Hole Punching

Strategy/Review

The Reverse-Fold Method (Gold Standard)

This is the technique that virtually all high scorers use. Instead of trying to visualize the whole folding process forward, you work backward from the final folded state.

Step 1 — Identify the Final Folded State. Look at the last image in the sequence. This is the paper after all folds are complete. There's a dot showing where the hole was punched.

Step 2 — Work Backward, One Fold at a Time. Take the last fold and mentally UNFOLD it. The hole(s) that exist after this unfold = the hole(s) before this unfold PLUS their mirror images reflected across the fold line.

Step 3 — Repeat. Continue unfolding backward through the sequence. For each unfold, reflect every existing hole across that fold line and add the new holes to your mental map. By the time you reach the original unfolded square, you'll have all hole positions.

Step 4 — Match to Answer Choices. Count the holes, check their positions, and verify no holes are missing or extra.

The Reflection Rule

Here's the core operation you need to master: when you unfold along a fold line, every existing hole is REFLECTED (mirrored) across that line to create a new hole on the other side. The original hole stays where it is. The fold line acts like a mirror.

This means:

• Holes come in pairs symmetric about fold lines.
• If folding then punching creates one hole in the final state, unfolding creates 2 holes after the first unfold, 4 after the second, 8 after the third, and so on — doubling each time (unless holes overlap).

Fold Types and Their Reflections

Horizontal Fold (across the middle, left to right): Reflect vertically. A hole at (x, y) creates a new hole at (x, mirror_y) where mirror_y is the same distance on the other side of the horizontal fold line.

Vertical Fold (across the middle, top to bottom): Reflect horizontally. A hole at (x, y) creates a new hole at (mirror_x, y) where mirror_x is on the other side of the vertical fold line.

Diagonal Fold (corner to opposite corner): Reflect across the diagonal. If the fold goes from bottom-left to top-right, every hole reflects to a position where x and y swap AND shift relative to the diagonal. For the 45° diagonals, this means the hole's perpendicular distance to the diagonal is preserved on the other side.

Corner Fold (a corner folded to the center or an edge): The fold line is at 45° to the edges. Reflect across this angled line — a hole near the folded corner creates a mirror hole also near the corner region.

Coordinate-Grid Examples

Example 1 — Single Vertical Fold

Imagine a 4×4 grid. The paper is folded left half over right half (vertical fold at x=2.5). A hole is punched at grid position (3, 3) — that's in the right half, third row from top.

Unfold: the fold line is vertical at x=2.5. The original hole is at (3, 3). Its mirror across x=2.5 is at (2, 3) — same row, but same distance on the left side (3 is 0.5 right of 2.5, so mirror is 0.5 left = 2).

Result: holes at (2, 3) and (3, 3).

Example 2 — Two Folds: Horizontal then Vertical

Grid: 4×4.

• Fold 1: bottom half folded up (horizontal fold at y=2.5).
• Fold 2: right half folded left (vertical fold at x=2.5).
• Punch hole at position that is now (3, 2) in the final folded state.

Working backward:

• First unfold (vertical): starting hole at (3, 2), mirror across x=2.5 → new hole at (2, 2). Now have holes at (2, 2) and (3, 2).
• Second unfold (horizontal): reflect both across y=2.5. Hole (2, 2) → mirror at (2, 3). Hole (3, 2) → mirror at (3, 3).
• Final: holes at (2, 2), (3, 2), (2, 3), (3, 3) — all four quadrants filled.

Common Traps

Trap 1 — Forward Thinking. Students try to mentally simulate the entire folding sequence forward, tracking where the punch goes through. This is vastly harder than the reverse method. The paper folds obscure the punch position, and you have to track layer stacking. Reverse thinking eliminates the need to imagine layers.

Trap 2 — Forgetting the Original Hole. When you unfold, the original hole stays. Some students reflect the hole and then treat the original as "moved." No — the original stays and a new copy appears on the other side.

Trap 3 — Overlapping Holes. If a hole reflects to exactly the same position as an existing hole, they overlap and you see only one hole, not two. This happens when the punch is ON the fold line. A punch exactly on a fold line does NOT double on that unfold (because the reflection lands on the original). But on subsequent unfolds involving other fold lines, it will still double.

Trap 4 — Diagonal Fold Confusion. Diagonal reflections are spatially harder. Students often misjudge the perpendicular distance to the diagonal fold line. Practice diagonal folds specifically — trace the perpendicular from the hole to the diagonal, then extend an equal distance on the other side.

Trap 5 — Losing Track on Multiple Folds. With 3+ folds, hole counts can reach 8 or 16. Tracking 8+ hole positions mentally is demanding. Use a systematic approach: number your unfolds, and with each step, explicitly list the new holes created. Better yet, lightly mark them on scratch paper or imagine a numbered grid.