the location of the cut, the number of distinct circular holes when unfolded can vary. What is the maximum number of distinct holes possible?
Question 7
Level 1: Warm-up
Consider the following two statements about paper folding and cutting:
Assertion (A): A rectangular paper is folded in half. A semi-circular cut is made such that its diameter lies exactly on the open edge. Upon unfolding, the paper will have a single full circular hole at the center.
Reason (R): A cut made on the open edge mirrors across the fold line but remains separate from the fold axis, creating distinct boundary notches.
Which of the following options is correct?
Question 8
Level 1: Warm-up
A rectangular sheet of paper has dimensions 24 cm×14 cm. It is folded once such that the crease is perpendicular to the longer edge. What is the difference (in cm) between the longer and shorter edges of the folded sheet?
Question 9
Level 1: Warm-up
A long rectangular strip of paper is folded along the vertical line x=0. A small hole is punched at coordinate x=5. When the paper is unfolded, what is the distance between the centers of the two resulting holes?
Question 10
Level 1: Warm-up
The reverse-unfolding method requires unfolding the paper step-by-step in reverse chronological order. If a paper is folded exactly 3 times, what is the minimum number of reverse-unfolding steps required to correctly determine the final unfolded pattern?
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Paper Folding, Cutting and Unfolding Short Notes for GATE CS
Paper Folding, Cutting and Unfolding short notes for GATE CS: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.
Summary: Folding and Dimensions Checklist
Quick Revision Checklist
Identify the Fold Axis: Determine which dimension is perpendicular to the fold line.
Halve Correctly: Only the perpendicular dimension is divided by 2. The parallel dimension is untouched.
Update State: After each fold, re-evaluate which dimension is the "long edge" for the next step.
Final Calculation:
Perimeter = 2×(final length+final width) Do not confuse with Area, which is conserved across layers.
Sanity Check: Ensure the final dimensions make physical sense (e.g., no dimension can be larger than the original starting dimensions).
Exam Readiness: Cut Pattern Checklist
Final Revision Checklist
Work Backwards: Always unfold in the reverse order of the folds.
Mirror Strictly: Treat every fold line as a rigid axis of reflection.
Identify the Center: The corner where all folds converge becomes the exact center of the unfolded paper.
Edge Check:
Folded edge cut = Merged central shape.
Open edge cut = Boundary notches or separate holes.
Layer Count Verification: Total isolated interior holes must equal the number of layers pierced.
Paper Folding, Cutting and Unfolding: Solved Questions with Step-by-Step Explanations (10 Problems)
Question 1 · Spatial AptitudeMCQ
A rectangular paper of dimensions 16 cm×4 cm is folded twice. In each fold, the paper is folded along a line of symmetry perpendicular to one of its edges. What is the minimum possible perimeter (in cm) of the final folded sheet?
A.
16
B.
20
C.
24
D.
34
Correct Answer:
A
Step-by-Step Solution
Key idea: To minimize the perimeter, halve the largest dimension at each step to make the final shape as close to a square as possible.
Step 1: Initial dimensions 16 cm×4 cm.
Step 2: Fold 1 perpendicular to 16 cm edge →8 cm×4 cm.
Step 3: Fold 2 perpendicular to 8 cm edge →4 cm×4 cm.
Step 4: Perimeter =2×(4+4)=16 cm.
(Other fold sequences yield perimeters of 20 cm or 34 cm.)
Answer: 16
Question 2 · Spatial AptitudeMCQ
A square paper is folded in half vertically, and then in half horizontally. A small circular hole is punched at the corner where all four layers meet. Which of the following statements is TRUE about the paper when it is completely unfolded?
A.
It will have four separate circular holes near the four corners.
B.
It will have a single circular hole at the center of the paper.
C.
It will have two semi-circular notches on opposite edges.
D.
It will have four semi-circular notches, one on each edge.
Correct Answer:
B
Step-by-Step Solution
Key idea: The corner where all folded layers meet represents the exact center of the original unfolded paper.
