SSC CGL Reasoning · Figure Counting

Counting Squares and Other Figures

Count the squares in a grid or a set of overlapping squares, the rectangles in a divided rectangle, and the circles or semicircles in a drawing.

All Figure Counting notes

Why this matters

Eleven questions in the bank, all from the 2024 shifts, with 1 EASY, 7 MODERATE and 3 HARD. Grids have exact formulas, so the work is splitting the figure into grids and overlaps and adding the parts with care.

Concept 1 of 2: Squares in grids and overlapping squares

A grid holds squares of every size: single cells, 2 by 2 blocks, 3 by 3 blocks, up to the whole grid. Count each size separately and add. Overlapping squares add one more square wherever two of them cross in a square shape.

Definition

An n by n grid holds $n^2$ single cells, $(n - 1)^2$ blocks of 2 by 2, and so on down to 1 square the size of the grid. A 3 by 3 grid has $9 + 4 + 1 = 14$ .
A grid of m rows and n columns (m less than n) holds mn single cells, $(m - 1)(n - 1)$ blocks of 2 by 2, and so on until the smaller side runs out.
Overlapping squares: count each drawn square, then each square formed where two of them overlap.
A tilted square drawn inside another square is one more square.
Fast route: split the figure into grids you can count by formula, then add the loose squares and the overlaps one by one.

Squares in an n by n grid

1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}

nthe number of cells along one side of the grid

Worked example

How many squares are there in a 3 by 3 grid of equal cells?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 24 Sep 2024, Shift 3 · Q8Moderate

Example 1 · Figure Counting · Counting Squares and Other Figures

How many squares are there in the figure shown below?

Counting only the single cells

A 3 by 3 grid has 9 cells but 14 squares. Count every size of square, from one cell up to the whole grid.

Using the n by n rule on a grid that is not square

A grid of 2 rows and 3 columns has no 3 by 3 square. In an m by n grid, stop at the size of the shorter side.

Drill 7 more on squares in grids and overlapping squares

Concept 2 of 2: Rectangles, circles and semicircles

A rectangle is fixed by choosing two of the horizontal lines and two of the vertical lines, so rectangles in a grid can be counted by choosing lines. Circles and semicircles have no formula: count them by kind, the large ones first and then the small ones.

Definition

Rectangles in a grid of m rows and n columns (squares included): choose 2 of the m + 1 horizontal lines and 2 of the n + 1 vertical lines. A 2 by 3 grid gives $3 \times 6 = 18$ .
A rectangle cut into k strips by k minus 1 lines holds k(k + 1) ÷ 2 rectangles.
A figure with an open side has fewer: a missing edge cannot close a shape.
Circles: count the nested ones from the outside in, then the separate small ones. A semicircle is half a circle; a full circle cut by a diameter makes 2.
Fast route: for rectangles, count the lines, not the shapes.

Rectangles in a grid

\binom{m+1}{2} \times \binom{n+1}{2}

mthe number of rows of cells
nthe number of columns of cells

Worked example

How many rectangles, squares included, are there in a grid of 2 rows and 3 columns of equal cells?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 23 Sep 2024, Shift 2 · Q23Moderate

Example 2 · Figure Counting · Counting Squares and Other Figures

How many circles are there in the given figure?

Forgetting that a square is a rectangle

When a question asks for rectangles, the squares count too. The line-choosing formula already includes them.

Closing an open figure in your head

If a side of the figure is not drawn, the shapes that would use it do not exist. Count only shapes whose every side is on the page.

Drill 2 more on rectangles, circles and semicircles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Squares in grids and overlapping squares
Squares in an n by n grid
$1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$
Rectangles, circles and semicircles
Rectangles in a grid
$\binom{m+1}{2} \times \binom{n+1}{2}$

Watch out for (4)

Counting only the single cells
→ Squares in grids and overlapping squares
Using the n by n rule on a grid that is not square
→ Squares in grids and overlapping squares
Forgetting that a square is a rectangle
→ Rectangles, circles and semicircles
Closing an open figure in your head
→ Rectangles, circles and semicircles

Test yourself on Figure Counting

15 past SSC CGL questions from this chapter, timed at 9 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Figure Counting chapter test Drill all 11 questions