5.2 Cube Nets (Küp Açılımları), Opposite-Face Parities & 3D Die Rotations

5.2 Cube Nets (Küp Açılımları), Opposite-Face Parities & 3D Die Rotations

Two of the most iconic spatial reasoning question types on the TR-YÖS Sayısal Yetenek Testi are Cube Nets (Küp Açılımları) — where an unfolded 6-square flat pattern must be matched to its folded 3D cube — and 3D Die Rotations (Küp / Zar Döndürme ve Yuvarlama) — where a die or patterned cube is rolled along a grid path or rotated in space. Attempting to mentally fold and spin a 3D object in working memory is slow and prone to mirror-image reversals. Instead, you can use face-pair and orientation checks to solve fully specified cube exercises.

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1. The 11 Valid Hexomino Cube Nets (11 Geçerli Küp Açılımı)

Slicing a hollow cube along 7 of its 12 edges unfolds its 6 square faces into a flat, edge-connected 6-square polygon called a hexomino. Out of 35 possible free hexominoes, exactly 11 fold into a valid closed cube without overlapping faces:

1. The $1\text{-}4\text{-}1$ Family (6 Nets): A central spine of 4 collinear squares with 1 wing square attached on the left/top side and 1 wing square attached on the right/bottom side (including the classic "cross" and "T" shapes). The two wings can be attached at any of the 4 spine squares without colliding.
2. The $2\text{-}3\text{-}1$ Family (3 Nets): A middle row of 3 squares, an offset row of 2 squares on one side (sharing only a single edge with an end square of the 3-row), and 1 wing square on the other side.
3. The $2\text{-}2\text{-}2$ Staircase (1 Net): Three parallel pairs of 2 squares offset like stairs.
4. The $3\text{-}3$ Double Strip (1 Net): Two parallel strips of 3 squares overlapping along only 1 end square.

Instant Invalid-Net Traps (Küp Oluşturmayan Şekiller): Any net containing a $2 \times 2$ block of 4 squares, a straight strip of 5 squares, a U-cup of 5 squares (C-shape), or two wing squares on the same side of a 4-square spine will cause two squares to fold onto the exact same 3D face, leaving the opposite face open.

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2. Finding All Three Opposite-Face Pairs (Karşılıklı Yüzey Kuralları)

Every 3D cube has 6 faces grouped into 3 mutually opposite pairs:

$$\text{Pair 1: } (A \leftrightarrow D), \qquad \text{Pair 2: } (B \leftrightarrow E), \qquad \text{Pair 3: } (C \leftrightarrow F)$$

Rule A — The Straight-Line Alternating Rule (1-Atla-3 Kuralı)

Whenever 3 or 4 squares lie in a contiguous straight row or column, skipping exactly one intermediate square always lands on the opposite face in 3D space:

• In a 4-square spine [1][2][3][4], square 1 is opposite 3 ($1 \leftrightarrow 3$) and square 2 is opposite 4 ($2 \leftrightarrow 4$).
• In a 3-square strip [A][B][C], the two outer squares fold up $90^\circ$ parallel to each other while B remains the base, so $A \leftrightarrow C$.

Rule B — The Z-Path Fold Rule (Z Kuralı / L-Atlama)

For a path of three hinges in a valid net, a Z pattern (for example East, South, East) places the end faces opposite. A straight three-square strip also makes its end faces opposite. A one-step-plus-two-step L path is not equivalent to a Z path and must not be used as a universal shortcut. For an unfamiliar net, propagate the face orientation across each actual shared hinge and verify all six faces rather than relying only on planar displacement.

• Even simpler for $1\text{-}4\text{-}1$ nets: Once you pair $(1 \leftrightarrow 3)$ and $(2 \leftrightarrow 4)$ along the 4-square spine, the two remaining wing squares on opposite sides of the spine are ALWAYS opposite each other regardless of which spine squares they touch!

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3. The 10-Second Elimination Rule (Aynı Anda Görünemezler)

In any perspective drawing of a solid cube, the viewer sees 3 faces meeting at a single central corner vertex (köşe noktası).

• Fundamental Visibility Law: Two opposite faces are parallel in 3D space and share no edge or vertex. Therefore, in the usual three-face solid-cube view, opposite faces cannot both be visible or both belong to the hidden three.
• Every valid 3D view must pick exactly one member from Pair 1 $(A \text{ or } D)$, one member from Pair 2 $(B \text{ or } E)$, and one member from Pair 3 $(C \text{ or } F)$.
• Exam Execution: As soon as you list the 3 opposite pairs, scan the five answer choices (A–E). Any 3D cube that shows both members of an opposite pair is immediately eliminated. The number of choices eliminated depends on the actual options.

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4. Resolving Surviving Options: Handedness & $90^\circ$ Edge Rolling

When two answer choices survive the opposite-face check, they display the same three valid faces $\{X, Y, Z\}$ but differ in either cyclic vertex order or internal symbol rotation.

Technique 1 — Clockwise Vertex Cycle (Saat Yönü Köşe Döngüsü)

Pick the shared vertex where faces $X, Y, Z$ meet on the 3D drawing and read their names going clockwise (saat yönünde) around that point: $X \to Y \to Z$ (which is identical to $Y \to Z \to X$ and $Z \to X \to Y$). Fold those three squares mentally around their common vertex in the 2D net (always folding away from you into the page so the printed face stays on the outside). If an option shows the counterclockwise order $X \to Z \to Y$, it is a mirror-inverted fake and must be eliminated.

Technique 2 — $90^\circ$ Corner Edge Rolling & 4-Square Wrap-Around

What if faces $X, Y, Z$ do not touch at a single vertex in the 2D net?

