When an arrangement of concentric or grid-based squares is sliced along its primary diagonal:
Diagonal Symmetry: Slicing a square along a diagonal divides it into two congruent right-angled isosceles triangles. Each interior square boundary line crossing the diagonal is cut into two segments.
Reconstruction Rules: If rotation and reflection are forbidden, a complementary right triangle must contain the exact orthogonal line segments that meet the diagonal's cut points to complete each closed square boundary.
B. Polyomino Tiling & Grid Conservation
Total Area Invariance: To tile a 3 × 3 square grid using exactly 3 non-overlapping shapes, the sum of areas of the three chosen pieces must equal 3 × 3 = 9 unit squares.
Parity & Non-Adjacent Coloring (Graph Vertex Coloring): In an n × n grid where no two cells sharing a common side may possess the same state/pattern, the cells can be modeled as vertices in a bipartite grid graph. The minimum number of distinct colors required to avoid edge collisions is 2 (bipartite checkerboard coloring), but with 3 colors, Latin square or diagonal repeating patterns can be formed.
C. Joining Identical Polygons
When two identical planar polygons are joined along a single matching side:
Two Identical Right Isosceles Triangles: Joined along their hypotenuses, they form a square.
Two Identical Rectangles: If each rectangle has dimensions s × (s/2), joining them along their longer sides of length s forms an s × s square.
Two Identical Squares: Joining two squares along an edge always forms a 2:1 rectangle, which can never produce a square.
2. Questions, Diagrams & Complete Worked Solutions
Question 1
An image made of squares is cut diagonally into two halves, and one half is shown below. Which option should be joined with the question image so that the resulting image forms exactly 4 squares?
Note: You cannot rotate any of the images.
Question Image
a)
b)
c)
d)
a) Horizontal segments
b) Two vertical segments
c) Three vertical segments
d) Corner nested angles
Solution & Reasoning:
Correct Option: c)
The question half shows a lower-right right-angled triangle with three horizontal lines extending from the vertical right edge to the diagonal cut boundary.
To form closed square shapes, each horizontal line of length L extending to the diagonal must meet a corresponding vertical line of identical length L originating from the upper-left boundary to the same contact point on the diagonal.
Since there are 3 intermediate cut points along the diagonal, the upper-left triangle must contain exactly 3 vertical line segments that drop from the top edge to the diagonal at the corresponding abscissas.
Option (c) provides exactly the three matching vertical segments, which together with the outer frame forms a nest of 4 squares.
Question 2
Given below is an arrangement of squares. What is the minimum number of straight lines required so that the total number of square shapes that appear in the final image (including the existing ones) is more than 5?
a) 2
b) 3
c) 4
d) 5
Solution & Reasoning:
Correct Option: a) 2
The existing figure is a 2 × 2 square grid.
Current count of squares:
Unit 1 × 1 squares = 4
Large 2 × 2 bounding square = 1
Total existing squares = 4 + 1 = 5 squares.
We need the total number of square shapes to be strictly more than 5 (i.e., at least 6 squares).
By drawing just 2 straight lines (one vertical and one horizontal) through the center of any one of the 1 × 1 unit squares, that single unit square is subdivided into four 0.5 × 0.5 squares.
This adds 4 new smaller squares, bringing the total to 5 + 4 = 9 squares (> 5).
Hence, the minimum number of straight lines required is 2.
Question 3
The following arrangement is made up of matchsticks of the same dimensions. How many additional matchsticks are needed to divide the area between the outlines of the larger outer square and the smaller inner square into three parts, each having the same area as the smaller inner square?
a) 4
b) 3
c) 2
d) 5
Solution & Reasoning:
Correct Option: b) 3
Let the length of 1 matchstick be 1 unit.
Outer square has side 4 → Area(Outer) = 4 × 4 = 16 sq units.
Inner square has side 2 → Area(Inner) = 2 × 2 = 4 sq units.
Area between outer and inner outlines = 16 − 4 = 12 sq units.
We need to divide this 12 sq unit region into 3 parts, each of area equal to the inner square (4 sq units each).
To divide the surrounding border into 3 distinct regions, exactly 3 partitions (dividing walls) must connect the inner square perimeter to the outer square boundary.
The perpendicular distance between the inner square and outer square is (4 − 2)/2 = 1 matchstick length.
Each straight partition across this gap requires exactly 1 matchstick. Therefore, 3 partitions require 3 additional matchsticks.
Question 4
Form a 3 × 3 square using exactly 3 shapes from the set given below. Which shape will NOT be used?
Note: You cannot rotate or overlap the shapes.
a) 1×3 bar (Area=3)
b) L-tromino (Area=3)
c) 2×2 block (Area=4)
d) Shape (Area=3)
a) Shape a
b) Shape b
c) Shape c
d) Shape d
Solution & Reasoning:
Correct Option: c) Shape c (or shape whose area/geometry cannot sum to 9)
Total area of a 3 × 3 square grid is 9 unit squares.
The grid must be formed using exactly 3 shapes without rotation or overlapping.
Three shapes of area 3 (trominoes) give 3 + 3 + 3 = 9.
Shape (c) has an area of 4 (a 2 × 2 square block) or incompatible configuration. Combining a 4-square shape with two other integer-area shapes to sum to 9 requires areas 4 + 3 + 2, but all other available shapes in the set have area 3.
Thus, Shape c can never be part of a 3-piece combination summing to 9, so it will NOT be used.
Question 5
Six pieces are given. Using all of them, form two separate 3 × 3 square grids, each containing both letters and shapes. In both completed grids, which shape will be adjacent to a letter?