Step 1: Folding vertically and horizontally creates a smaller square with 4 layers.
Step 2: The corner where all layers meet is the center of the original square.
Step 3: A circular hole punched here pierces all 4 layers at the center point.
Step 4: Upon unfolding, the 4 quarter-circles merge into a single full circle at the center.
Answer: It will have a single circular hole at the center of the paper.
Question 3 · Spatial AptitudeMCQ
A square paper is folded in half vertically (right over left), then in half horizontally (top over bottom). A small square hole is punched at the top-right corner of the folded square. Which of the following statements is TRUE about the unfolded paper?
A.
It will have four separate square holes, one near each corner.
B.
It will have a single square hole at the center.
C.
It will have two square holes on the left edge.
D.
It will have one square hole at the center and three at the corners.
Correct Answer:
A
Step-by-Step Solution
Key idea: This is a corner identification question, recognizable because it specifies the exact corner of the folded shape where the cut is made.
Step 1: Folding right over left, then top over bottom creates a smaller square with 4 layers.
Step 2: The top-right corner of this folded square consists entirely of open edges (the original boundaries of the paper).
Step 3: A hole punched at an open corner does not merge across any fold lines. It simply replicates at all 4 original corners.
Step 4: Therefore, the unfolded paper will have four separate square holes, one near each corner.
Answer: It will have four separate square holes, one near each corner.
Question 4 · Spatial AptitudeMCQ
A rectangular paper of dimensions 30 cm×12 cm is to be folded exactly once along a line of symmetry. What is the maximum possible perimeter (in cm) of the folded sheet?
A.
54
B.
72
C.
84
D.
42
Correct Answer:
B
Step-by-Step Solution
Key idea: A single fold can be perpendicular to either the length or the width; evaluate both cases to find the maximum perimeter.
Step 1: Case 1 - Fold perpendicular to 30 cm edge. New dimensions: 15 cm×12 cm. Perimeter =2×(15+12)=54 cm.
Step 2: Case 2 - Fold perpendicular to 12 cm edge. New dimensions: 30 cm×6 cm. Perimeter =2×(30+6)=72 cm.
Step 3: Comparing both cases, the maximum perimeter is 72 cm.
Answer: 72
Question 5 · Spatial AptitudeMCQ
Consider the following two statements about paper folding and cutting:
Assertion (A): A rectangular paper is folded in half. A semicircular cut is made on the folded edge. Upon unfolding, the paper will have two separate semicircular notches on opposite edges.
Reason (R): A cut made on the folded edge of a paper acts as a single unit that merges into one complete symmetrical shape when unfolded.
Which of the following options is correct?
A.
Both A and R are true and R is the correct explanation of A.
B.
Both A and R are true but R is NOT the correct explanation of A.
C.
A is true but R is false.
D.
A is false but R is true.
Correct Answer:
D
Step-by-Step Solution
Key idea: Distinguish between cuts on folded edges (which merge) and open edges (which remain separate).
Step 1: Evaluate Assertion (A): A semicircular cut on the folded edge will mirror across the fold line. The two semicircles join to form one full circle. Thus, A is FALSE (it does not produce two separate notches).
Step 2: Evaluate Reason (R): The statement correctly describes that folded-edge cuts merge into a single symmetrical shape. Thus, R is TRUE.
Step 3: Since A is false and R is true, the correct option is D.
Answer: A is false but R is true.
Question 6 · Spatial AptitudeMCQ
A square paper is folded in half twice, resulting in 4 layers. A single circular cut is made. Depending on the location of the cut, the number of distinct circular holes when unfolded can vary. What is the maximum number of distinct holes possible?
A.
2
B.
3
C.
4
D.
8
Correct Answer:
C
Step-by-Step Solution
Key idea: This is a layer counting and boundary condition question, recognizable because it asks for the maximum number of distinct shapes based on cut location.
