Question 1

A rectangle is divided into 12 equal smaller rectangles arranged in 3 rows and 4 columns. Count the total number of rectangles of all possible sizes that can be formed by combining these smaller rectangles.

[A 3×4 grid of rectangles is shown.]

Accepted Answer

This question tests counting of rectangles (not just squares) of multiple sizes in a grid, requiring understanding of how rectangles can be formed by combining adjacent cells.

Question 2

In the given figure, a rectangle is divided into smaller sections by horizontal and vertical lines. There are 3 horizontal lines and 4 vertical lines drawn inside the rectangle (including the borders). How many rectangles of all sizes can be formed in this grid?

Accepted Answer

To count rectangles in a grid, we use the formula: Number of rectangles = C(h,2) × C(v,2), where h is the number of horizontal lines and v is the number of vertical lines.

Given: 3 horizontal lines and 4 vertical lines (including borders)

This creates a grid of 2 rows (3-1) and 3 columns (4-1).

Number of ways to choose 2 horizontal lines from 3 = C(3,2) = 3!/(2!×1!) = 3
Number of ways to choose 2 vertical lines from 4 = C(4,2) = 4!/(2!×2!) = 6

Total rectangles = 3 × 6 = 18

Breakdown verification:
- 1×1 rectangles: 2 × 3 = 6
- 1×2 rectangles: 2 × 2 = 4
- 1×3 rectangles: 2 × 1 = 2
- 2×1 rectangles: 1 × 3 = 3
- 2×2 rectangles: 1 × 2 = 2
- 2×3 rectangles: 1 × 1 = 1
Total = 6 + 4 + 2 + 3 + 2 + 1 = 18

Question 3

A square grid of 5×5 contains some cells shaded black and others white. A 'region' is defined as a connected group of cells of the same colour (horizontally or vertically adjacent, not diagonally). If the grid has exactly 8 black regions and 6 white regions, what is the minimum number of black cells required?

Accepted Answer

This tests logical reasoning about connected components and optimisation. With 8 separate black regions and 6 white regions in a 5×5 grid (25 cells total), we need to find the minimum black cells while maintaining the region count constraint.

Question 4

A figure shows a hexagon divided by internal lines into smaller polygons. The hexagon has 2 diagonals drawn from one vertex, creating internal divisions. Additionally, 3 horizontal lines cross the entire hexagon. How many distinct polygons (triangles, quadrilaterals, pentagons, or hexagons) are formed in total?

Accepted Answer

This requires careful counting of all polygon types formed by the intersection of diagonals and horizontal lines within a hexagon—a complex spatial reasoning task.

Question 5

In a complex figure, there is a large triangle subdivided by internal lines into smaller triangles. The internal lines create 3 horizontal divisions and 2 vertical divisions within the triangle. Count the total number of triangles of all sizes visible in the figure.

Accepted Answer

This requires systematic counting of triangles at multiple scales. With 3 horizontal and 2 vertical divisions in a triangle, the counting must account for upward-pointing and downward-pointing triangles separately.

Question 6

A rectangular grid contains horizontal and vertical lines. There are 5 horizontal lines and 6 vertical lines. How many rectangles of all possible sizes can be formed by these lines?

Accepted Answer

This question tests combinatorial counting of rectangles in a grid, requiring understanding that each rectangle is formed by selecting 2 horizontal and 2 vertical lines.

Question 7

A figure consists of a square subdivided by 2 horizontal lines and 2 vertical lines, creating a 3×3 grid of 9 cells. Additionally, both diagonals of the main square are drawn. How many quadrilaterals (including squares and rectangles) of all sizes are present in the figure?

Accepted Answer

This question tests counting quadrilaterals in a grid with diagonal lines, requiring careful distinction between quadrilaterals formed by grid lines versus those partially bounded by diagonals.

Question 8

A square is divided into 16 equal smaller squares (4×4 grid). Additionally, there are diagonal lines drawn from two opposite corners of the large square, creating further subdivisions. Count the total number of triangles of all sizes that can be identified in this figure.

Accepted Answer

This tests the ability to count triangles formed by both grid lines and diagonal lines, requiring careful tracking of overlapping regions.

Question 9

A hexagon is divided into 6 equilateral triangles by drawing lines from the center to each vertex. Additionally, lines are drawn connecting alternate vertices, creating a smaller hexagon in the center. Count the total number of triangles of all sizes visible in the final figure.

Accepted Answer

This tests the ability to count triangles in a complex geometric figure with multiple overlapping layers and symmetry.

Question 10

A rectangle is divided into a 3×4 grid of smaller rectangles. Additionally, one diagonal is drawn from the top-left corner to the bottom-right corner of the entire large rectangle. Count the total number of triangles formed by this diagonal and the grid lines.

Accepted Answer

This tests the ability to count triangles created when a diagonal intersects a rectangular grid, requiring careful tracking of intersection points.

Question 11

A figure shows a cube with all six faces visible. Some faces contain patterns. If the cube is unfolded into a net, and we know that opposite faces never touch in the net, how many distinct nets can be formed from a standard cube?

Accepted Answer

This tests understanding of cube nets and spatial visualisation. There are exactly 11 distinct nets that can fold into a cube, but the question asks for nets where opposite faces don't touch in the unfolded pattern.

Question 12

A figure shows a large triangle subdivided by internal lines into smaller triangles. The large triangle has 2 internal parallel lines to its base, creating 3 rows of smaller triangles. The top row has 1 triangle, the middle row has 3 triangles, and the bottom row has 5 triangles. Count the total number of triangles of all sizes (including those formed by combining smaller triangles).

Accepted Answer

This tests the ability to count overlapping triangles of multiple sizes, not just the unit triangles visible in the figure.

Question 13

A hexagon is divided into smaller regions by drawing 3 lines from the center to alternate vertices, and 2 additional lines parallel to two of its sides. How many triangles of all sizes are formed? (Assume the hexagon is regular and all internal divisions are clearly defined.)

Accepted Answer

This question tests counting triangles in a complex irregular subdivision with both radial and parallel divisions, requiring systematic enumeration while avoiding miscounting overlapping regions.

Agniveer Air Force Counting Figures

Counting Figures— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Counting Figures — Practice Questions

60-Second Revision — Counting Figures