Fold Line
The crease is a mirror axis; folded points move to equal perpendicular distance across it.
Note 27
Paper Folding, Hole Punching and Figure Completion
Complete notes for SSC Combined Graduate Level examinations
1The Big Picture — Graphic Mind Map
Punch Point
A punch affects every paper layer lying beneath that point.
Reflection
Each reverse unfold reflects all current holes across the reopened crease.
Multiplicity
A general off-crease punch can double after each independent unfold.
Crease Exception
A punch exactly on a crease maps onto itself and does not create a distinct mirror partner.
Diagonal Fold
Reflect using perpendicular distance to the diagonal—not horizontal/vertical guessing.
Figure Completion
Continue lines, counts, shading, rotation and symmetry across the missing region.
Verification
Refold the candidate mentally; all predicted holes/lines must collapse to the shown punch/pattern.
1. Number
folds in order
➜folds in order
2. Mark
crease & direction
➜crease & direction
3. Punch
location / boundary
➜location / boundary
4. Reverse
last fold first
➜last fold first
5. Audit
symmetry & count
symmetry & count
2The Foundation — Folding, Unfolding and Completion
2.1 Anatomy of a Fold
- Crease/fold line: reflection axis.
- Fold arrow: shows which side moves.
- Stationary side: receives moving layers.
- Packet: final stacked paper.
- Layer count: number of sheets under a punch.
2.2 Fold as Reflection
For a vertical crease \(x=c\):
\[(x,y)\mapsto(2c-x,y)\]
For a horizontal crease \(y=d\):
\[(x,y)\mapsto(x,2d-y)\]
The crease itself is fixed.
2.3 Fold Types
Vertical
Left/right positions mirror; top/bottom positions remain.
Horizontal
Top/bottom mirror; left/right remain.
Diagonal
Reflect across the diagonal using equal perpendicular distance.
Radial / sector
Fold sectors around a centre; angular separation governs copies.
2.4 Unfolding Principle
Reverse order
If folds are \(F_1,F_2,\ldots,F_k\), unfold in order \(F_k,\ldots,F_2,F_1\).
Duplicate by mirror
When a crease opens, retain the current hole and add its reflection across that crease.
Fixed-point exception
If the hole lies on the reopened crease, its reflection coincides; no new distinct hole appears.
2.5 Hole-Count Logic
| Punch position | Effect on one unfold | Reason |
|---|---|---|
| General off-crease interior point | Usually doubles | Mirror point is distinct |
| Exactly on crease | Does not double | Point maps to itself |
| At intersection of two reopened creases | Can remain one | Fixed by both reflections |
| On one crease but off another | Doubles only for the other | One fixed, one distinct mirror |
| Part of packet absent/cut away | May produce fewer copies | No paper layer exists there |
2.6 Maximum Copy Count
Independent off-crease folds
With \(k\) effective mirror unfoldings, a general punch can generate:
\[N_{\max}=2^k\]
This is an upper bound, not an automatic answer.
Effective reflections
If a punch is fixed by \(s\) independent reopened creases, only the other \(k-s\) create new locations:
\[N=2^{k-s}\]
Subject to layer/cut geometry.
2.7 Layer Count and Punch Shape
Packet layers
A punch cuts all paper layers present at its location. Unequal/partial folds can produce nonuniform layer counts.
Edge punch
A semicircular notch on a folded edge may unfold into a full circular hole or paired notches, depending on whether the edge is a crease or outer boundary.
Corner cut
Identify whether the packet corner represents an original corner, one crease plus boundary, or intersecting creases.
2.8 Boundary Classification
| Packet edge/corner | After unfolding |
|---|---|
| Original outer edge | Remains an outer boundary; no mirror beyond missing paper |
| Folded crease edge | Opens and mirrors the cut/punch |
| Intersection of two crease edges | Can produce four-way symmetry |
| Original corner | Stays at an original sheet corner |
| Mixed crease–outer corner | Copies across crease only |
2.9 Diagonal Reflection
Across \(y=x\)
\[(x,y)\mapsto(y,x)\]
Horizontal and vertical offsets exchange.
Across \(y=-x\)
\[(x,y)\mapsto(-y,-x)\]
Coordinates exchange and signs reverse.
Perpendicular rule
The segment joining a point to its image is perpendicular to the crease and bisected by it.
2.10 Compound Folds
Order matters
Two different reflections generally do not commute. Track fold sequence and reverse it exactly.
Reflections in perpendicular central axes together act like a \(180^\circ\) rotation on positions.
Angular sector folds
When sectors divide a full circle equally into \(m\) parts, a general off-axis punch may repeat \(m\) times after complete unfolding.
2.11 Folded-Figure Selection
Symmetry test
Every unfolded copy must be symmetric across each reopened crease that generated it.
Distance test
Mirror partners must be at equal perpendicular distance from their crease.
Orientation test
Asymmetric punches/cuts reverse handedness in reflected copies.
2.12 Figure Completion — Feature Ledger
| Feature | Questions | Common completion rule |
|---|---|---|
| Line continuity | Which segments enter missing region? | Continue with same angle/curvature |
| Count | Dots, strokes, sides, compartments? | Arithmetic or alternating progression |
| Orientation | How much rotation each step? | Constant angular shift |
| Shading | Which sector moves or alternates? | Cyclic/inverting shade |
| Symmetry | Is an axis/centre implied? | Reflect or rotate missing part |
| Combination | Do rows/columns superpose/subtract? | Union, XOR, cancellation |
2.13 Missing-Part Completion Method
Boundary clues
Start with lines touching the missing boundary; the correct option must connect them exactly.