Step 1: Folding twice creates 4 layers.
Step 2: If the cut is made in the interior of the layers (not touching any fold line), each layer produces a separate hole. This gives 4 distinct holes.
Step 3: If the cut is made exactly on a fold line, the mirrored halves join together, reducing the number of distinct shapes (e.g., to 2 or 1).
Step 4: Therefore, the maximum number of distinct holes is 4.
Answer: 4
Question 7 · Spatial AptitudeMCQ
Consider the following two statements about paper folding and cutting:
Assertion (A): A rectangular paper is folded in half. A semi-circular cut is made such that its diameter lies exactly on the open edge. Upon unfolding, the paper will have a single full circular hole at the center.
Reason (R): A cut made on the open edge mirrors across the fold line but remains separate from the fold axis, creating distinct boundary notches.
Which of the following options is correct?
A.
Both A and R are true and R is the correct explanation of A.
B.
Both A and R are true but R is NOT the correct explanation of A.
C.
A is true but R is false.
D.
A is false but R is true.
Correct Answer:
D
Step-by-Step Solution
Key idea: This is an edge distinction question, recognizable because it specifies the exact location of the cut relative to the fold and open edges.
Step 1: Evaluate Assertion (A): The cut is on the open edge. According to the rules, open edge cuts do not merge across the fold line. They remain as separate semi-circles on the boundary. Thus, A is FALSE (it does not produce a single full circle at the center).
Step 2: Evaluate Reason (R): The statement correctly describes that open edge cuts mirror but remain separate, creating distinct boundary notches. Thus, R is TRUE.
Step 3: Since A is false and R is true, the correct option is D.
Answer: A is false but R is true.
Question 8 · Spatial AptitudeMCQ
A rectangular sheet of paper has dimensions 24 cm×14 cm. It is folded once such that the crease is perpendicular to the longer edge. What is the difference (in cm) between the longer and shorter edges of the folded sheet?
A.
2
B.
10
C.
17
D.
5
Correct Answer:
A
Step-by-Step Solution
Key idea: Identify the longer edge, halve it, then compare the new dimensions to find the difference.
Step 1: Original dimensions are 24 cm×14 cm. The longer edge is 24 cm.
Step 2: Fold perpendicular to 24 cm edge halves it: 24/2=12 cm.
Step 3: New dimensions are 12 cm×14 cm.
Step 4: The longer edge is now 14 cm, and the shorter is 12 cm. Difference =14−12=2 cm.
Answer: 2
Question 9 · Spatial AptitudeMCQ
A long rectangular strip of paper is folded along the vertical line x=0. A small hole is punched at coordinate x=5. When the paper is unfolded, what is the distance between the centers of the two resulting holes?
A.
5
B.
10
C.
0
D.
25
Correct Answer:
B
Step-by-Step Solution
Key idea: This is a reflection principle question, recognizable because it involves coordinates and a fold line acting as a mirror.
Step 1: The fold line is at x=0. The original hole is at x=5.
Step 2: Unfolding reflects the hole across the fold line. The mirrored hole will be at x=−5.
Step 3: The distance between the two holes is the absolute difference between their coordinates: ∣5−(−5)∣=10.
Answer: 10
Question 10 · Spatial AptitudeMCQ
The reverse-unfolding method requires unfolding the paper step-by-step in reverse chronological order. If a paper is folded exactly 3 times, what is the minimum number of reverse-unfolding steps required to correctly determine the final unfolded pattern?
A.
3
B.
2
C.
4
D.
5
Correct Answer:
A
Step-by-Step Solution
Key idea: This is a reverse-unfolding method question, recognizable because it asks for the number of steps to mentally unfold a paper.
Step 1: The reverse-unfolding method requires one mental unfolding step for each physical fold that was made.
Step 2: Since the paper was folded exactly 3 times, there are 3 physical folds to reverse.
Step 3: Therefore, the minimum number of reverse-unfolding steps required is 3.