Paper Folding — The Fold-and-Punch Rule
Paper Folding — fold karo, punch karo, ginti karo
Paper Folding — The Fold-and-Punch Rule
- Paper Folding & Cutting
- Paper Folding — The Fold-and-Punch Rule
Predict the number and position of holes when a folded-and-punched paper is opened.
🎯 Learning Objective
Predict the number and position of holes when a folded-and-punched paper is opened.
💡 Concept
- Every fold DOUBLES the layers: after f folds there are 2^f layers of paper
- A punch goes through all layers, so total holes = punches × 2^folds
- 1 fold + 1 punch = 2 holes; 2 folds + 1 punch = 4 holes; 2 folds + 2 punches = 8 holes
- Positions: unfold in REVERSE order — at every unfold, each hole gets a mirror copy across that crease
- Every crease acts as a mirror line for the holes on the opened sheet
- Caution: a punch sitting exactly ON a crease or missing some layers breaks the 2^f count
🧮 Key Formulas
Holes = Punches × 2^(number of folds)
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Layers after f folds = 2^f
✏️ Easy Example
Q. A sheet is folded in half once and one hole is punched through it. How many holes appear on opening?
- Folds f = 1 → layers = 2^1 = 2
- Holes = 1 punch × 2 = 2
Answer: 2 holes, mirror images of each other across the fold line
🇮🇳 Real-Life Example
The ticket-checker's punch through a folded bus ticket leaves two holes — open the ticket and both halves carry the mark at mirrored spots.
📝 Exam-Level Example
Q. A square sheet is folded left-half onto right-half, then top-half onto bottom-half. One hole is punched at the centre of the small folded square. How many holes on opening, and where?
- Folds = 2 → holes = 1 × 2^2 = 4
- Unfold top fold: hole mirrors across the horizontal centre crease
- Unfold side fold: both holes mirror across the vertical centre crease
Answer: 4 holes — one at the centre of each quarter of the sheet
📝 Exam-Level Example
Q. A paper is folded 3 times and 2 holes are punched through all layers. Total holes on opening?
- Layers = 2^3 = 8
- Holes = 2 × 8 = 16
Answer: 16
🪄 Memory Trick
Count first, place later: write 2^folds × punches for the number, then use 'every crease = mirror' to reject options with wrong positions.
⚠️ Common Mistakes
- ❌ Multiplying by the number of folds instead of 2^folds (3 folds → ×8, not ×3)
- ❌ Mirroring holes in the same direction for every unfold instead of across each fold's own crease
- ❌ Ignoring that a punch on the crease itself gives fewer, merged holes
🏆 Exam Tips
- ✅ Unfold in exactly the reverse order of folding — last fold opens first
- ✅ Eliminate options by hole COUNT before checking hole positions — it kills 2 options instantly
📌 Summary
- Each fold doubles layers: 2^f
- Holes = punches × 2^folds
- Unfold in reverse; mirror holes across each crease
- Wrong count in options = instant elimination