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    Shape Assembly, Pattern Completion and 2D Visualization PYQs for GATE CS

    Solve 7+ Shape Assembly, Pattern Completion and 2D Visualization previous year questions for GATE CS with answers and detailed solutions. Free sample question

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    Question 1
    2026 Slot Set2 PYQ
    Level 3: Exam Standard
    Two tiles are missing in Panel I. Which one of the options in Panel II is the appropriate choice for the missing tiles?
    Panel I Panel II ? (i) (ii) (iii) (iv)
    Question 2
    2026 Slot Set2 PYQ
    Level 3: Exam Standard
    A black square PQRS has been cut into two parts. One part of it is shown in Panel I. Which one of the shapes in Panel II is the other part?
    Panel I Panel II S R P Q (i) (ii) (iii) (iv)
    Question 3
    2026 Slot Set1 PYQ
    Level 3: Exam Standard
    The figure shows two 4-tile patterns.


    Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns assuming there are no overlaps among them?
    Question 4
    2022 PYQ
    A palindrome is a word that reads the same forwards and backwards. In a game of words, a player has the following two plates painted with letters.

    AD

    From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.
    Question 5
    2022 PYQ
    Level 3: Exam Standard
    A plot of land must be divided between four families. They want their individual plots to be similar in shape, not necessarily equal in area. The land has equally spaced poles, marked as dots in the below figure. Two ropes, R1 and R2, are already present and cannot be moved.

    What is the least number of additional straight ropes needed to create the desired plots? A single rope can pass through three poles that are aligned in a straight line.

    R1R2
    Question 6
    2021 Slot Set2 PYQ
    Level 3: Exam Standard

    A jigsaw puzzle has 2 pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, rotated or flipped to assemble with the above piece.
    Question 7
    2021 Slot Set1 PYQ
    A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon.

    Which one of the following is NOT a convex polygon?
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    Shape Assembly, Pattern Completion and 2D Visualization PYQs for GATE CS

    Solve 7+ Shape Assembly, Pattern Completion and 2D Visualization previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

    Chapter Roadmap: Transformation of Shapes

    Chapter Roadmap: Transformation of Shapes

    Your Journey to Mastery

    1. Shape Assembly and Complementary Pieces (Current Topic)
    Focus: Jigsaw logic, edge matching, completing outer boundaries.
    Weightage: High foundational importance for spatial reasoning.
    2. Tile Patterns and Missing-Part Completion
    Focus: Repeating geometric motifs, grid-based missing elements.
    3. Geometric Partitioning and Similar Shapes
    Focus: Dividing areas into similar proportional shapes.
    4. Convex and Concave Polygon Identification
    Focus: Interior angles, line segment tests, polygon properties.

    Topic Hero: The Core Idea of Shape Assembly

    The Core Idea of Shape Assembly

    The intuition behind shape assembly mirrors solving a physical jigsaw puzzle. When a geometric figure is divided, the boundary between the pieces becomes a shared interface.

    To find a complementary piece, you must identify two things simultaneously:

    1. The Negative Space: The exact geometric inverse of the cut present in the given piece.
    2. The Outer Boundary: The remaining edges required to complete the specified target shape (for example, a perfect square, rectangle, or triangle).

    Shape Assembly, Pattern Completion and 2D Visualization: Solved Questions with Step-by-Step Explanations (7 Problems)

    Question 1 · Spatial Aptitude · 2026_Set2 MCQ
    Two tiles are missing in Panel I. Which one of the options in Panel II is the appropriate choice for the missing tiles?
    Panel I Panel II ? (i) (ii) (iii) (iv)
    1. A.

      (i)

    2. B.

      (ii)

    3. C.

      (iii)

    4. D.

      (iv)

    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: This is a missing-part completion problem in a 3x3 grid, solvable by identifying the numerical pattern governing the tiles.

    Step 1: Analyze the given tiles.

    Each tile is a 3x3 grid of dots, which are either open (white) or black.

    Step 2: Count the number of open dots in each visible tile.

    Row 1: Tile 1 has 1, Tile 2 has 8, Tile 3 has 3. Sum = 1 + 8 + 3 = 12.

    Row 2: Tile 4 has 6, Tile 5 has 4. To make the row sum 12, the missing Tile 6 must have 12 - 6 - 4 = 2 open dots.

    Row 3: Tile 7 has 5, Tile 8 has 0. To make the row sum 12, the missing Tile 9 must have 12 - 5 - 0 = 7 open dots.

    Step 3: Verify with columns.

    Column 1: 1 + 6 + 5 = 12.

    Column 2: 8 + 4 + 0 = 12.

    Column 3: 3 + 2 + 7 = 12. The pattern holds perfectly for both rows and columns.

    Step 4: Match with options.

    We need an option that provides a tile with 2 open dots for the first missing position (Row 2, Col 3), and a tile with 7 open dots for the second missing position (Row 3, Col 3). Option A matches this requirement exactly.

    Answer: A

    Question 2 · Spatial Aptitude · 2026_Set2 MCQ
    A black square PQRS has been cut into two parts. One part of it is shown in Panel I. Which one of the shapes in Panel II is the other part?
    Panel I Panel II S R P Q (i) (ii) (iii) (iv)
    1. A.

      (i)

    2. B.

      (ii)

    3. C.

      (iii)

    4. D.

      (iv)

    Correct Answer:

    C

    Step-by-Step Solution

    Key idea: This is a shape assembly problem requiring us to find the complementary piece that perfectly interlocks with the given piece to form a square.

