Lessons / Logical reasoning
Logical reasoning · Day 12

Counting Figures

Count systematically by size: all 1-piece shapes, then all 2-piece shapes, and so on. Or use the formulas below for standard grids.

Practise (7 questions) Present as slidesCheat sheet: Counting shortcuts (no listing by hand)

Why this topic matters

How many triangles are in a triangle with three lines drawn from the top? Not 4. Count by size, and it is 10.

Counting figures looks visual but it is really counting in a system. Label every point and count shapes by how many small pieces they use.

Label, then count by size

Label every corner and meeting point with a letter. Count the smallest shapes first, then shapes made of 2 pieces, 3 pieces and so on.

Triangle split by n lines from the top corner: the base has n + 1 pieces, and the number of triangles = (n + 1)(n + 2)/2.

PQRSTA
Count by size: 4 single pieces, 3 made of two, 2 of three, 1 of four = 10
A 3 × 3 grid has 9 + 4 + 1 = 14 squares
Lines from the apex splitting the base into k parts
Triangles = k(k + 1)/2
Squares in an n × n grid
12 + 22 + ... + n2
Rectangles in an m × n grid
(m+1)C2 × (n+1)C2
Example 1A triangle has 3 lines drawn from the top corner to the base (4 small triangles). How many triangles in total?
  1. Base split into 4 parts.Each triangle is picked by choosing two points on the base.
  2. Choose 2 of the 5 base points: 5C2 = 10.Every pair of base points with the apex makes a triangle.
  3. 10.
Example 2How many squares are there on a chessboard (8 × 8)?
  1. 1 × 1: 64, 2 × 2: 49, 3 × 3: 36 ... 8 × 8: 1.An n × n grid has (n - k + 1)2 squares of size k.
  2. 1 + 4 + 9 + ... + 64 = 204.

Formula summary

Squares in an n × n grid
12 + 22 + ... + n2 = n(n + 1)(2n + 1)/6.
Rectangles in an m × n grid
C(m + 1, 2) × C(n + 1, 2).
Triangle with n lines from one vertex to the base
n(n + 1)/2 triangles (n lines including sides).
Triangles in a triangle divided into n rows (all upright + inverted)
For n = 1, 2, 3, 4: 1, 5, 13, 27.

Shortcuts and tricks

Squares in a chessboard

Use it when: 8 × 8.

  1. 1 + 4 + 9 + ... + 64 = 204.
ExampleSquares on a chessboard?
  1. 8 × 9 × 17 / 6.

Before you move on, you should be able to...

  • label every point before counting
  • count by number of pieces
  • use the grid formulas

Practice questions (7)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy Squares in a 4 × 4 grid?
  1. 1 + 4 + 9 + 16.

Answer: 30

easy Squares on a 6 × 6 board?
  1. 6 × 7 × 13 / 6.

Answer: 91

medium Rectangles (including squares) in a 3 × 3 grid?
  1. C(4,2) × C(4,2) = 6 × 6.

Answer: 36

medium Rectangles in a 2 × 4 grid?
  1. C(3,2) × C(5,2) = 3 × 10.

Answer: 30

medium A triangle has 4 lines drawn from the top vertex to the base (including the two sides, so 4 lines total). Triangles?
  1. Choose 2 of the 4 lines: C(4,2).

Answer: 6

hard Squares in a 5 × 5 grid?
  1. 1 + 4 + 9 + 16 + 25.

Answer: 55

hard Rectangles (including squares) in a 4 × 4 grid?
  1. C(5,2) × C(5,2).

Answer: 100

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
30 minLabelling and counting by size.
25 minFormulas for triangles, squares, rectangles.
15 minMixed figures.
35 minPractice set.
5 minRecap.
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