5.3 Paper Folding & Hole Punching (Kağıt Katlama-Kesme), Mirror & Water Reflections

5.3 Paper Folding & Hole Punching (Kağıt Katlama-Kesme), Mirror & Water Reflections

The final pillar of 2D/3D spatial transformation on the TR-YÖS Sayısal Yetenek Testi combines Paper Folding and Cutting (Kağıt Katlama ve Kesme / Delme) with Planar Reflections (Ayna ve Su Yansıması). Both topics are governed by the exact same geometric operation: orthogonal reflection across a crease or mirror line (simetri eksenine göre dik yansıma). Once you replace mental guesswork with coordinate-reflection laws, even 4-fold diagonal hole-punch problems and alphanumeric mirror codes become effortless.

---

1. Paper Folding & Hole Punching (Kağıt Katlama ve Kesme)

In a paper-folding question, a flat sheet (usually a square or circle) undergoes $k$ sequential folds along dashed crease lines (kat çizgileri), after which one or more holes or notches are punched through all stacked layers. You must identify how the sheet looks when unfolded completely flat.

Rule 1 — Reverse-Order Unfolding (Son Katlanan İlk Açılır — LIFO Principle)

If a sheet is folded in the sequence $\text{Fold } 1 \to \text{Fold } 2 \to \text{Fold } 3$, you must unfold it in the exact reverse sequence:

$$\text{Unfold } 3 \longrightarrow \text{Unfold } 2 \longrightarrow \text{Unfold } 1$$

At each step, treat the active crease line as a flat mirror. Copy every hole and notch currently in the folded packet across that crease line using the two mirror invariants:

1. Distance Invariant (Yakın-Uzak Kuralı): Features close to the crease reflect to positions close to the crease; features far from the crease reflect to positions far from the crease.
2. Orientation Reversal: A cut pointing toward the crease reflects into a cut pointing toward the crease from the opposite side.

Rule 2 — The $2^k$ Layer Multiplication Rule & Crease-Cut Merging

Every time a sheet is folded in half across an axis of symmetry, the number of stacked paper layers doubles. After $k$ independent half-folds, the packet contains $2^k$ layers ($k = 1 \implies 2$ layers; $k = 2 \implies 4$ layers; $k = 3 \implies 8$ layers).

• Case A — Strictly Interior Punch (İç Bölge Deliği): If a hole (circle, triangle, square) is punched completely inside the boundaries of the folded packet without touching any crease line or free outer edge, it pierces all $2^k$ layers separately and produces exactly $2^k$ distinct holes on the unfolded sheet.
• Case B — Cut on a Folded Crease Line (Kat Çizgisi Üzerinden Kesim): When a notch (such as a semicircle, V-cut, or quarter-circle) is cut into a folded crease edge, unfolding across that crease causes the cut and its mirror image to fuse together (birleşir) along the crease:
  • A semicircle cut into a single crease unfolds into 1 full circle centered on the crease.
  • A right-angled V-notch cut into a single crease unfolds into 1 diamond (rhombus) centered on the crease.
  • A quarter-circle cut at the central corner vertex where two perpendicular creases meet unfolds across both creases into 1 central circle (instead of 4 separate holes!).
• Case C — Cut on an Unfolded Outer Edge (Açık Dış Kenar Kesimi): A notch cut into an original outer perimeter edge never merges into an interior hole; it produces open notches along the perimeter of the unfolded sheet.

Rule 3 — Diagonal Folds ($45^\circ$ Creases Swap Horizontal $\leftrightarrow$ Vertical)

When a square sheet is folded along a $45^\circ$ diagonal crease ($y = x$ or $y = -x$), reflecting across the diagonal swaps the horizontal and vertical coordinates!

• A horizontal slot (—) cut near the top edge reflects across a $45^\circ$ diagonal crease into a vertical slot (\lvert) near the lateral edge.
• An arrow pointing North ($\uparrow$) reflects across the main diagonal $y = x$ into an arrow pointing East ($\rightarrow$), and across $y = -x$ into an arrow pointing West ($\leftarrow$).