1. $90^\circ$ Corner Roll (Köşe Etrafında 90° Yuvarlama): Any square adjacent to an empty L-shaped corner gap between two perpendicular edges can be "rolled" $90^\circ$ across that corner so those two edges zip together. Crucial Rule: Rolling a square $90^\circ$ clockwise (or counterclockwise) across an empty corner rotates both the square AND any asymmetric symbol inside it (an arrow $\uparrow$, a shaded right triangle, a letter L, or an off-center dot) by $90^\circ$ in the exact same direction!
2. 4-Square Spine Wrap-Around (4'lü Şerit Kaydırma): In a 4-square strip [1][2][3][4], detaching square 1 from the left end and reattaching it to the right of square 4 ([2][3][4][1]) preserves its orientation with $0^\circ$ rotation because four $90^\circ$ folds wrap $360^\circ$ around the cube.

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5. 3D Die Rolling on a Grid Track (Küp / Zar Yuvarlama)

In grid-rolling problems, a die sits on a starting square with specified Top (Üst), Front/South (Ön), and Right/East (Sağ) faces, and rolls $90^\circ$ step by step along a path of connected grid squares.

• Standard Die Rule (Standart Zar): Opposite faces always sum to $7$: $(1 \leftrightarrow 6),\; (2 \leftrightarrow 5),\; (3 \leftrightarrow 4)$. Thus, knowing Top, North (Back), and East (Right) automatically locks Bottom $= 7 - \text{Top}$, South (Front) $= 7 - \text{North}$, and West (Left) $= 7 - \text{East}$.
• 4-Roll Straight-Line Periodicity: Rolling a die 4 times in the same straight direction returns every face to its original position.

Reference Table: 3D Die $90^\circ$ Roll Transitions & Net Orientation Laws

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Worked Examples

Worked Example 5.2.1 — Cube Net Elimination & $90^\circ$ Symbol Rotation

Problem: A $1\text{-}4\text{-}1$ cube net has a horizontal 4-square spine [A][B][C][D] from left to right. Wing square [E] is attached above [B], and wing square [F] is attached below [C]. Inside square [E] is an arrow pointing Up (North, away from [B]). When the net is folded into a cube with printed symbols on the outside: (a) Find the three opposite-face pairs. (b) If the folded cube is placed with [C] on the Front face and [B] on the Left face, which face is on Top, and in which direction does the arrow on [E] point?

Solution:

1. (a) Opposite-Face Pairs:
   • Along the 4-square spine [A][B][C][D], the $1\text{-skip-}3$ rule gives $\mathbf{A \leftrightarrow C}$ and $\mathbf{B \leftrightarrow D}$.
   • The two remaining wing squares on opposite sides of the spine form the third pair: $\mathbf{E \leftrightarrow F}$.
2. (b) Top Face and Arrow Orientation via $90^\circ$ Corner Roll:
   • In the net, [B] is to the left of [C], and [E] is directly above [B]. When [C] is placed at the Front and [B] folds $90^\circ$ back to become the Left face, [E] (which is above [B] and adjacent to the empty corner above [C]) folds over the top to become the Top face.
   • To see how the arrow inside [E] orients relative to [C], roll [E] by $90^\circ$ clockwise across the empty corner between [E] and [C] so that [E] sits directly above [C] in the 2D net.
   • Because [E] rotated $90^\circ$ clockwise, its internal arrow (originally pointing Up/North) also rotates $90^\circ$ clockwise to point Right (East)—which, relative to [C] at the Front and [D] on the Right face, points directly toward face [D] (Right/East)!

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Worked Example 5.2.2 — Multi-Step Die Rolling Along an L-Shaped Grid Track

Problem: A standard die (opposite faces sum to $7$) starts on cell $(0, 0)$ with $\text{Top} = 1$, $\text{Front (South)} = 2$, and $\text{Right (East)} = 3$. It rolls 2 steps East, then 2 steps North, and finally 1 step West. Which number is on the Top face and which is on the Front (South) face at the end of the path?

Solution:

1. Initial 6-Face State at $(0, 0)$:
   • Since opposite faces sum to $7$: $B = 7 - 1 = 6$, $N = 7 - 2 = 5$, $W = 7 - 3 = 4$.
   • State $(T, B, S, N, E, W) = (1, 6, 2, 5, 3, 4)$.
2. Roll 2 Steps East (Right):
   • Shortcut: Rolling $180^\circ$ (2 steps) along the East–West axis swaps $T \leftrightarrow B$ and $E \leftrightarrow W$, while keeping $S$ and $N$ fixed!
   • After 2 steps East at $(2, 0)$: $(T, B, S, N, E, W) = (6, 1, 2, 5, 4, 3)$.
3. Roll 2 Steps North (Up the Grid):
   • Shortcut: Rolling $180^\circ$ (2 steps) along the North–South axis swaps $T \leftrightarrow B$ and $S \leftrightarrow N$, while keeping $E$ and $W$ fixed!
   • After 2 steps North at $(2, 2)$: $(T, B, S, N, E, W) = (1, 6, 5, 2, 4, 3)$.
4. Roll 1 Step West (Left):
   • Rolling West replaces $T$ with old $E$ ($4$), $W$ with old $T$ ($1$), $B$ with old $W$ ($3$), and $E$ with old $B$ ($6$), while $S = 5$ and $N = 2$ remain unchanged.
   • Final State at $(1, 2)$: $\mathbf{\text{Top} = 4}$ and $\mathbf{\text{Front (South)} = 5}$.

Review the Method

Rolling a die 90 degrees along a grid axis leaves the two perpendicular side faces unchanged while cycling the four faces in the rolling plane: Top -> Roll Direction -> Bottom -> Opposite Direction -> Top.