Paper Folding and Cutting
About 7 min · Logical Reasoning · English
Time worked out from the lesson's 492 words at 150 words a minute, plus a minute for each question.
Paper Folding and Cutting is a spatial reasoning topic that tests your ability to visualize transformations in three-dimensional space. In these problems, you are shown a sequence of folds applied to a square or rectangular sheet of paper, followed by one or more cuts (punches or holes). Your task is to predict what the paper will look like when completely unfolded.
Key ideas
- Fold creates symmetry: Every fold acts as a mirror line. A cut on the folded side will appear on both sides of the fold line when unfolded.
- Layers multiply holes: If paper is folded once, a single punch creates 2 holes (one on each half). Two folds create up to 4 layers, so one punch creates 4 holes.
- Order of folds matters: The sequence in which folds are made affects the final symmetry axes. Always process folds in the given order.
- Punch position is critical: A hole near a fold line creates holes close together when unfolded. A hole far from fold lines creates widely spaced holes.
- Common fold types: Horizontal fold (top to bottom or bottom to top), vertical fold (left to right or right to left), diagonal fold (corner to corner).
Watch out for
- Ignoring the order of folds: Students apply symmetry for both folds simultaneously instead of sequentially. Fix: Process each fold in the given order. Unfold in reverse order mentally (last fold first).
- Miscounting layers: Assuming two folds always create exactly 4 holes. If the punch is on a fold line, it may produce fewer holes. Fix: Check whether the punch position overlaps any fold line.
- Wrong symmetry axis: Mixing up horizontal and vertical folds. Fix: Clearly mark the fold direction with an imaginary line on your mental image or rough sketch.
Quick reference
- Each fold doubles the number of potential holes from one punch (if all layers are penetrated).
- Horizontal fold → top-bottom symmetry; vertical fold → left-right symmetry; diagonal fold → corner-to-corner symmetry.
- Punch on the fold line = fewer holes than punch away from the fold.
- Two perpendicular folds + one punch typically = 4 holes in a symmetric pattern.
- Always process folds in order and unfold in reverse to reconstruct the pattern.
Check yourself: 3 questions
1. A square sheet of paper is folded once vertically down the middle, then a circular hole is punched through all layers at the centre of the folded paper. When the paper is completely unfolded, how many holes will be visible?
2. A rectangular sheet of paper is folded in half horizontally, then folded in half vertically. A small triangular piece is cut from the corner where all the folded edges meet. When the paper is completely unfolded, what shape will be seen?
3. A square piece of paper is folded twice and then a circular hole is punched through all layers. When the paper is completely unfolded, how many holes will appear?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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