---

Character symmetry depends on the actual font. The letter lists below assume ideal geometric block capitals and equally sized digit lobes; use the drawn glyph rather than treating the lists as font-independent facts.

2. Mirror Reflection (Ayna Yansıması) vs. Water Reflection (Su Yansıması)

TR-YÖS tests two perpendicular types of planar reflections on geometric figures, alphanumeric strings, and analog clocks:

A. Vertical Mirror (Dikey Ayna — Mirror Placed on the Left or Right)

A vertical mirror line (parallel to the $y$-axis) performs the coordinate reflection $(x, y) \mapsto (-x, y)$, flipping Left $\leftrightarrow$ Right (East $\leftrightarrow$ West) while leaving Top and Bottom unchanged:

1. String Order Reverses Completely: In a word or code such as YÖS2026, the rightmost character (6) is closest to a right-hand mirror and therefore appears first (leftmost) in the mirror image, reversing the sequence from right to left.
2. Vertically Symmetric Characters Stay Unchanged: The 11 uppercase letters and 2 digits that possess a vertical axis of symmetry look identical in a vertical mirror:

$$\mathbf{A,\; H,\; I,\; M,\; O,\; T,\; U,\; V,\; W,\; X,\; Y} \quad (\text{plus Turkish } \mathbf{\ddot{O},\; \ddot{U}} \text{ and digits } \mathbf{0,\; 8})$$

1. Horizontally Asymmetric Characters Flip Left-to-Right: Letters B, C, D, E, F, G, J, K, L, N, P, Q, R, S, Z and digits 1, 2, 3, 4, 5, 6, 7, 9 reverse their horizontal facing (for instance, L $\to$ \rfloor, S $\to$ 2-like curve, p $\leftrightarrow$ q, b $\leftrightarrow$ d).

B. Horizontal Mirror / Water Reflection (Yatay Ayna / Su Yansıması — Mirror at Bottom or Top)

A horizontal mirror line or water surface placed beneath the object performs the coordinate reflection $(x, y) \mapsto (x, -y)$, flipping Top $\leftrightarrow$ Bottom (North $\leftrightarrow$ South) while leaving Left and Right in their exact original order:

1. String Order Remains Left-to-Right: The 1st character on the left stays 1st on the left! Any answer choice that reverses the character sequence in a water reflection is an instant trap.
2. Horizontally Symmetric Characters Stay Unchanged: In standard symmetrical block font, the 9 uppercase letters with a horizontal midline of symmetry look unchanged in a water reflection:

$$\mathbf{B,\; C,\; D,\; E,\; H,\; I,\; K,\; O,\; X}$$

(Note: Both H, I, O, X and digits 0, 8 are doubly symmetric across both vertical and horizontal axes!) 3. Vertically Asymmetric Characters Invert Upside Down: Letters A, F, G, J, L, M, N, P, Q, R, S, T, U, V, W, Y, Z flip upside down (M $\leftrightarrow$ W, V $\leftrightarrow$ \wedge, U $\leftrightarrow$ \cap, T $\leftrightarrow$ $\perp$).

---

3. The Mirror Clock Time Formula (Aynadaki Saat Formülü)

When a standard 12-hour analog wall clock is viewed in a vertical mirror, the vertical $12\text{--}6$ axis acts as the line of reflection. Every clockwise angle $\theta$ from $12\text{:}00$ is reflected to $360^\circ - \theta$, meaning the sum of the real time and the mirror time always equals $12\text{:}00$ (or $24\text{:}00$ if the time is between $11\text{:}01$ and $12\text{:}59$). Borrowing 1 hour as 60 minutes gives the universal subtraction shortcut:

$$\text{Real Time} = \mathbf{11\text{:}60} - \text{Mirror Time} \qquad \text{(or } \mathbf{23\text{:}60} - \text{Mirror Time \text{ when hours } = 11 \text{ or } 12)}$$

• Example 1: Mirror clock reads $04\text{:}25 \implies \text{Real Time} = 11\text{:}60 - 04\text{:}25 = \mathbf{07\text{:}35}$.
• Example 2: Mirror clock reads $12\text{:}18 \implies \text{Real Time} = 23\text{:}60 - 12\text{:}18 = \mathbf{11\text{:}42}$.

