Question 1

A large square is divided into 16 equal smaller squares arranged in a 4×4 grid. How many times does the basic square shape appear as an embedded figure when you count all possible squares of any size that can be formed within this grid?

Accepted Answer

In a 4×4 grid, the total count of squares is 4² + 3² + 2² + 1² = 16 + 9 + 4 + 1 = 30, using the formula for embedded squares in an n×n grid.

Question 2

A rectangle is divided into 12 equal smaller rectangles arranged in a 3×4 grid (3 rows and 4 columns). Counting all possible rectangles of any size that can be formed by combining adjacent cells, how many embedded rectangles exist in total?

Accepted Answer

For an m×n grid of rectangles, the total embedded rectangles = [m(m+1)/2] × [n(n+1)/2] = [3×4/2] × [4×5/2] = 6 × 10 = 60.

Question 3

In a figure with 3 intersecting straight lines (no two parallel, no three concurrent), how many distinct triangles are formed?

Accepted Answer

Three non-parallel, non-concurrent lines create exactly 4 triangular regions: one central triangle and three corner triangles formed by the intersection points.

Question 4

When the number '12345' is reflected in a horizontal water mirror, how does it appear?

Accepted Answer

A horizontal water mirror creates a vertical flip (upside-down reflection), reversing the orientation of digits top-to-bottom.

Question 5

If the word 'LOGIC' is reflected in a vertical mirror, which of the following represents the mirror image?

Accepted Answer

In a vertical mirror image, letters are horizontally flipped from right to left, so LOGIC becomes CIGOL.

Question 6

A square sheet of paper is folded diagonally to form a triangle, then folded diagonally again in the same direction. If a small corner is cut from the folded paper, how many separate pieces will result when the paper is completely unfolded?

Accepted Answer

Two diagonal folds create 4 layers of paper; cutting one corner through all 4 layers results in 4 separate pieces when unfolded.

Question 7

A rectangular grid has 4 rows and 5 columns of small squares. How many total rectangles can be formed by combining adjacent squares in this grid?

Accepted Answer

Number of rectangles = (rows × (rows+1)/2) × (columns × (columns+1)/2) = (4×5/2) × (5×6/2) = 10 × 15 = 150... recalculating: rectangles = C(5,2) × C(4,2) = 10 × 6 = 60.

Question 8

In a 3×3 figure matrix, the first row contains: Circle, Square, Triangle. The second row contains: Square, Triangle, Circle. The third row contains: Triangle, Circle, ?. Each shape appears exactly once in every row and column. What should replace the question mark?

Accepted Answer

Each row and column must contain all three shapes (Circle, Square, Triangle) exactly once; Column 3 is missing Square, so the answer is Square.

Question 9

In a 2×2 figure matrix, the top-left cell shows 4 dots, top-right shows 8 dots, and bottom-left shows 6 dots. If the pattern follows that each row's sum equals 14, how many dots should be in the bottom-right cell?

Accepted Answer

Top row: 4 + 8 = 12, bottom row must sum to 14; therefore 6 + ? = 14, so ? = 2 dots.

Question 10

A rectangular sheet of paper is folded in half along its length, then folded in half again along its width. If the sheet is then unfolded completely, how many rectangular sections will be visible on the paper?

Accepted Answer

Each fold doubles the sections created; with 2 folds (one along length, one along width), the paper is divided into 2 × 2 = 4 rectangular sections when completely unfolded.

Question 11

A die is placed on a table with face 3 touching the surface. Face 5 is facing north. Which face is facing south?

Accepted Answer

On a standard die, opposite faces sum to 7; since face 5 faces north, face 2 (opposite to 5) faces south.

Question 12

A standard cube has its faces numbered 1 to 6, where opposite faces always sum to 7. If face 1 is on top and face 2 is facing you, which face is on the bottom?

Accepted Answer

Since opposite faces sum to 7, the face opposite to 1 is 6 (1 + 6 = 7), so the bottom face is 6.

Question 13

A 3×3 matrix shows that each cell contains a number. Row 1: 2, 4, 6. Row 2: 3, 6, 9. Row 3: 5, 10, ?. The pattern in each row is that every element is a multiple of the first element. What is the missing number?

Accepted Answer

In each row, elements are multiples of the first number; row 3 starts with 5, and the third element should be 5×3 = 15, following the same multiplier pattern as rows 1 and 2.

Question 14

In a 2×2 figure completion matrix, the top-left cell has 3 dots arranged in a triangle. The top-right cell has 4 dots in a square. The bottom-left cell has 5 dots in a pentagon shape. Based on the pattern of increasing dots and polygon sides, how many dots should appear in the bottom-right cell?

Accepted Answer

The pattern shows dots increase by 1 and polygon sides increase by 1 per cell (3→4→5→6 dots, triangle→square→pentagon→hexagon), so the missing cell contains 6 dots arranged in a hexagon.

Question 15

A rectangle measuring 12 cm × 6 cm is divided into 12 identical smaller rectangles. If a hexagon is embedded within this grid such that it covers exactly 5 of these smaller rectangles, what is the maximum number of smaller rectangles that can be completely embedded within the hexagon?

Accepted Answer

A hexagon covering 5 rectangles can completely contain at most 3 smaller rectangles due to the geometric constraint that irregular hexagons cannot fully enclose more than 60% of their boundary area.

Question 16

A rectangular piece of paper is folded along its diagonal, and then one corner is cut off with a straight cut perpendicular to the fold. When the paper is unfolded, how many separate pieces result?

Accepted Answer

When paper is folded diagonally and one corner is cut perpendicular to the fold, the cut affects both layers; upon unfolding, the two cuts create 3 separate pieces (one corner removed completely and the main body split).

Question 17

A square sheet is folded in half twice (first horizontally, then vertically), creating 4 layers. A triangular piece is cut from the folded corner. How many triangular pieces appear when the paper is completely unfolded?

Accepted Answer

Two folds create 4 layers; cutting one triangular piece from the folded corner means the cut passes through all 4 layers, producing 4 identical triangular pieces when unfolded (not 2, 8, or 16).

Question 18

A rectangular paper is folded into thirds along its length, creating 3 layers. A small circle is punched through the middle layer only (not the top or bottom). When unfolded, how many circular holes are visible?

Accepted Answer

A punch through only the middle layer of a tri-fold creates exactly 1 hole; it does not pass through the top and bottom layers, so only 1 circular hole appears upon unfolding.

Question 19

When a word is reflected in a mirror, each letter is horizontally flipped. If the word 'BRAVE' is reflected in a mirror, which option shows the correct mirror image?

Accepted Answer

In a mirror image, letters are horizontally flipped and the sequence is reversed; B→ᙠ, R→Я, A→Λ, V→Ʌ, E→Ǝ, giving the mirror image from right to left.

Question 20

A water image of a number flips both horizontally and vertically. If the number '2' is reflected in water, which of the following correctly represents its water image?

Accepted Answer

A water image involves both horizontal and vertical reflection; the digit '2' becomes 'Ƨ' when flipped both ways.

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