Global rule
Check symmetry and whole-figure structure, not merely local line matches.
Feature independence
Line movement, count and shading may follow separate simultaneous rules.
2.14 Pattern Combination Rules
Superposition
Combine all lines/features from source panels.
Cancellation
Common lines disappear; unique lines remain.
Alternation
Two forms or transformations take turns.
Movement
A fixed feature cycles through positions.
2.15 Verification Audit
| Audit | Pass condition |
|---|---|
| Fold order | Unfolded in exact reverse sequence |
| Crease symmetry | Partners are equidistant and perpendicular |
| Hole count | Matches effective reflections/layers |
| Boundary | No hole appears outside original paper |
| Shape orientation | Asymmetric cuts are mirrored correctly |
| Completion continuity | Every entering line exits/matches correctly |
| Global pattern | Count, rotation, shade and symmetry all agree |
3Short Tricks & Visual Speed Tools
3.1 Number the Creases
Write \(F_1,F_2,\ldots\); unfold in descending order. This prevents the most common mistake.
3.2 Mirror, Don’t Slide
Copy the punch across the crease at equal perpendicular distance—never translate it parallel to the crease.
3.3 Power-of-Two Screen
Off all creases, \(k\) effective folds suggest up to \(2^k\) holes. Use as a quick option filter.
3.4 Crease Punch Stays Single
A point on the fold axis is its own mirror and does not double at that unfold.
3.5 Diagonal Coordinate Swap
For the main diagonal, swap horizontal and vertical offsets.
3.6 Count Layers at Punch
Do not use total fold count blindly; inspect how many layers actually cover the punch position.
3.7 Boundary-First Completion
Reject any option whose incoming lines fail to join the surrounding figure.
3.8 XOR Line Trick
If identical lines cancel, keep only lines appearing in exactly one source panel.
3.9 Symmetry Kill
A predicted reopened crease must be an axis of the generated copies; asymmetric options die immediately.
3.10 Edge vs Crease Label
Mark packet boundaries \(O\) (outer) or \(C\) (crease). Only \(C\) creates a mirror on opening.
3.11 Track Features Separately
In completion, make mini-rows for count, direction, position and shading.
3.12 Refold the Option
A correct unfolded option collapses back to the exact shown punch/cut without extra locations.
3.13 Compact Rule Bank
| Situation | Rule |
|---|---|
| Vertical crease \(x=c\) | \((x,y)\mapsto(2c-x,y)\) |
| Horizontal crease \(y=d\) | \((x,y)\mapsto(x,2d-y)\) |
| Main diagonal | \((x,y)\mapsto(y,x)\) |
| General off-axis punch after \(k\) effective independent unfolds | Maximum \(2^k\) locations |
| Punch on \(s\) effective crease axes | Potential \(2^{k-s}\) distinct locations |
| Two perpendicular central reflections | Equivalent positional effect to \(180^\circ\) rotation |
4The SSC / TCS Traps — Red Flags
🚩 4.1 Unfolding in Fold Order
Opening must reverse the sequence: last fold opens first.
🚩 4.2 Sliding Instead of Mirroring
Copies cross the crease perpendicularly; they do not move along it.
🚩 4.3 Doubling a Crease Punch
A point on the crease maps to itself; duplicate drawings overlap.
🚩 4.4 Assuming Uniform Layers
Partial and unequal folds create different layer counts across the packet.
🚩 4.5 Outer Edge Treated as Crease
No paper exists beyond an original boundary, so no mirrored copy appears there.
🚩 4.6 Diagonal Guessing
Use perpendicular distance/coordinate swap, not horizontal visual estimates.
🚩 4.7 Right Count, Wrong Distance
Copies can have correct multiplicity but unequal crease distances.
🚩 4.8 Corner Type Ignored
An original corner, crease corner and mixed corner unfold differently.
🚩 4.9 Local Match Only
An option may connect boundary lines yet violate the global symmetry/count rule.
🚩 4.10 Union vs XOR
Know whether common lines accumulate or cancel; distractors swap these rules.
🚩 4.11 Shading Forgotten
Correct geometry with wrong shade/texture is still wrong.
🚩 4.12 No Refold Check
Extra unfolded holes must collapse onto the shown punch; otherwise the option is impossible.
Visual guardrail: every new hole is a legal reflection of an existing hole across a specific reopened crease—never an independently invented mark.
5Memory Hooks & Mnemonics
5.1 “CREASE” Workflow
Catalog folds, Reverse, Equidistance, Axis, Symmetry, Examine.
5.2 “Open the Last Door First”
Unfolding reverses the fold order.
5.3 “Mirror Walks Straight Across”
A hole crosses perpendicularly and keeps equal distance.
5.4 “On the Crease, Stay in Place”
A crease-point is its own reflection.
5.5 “Off Every Fold, Double Told”
Each effective off-axis unfold can double distinct copies.
5.6 “Edge Needs a Name”
Label every packet edge Outer or Crease before mirroring a cut.
5.7 “Meet the Border First”
Boundary line continuity eliminates completion options fastest.
5.8 “Fold the Answer Back”
The correct output must collapse to the exact packet punch.
Unfolding Recall
- Number folds.
- Open in reverse.
- Mirror every current hole.
- Respect crease fixed points.
- Check boundaries and layers.
Completion Recall
- Connect boundary lines.
- Continue count and rotation.
- Track shading independently.
- Test symmetry/combination.
- Verify the complete figure.
Final recall line: “Number the folds, open backward, mirror across each crease, and refold the answer to verify.”