Summary Reference Table: Paper Folding & Reflection Transformations

---

Worked Examples

Worked Example 5.3.1 — Multi-Fold Hole Counting with Crease Cuts

Problem: A square sheet of paper is folded in half 3 times in sequence: (1) bottom half folded up over the horizontal midline, (2) right half folded left over the vertical midline, and (3) the remaining quadrant folded in half along the diagonal from the original centre to its opposite outer corner, forming an 8-layer triangle with a 45-degree angle at the original centre. Then three cuts are made:

• Cut 1: A small circular hole is punched strictly in the interior of the triangle (touching no edges).
• Cut 2: A semicircular notch is cut into the vertical folded crease edge (away from the corners).
• Cut 3: A small 45-degree circular sector is cut out at the vertex corresponding to the original centre. Cuts are small and separated so their unfolded copies do not overlap. When the paper is completely unfolded, how many total circular holes appear in the interior of the sheet?

Solution:

1. Cut 1 (Strictly interior circle across $k = 3$ half-folds / 8 layers):
   • Because the punch touches no crease, it appears once on each of the $2^3 = 8$ layers $\implies \mathbf{8}$ small circular holes.
2. Cut 2 (Semicircle on the vertical midline crease):
   • The vertical edge of the folded triangle consists of 4 folded layers (2 pairs of layers joined along the vertical and horizontal midline creases). Unfolding across the $45^\circ$ diagonal reflects the semicircle onto the horizontal midline crease; unfolding across the vertical and horizontal midlines fuses each semicircle with its mirror image across the crease into a full circle $\implies \frac{8}{2} = \mathbf{4}$ full circular holes along the midlines.
3. Cut 3 (45-degree sector at the center vertex of the original sheet):
   • All 8 layers meet at the center point $(0, 0)$ of the square ($45^\circ \times 8 = 360^\circ$). Unfolding all 8 sectors fuses the eight $45^\circ$ sectors into $\mathbf{1}$ single circular hole right at the center of the sheet.
4. Total Interior Circular Holes: $8 + 4 + 1 = \mathbf{13\text{ circular holes}}$.

---

Worked Example 5.3.2 — Vertical Mirror vs. Water Reflection of an Alphanumeric Code

Problem: Consider the 6-character code MATH83. Compare its reflection in (a) a vertical mirror placed to its right (Dikey Ayna) and (b) a horizontal water surface placed beneath it (Su Yansıması). Which characters retain both their exact shape and orientation in each reflection?

Solution:

1. (a) Vertical Mirror to the Right (Dikey Ayna):
   • The left-to-right sequence reverses to read the mirror images of 3 - 8 - H - T - A - M from left to right.
   • Check vertical symmetry of each character: M, A, T, H, and 8 all belong to the vertically symmetric set {A, H, I, M, O, T, U, V, W, X, Y, 0, 8}, so their individual shapes remain completely unchanged! Only 3 becomes its reflected glyph; do not substitute a different printed character for that glyph.
   • Thus, the mirror string looks like the reflected glyph of 3, followed by 8HTAM, where 5 characters (M, A, T, H, 8) keep their individual upright shape (though in reversed positions).
2. (b) Horizontal Water Reflection Below (Su Yansıması):
   • The left-to-right position sequence stays M - A - T - H - 8 - 3.
   • Check horizontal symmetry of each character: H, 8, and 3 (assuming symmetric top and bottom loops on 8 and 3) have horizontal symmetry and keep their exact shape AND position! Meanwhile, M, A and T invert vertically; their exact reflected shapes depend on the supplied font.

Review the Method

A horizontal mirror or water reflection (Su Yansıması) flips North <-> South ((x, y) -> (x, -y)) without reversing string order, while analog mirror clock problems are solved via Real Time = 11:60 - Mirror Time.