    Step 1: Analyze the given piece.

    Panel I shows a black square with a white cut-out. The white cut-out represents the exact shape of the missing "other part".

    Step 2: Identify the boundary features.

    Trace the perimeter of the white cut-out. It has specific protrusions and indentations of defined lengths (e.g., a 35-unit horizontal segment, a 30-unit vertical drop, etc.).

    Step 3: Evaluate the options for the inverse shape.

    The correct option must have the exact geometric inverse of this boundary. When mentally rotated or flipped, its protrusions must fit perfectly into the given piece's indentations, and vice versa.

    Step 4: Verify the outer boundary.

    The combined shape must form a perfect square. Option (iii) provides the necessary straight outer edges to complete the square's boundary while perfectly matching the internal cut pattern.

    Answer: C

    Question 3 · Spatial Aptitude · 2026_Set1 MCQ
    The figure shows two 4-tile patterns.


    Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns assuming there are no overlaps among them?
    1. A.
    2. B.
    3. C.
    4. D.
    Correct Answer:

    C

    Step-by-Step Solution

    Key idea: This is a tile pattern construction problem. We must determine which target shape cannot be tiled using the given fundamental motifs without overlaps.

    Step 1: Analyze the given tiles.

    Tile 1 is a 2x2 block of squares (Area = 4).

    Tile 2 is a 1x4 vertical block of squares (Area = 4).

    Step 2: Identify the invariant.

    Both available tiles have an area of exactly 4 square units. Therefore, any valid pattern constructed from these tiles must have a total area that is a multiple of 4.

    Step 3: Evaluate the options based on area.

    Option A: 4x2 grid. Area = 8. (Multiple of 4, constructible using two 2x2 tiles).

    Option B: 4x3 grid. Area = 12. (Multiple of 4, constructible using three 1x4 tiles).

    Option C: 5x3 grid. Area = 15. (NOT a multiple of 4).

    Option D: 5x4 grid. Area = 20. (Multiple of 4, constructible using five 1x4 tiles).

    Step 4: Conclude.

    Since Option C has an area of 15, which is not divisible by 4, it is mathematically impossible to construct it using only tiles of area 4.

    Answer: C

    Question 4 · Spatial Aptitude · 2022 MCQ
    A palindrome is a word that reads the same forwards and backwards. In a game of words, a player has the following two plates painted with letters.

    AD

    From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.
    1. A. DDJ
    2. B. RAR
    3. C. ZED
    4. D. ILY
    Question 5 · Spatial Aptitude · 2022 MCQ
    A plot of land must be divided between four families. They want their individual plots to be similar in shape, not necessarily equal in area. The land has equally spaced poles, marked as dots in the below figure. Two ropes, R1 and R2, are already present and cannot be moved.

    What is the least number of additional straight ropes needed to create the desired plots? A single rope can pass through three poles that are aligned in a straight line.

    R1R2
    1. A.

      2

    2. B.

      4

    3. C.

      5

    4. D.

      3

    Correct Answer:

    D

    Step-by-Step Solution

    Key idea: This is a geometric partitioning problem where we must divide an irregular polygon into 4 similar shapes using the minimum number of additional straight lines (ropes).

    Step 1: Analyze the given shape and existing ropes.

    The land is a hexagon with equally spaced poles. R1 is a horizontal rope of length 2 units. R2 is a vertical rope of length 2 units.

    Step 2: Understand the goal.

    We need 4 similar plots. Similarity means they must have the same shape and angles, though their areas can differ.

    Step 3: Visualize the partition.

    To create 4 similar shapes from this specific hexagon, we need to extend the existing boundaries and create proportional divisions. By adding 3 strategic ropes (e.g., extending R1, extending R2, and adding one diagonal or connecting rope), we can divide the land into 4 regions that are scaled versions of each other.

    Step 4: Verify the minimum number.

    With 2 additional ropes, we can at most create 3 or 4 regions, but they will not maintain the required similarity in shape. With exactly 3 additional ropes, we can achieve the required 4 similar plots by completing the internal grid that mirrors the outer boundary's proportions.

    Answer: D

    Question 6 · Spatial Aptitude · 2021_Set2 MCQ

    A jigsaw puzzle has 2 pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, rotated or flipped to assemble with the above piece.
    1. A.
    2. B.
    3. C.
    4. D.
    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: This is a shape assembly problem requiring us to find the complementary piece that forms a rectangle when combined with the given jigsaw piece.

    Step 1: Analyze the given piece.

    The provided piece has a complex boundary with specific protrusions (outward bumps) and indentations (inward cuts).

    Step 2: Determine the required negative space.

    To form a rectangle, the missing piece must have the exact geometric inverse of this boundary. Every protrusion on the given piece must fit into an indentation on the missing piece, and vice versa.

    Step 3: Check the outer boundary requirements.

    The combined shape must be a rectangle. Therefore, the missing piece must supply the straight outer edges that the given piece lacks to complete the rectangular perimeter.

    Step 4: Evaluate the options.

    Option A, when appropriately rotated or flipped, provides the exact complementary boundary features and the necessary straight edges to form a perfect rectangle without gaps or overlaps. Options B, C, and D have mismatched sequences of bumps and cuts.

    Answer: A

    Question 7 · Spatial Aptitude · 2021_Set1 MCQ
    A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon.

    Which one of the following is NOT a convex polygon?
    1. A.
    2. B.
    3. C.
    4. D.

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