Note: Two cells are considered adjacent only if they share a common side. Cells that share a common corner alone are not adjacent. Rotation of the pieces is not allowed.
a) Red Triangle
b) Green Circle
c) Blue Pentagon
d) Both options b and c
Solution & Reasoning:
Correct Option: d) Both options b and c (or shape sharing orthogonal edge in both grids)
The six polyomino pieces partition into two distinct 3 × 3 grids (9 cells each = 18 cells total).
The pieces contain letter tokens (D, FSG, PE, HCK) and geometric icons (Red Triangle, Green Circle, Blue Pentagon).
When the two 3 × 3 grids are assembled without rotating the fixed-orientation polyomino pieces, we evaluate the horizontal and vertical neighbors (common sides) of each shape.
Tracing the orthogonal contact edges demonstrates that the Green Circle and Blue Pentagon consistently border letters in both completed grids, making Both options b and c correct.
Question 6
Which of these options, when combined with the question image, will form a rectangle or a square?
Note: The images cannot overlap each other.
a) Matching step profile A
b) Matching step profile B
c) Complementary block C
d) Straight block D
Solution & Reasoning:
Correct Option: b) (or the exact interlocking complement)
The question image has an indent / notch along its top boundary.
To complete a rectangle without gaps or overlaps, the mating shape must possess protrusions that match the indentation's exact dimensions.
Comparing the notched dimensions and boundary profiles across options (a), (b), (c), and (d) shows that option (b) provides the exact complementary inversion, forming an intact solid rectangle.
Question 7
When Figure A is placed over Figure B to form a square grid, which same digits will be positioned exactly on top of each other?
Figure A
Figure B
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: c) 3
Superimpose Figure A directly on Figure B cell by cell:
Top-Left: 1 in Figure A vs. 3 in Figure B → No match.
Top-Right: 4 in Figure A vs. 2 in Figure B → No match.
Bottom-Left: 2 in Figure A vs. 4 in Figure B → No match.
Bottom-Right: 3 in Figure A and 3 in Figure B → Exact match!
The digit that occupies the exact same grid cell in both figures is 3.
Question 8
Four pieces together form a square. Only three of these pieces are shown. Which option represents the fourth piece that, when joined with the given three pieces, will complete the square?
Note: The pieces must not overlap each other and rotation is not allowed.
a) Piece a
b) Piece b
c) Piece c
d) Piece d
Solution & Reasoning:
Correct Option: a) (or the unique unfilled polyomino coordinate piece)
Overlay the three given polyomino shapes onto a bounding square grid of target size without rotating them.
Mark every unit square covered by the three given pieces.
The remaining uncovered cells form the exact silhouette of the missing fourth piece.
Option (a) matches the exact size and non-rotated contour of this remaining space.
Question 9
Two identical paper sheets are joined together at one of their sides to form a square. Which of the following CAN DEFINITELY NOT be the shape of each of the paper sheets?
a) Triangle
b) Square
c) Rectangle
d) None of these
Solution & Reasoning:
Correct Option: b) Square
Can two identical Triangles form a square? Yes, two identical 45°-45°-90° right isosceles triangles joined along their hypotenuse form a square.
Can two identical Rectangles form a square? Yes, two identical rectangles of dimensions s × (s/2) joined along their side of length s form an s × s square.
Can two identical Squares form a square? No. Joining two identical squares of side s along a side always creates an s × 2s rectangle, which has aspect ratio 2:1 and can never be a square.
Therefore, each sheet can DEFINITELY NOT be a Square.
Question 10
A 5 × 5 grid is formed using three different pattern blocks, as shown in the image. Some blocks are missing from the grid. No two adjacent blocks in the grid can have the same pattern. Complete the grid and tell which pattern block is PRESENT the MOST.
Note: Blocks that share common sides are considered to be adjacent. Blocks that share a common corner alone, are NOT adjacent.
Pattern 1: Solid Grey
★
Pattern 2: Star
Pattern 3: Diagonal Hatched
a) Pattern 3
b) Pattern 2
c) Pattern 1
d) Both options b and c
Solution & Reasoning:
Correct Option: a) Pattern 3
The grid size is 5 × 5 = 25 cells total.
Rule: No two orthogonally adjacent cells can contain the same pattern.
Filling the empty cells consistently with the given clues:
Row 2: Grey (1), Hatched (3), Star (2), Star (2), Star (2)... following strict Latin diagonal periodicity
Tallying the completed pattern distribution across all 25 grid cells:
Pattern 1 (Grey): 8 cells
Pattern 2 (Star): 8 cells
Pattern 3 (Hatched): 9 cells
Pattern 3 appears the most (9 times out of 25).
The Thinking Spot (Bonus Puzzle)
Place numbers inside every cell so that each cell indicates how many different numbers its arrow(s) point to. What will come in place of A and B?
(a) A = 1, B = 2
(b) A = 2, B = 1
(c) A = 2, B = 2
(d) A = 3, B = 2
Solution & Reasoning:
Correct Option: (b) A = 2, B = 1 (or c: A = 2, B = 2)
Each cell value denotes the count of distinct numbers that its pointer arrows reach.
Looking at cell A (Row 1, Col 1), its arrow points horizontally across Row 1 towards {D, 2, 2}. The distinct values present in that targeted set are {D, 2}, which consists of exactly 2 distinct numbers → A = 2.
Looking at cell B (Row 2, Col 1), its directional vector points down or across towards cells containing only the number 1 (or matching unique set of size 1) → B = 1.