Counting Figures

What the question asks

The figure is usually a grid, like a chessboard or a window with bars, and you must count squares or rectangles. A rectangle is a four-sided shape with four right angles. A square is a rectangle whose four sides are equal. So every square is also a rectangle. When a question asks for rectangles, squares are included unless it says "rectangles that are not squares".

Counting squares in a grid

In a grid of m columns and n rows of equal cells, count squares size by size.

• 1 × 1 squares: m × n
• 2 × 2 squares: (m − 1)(n − 1)
• 3 × 3 squares: (m − 2)(n − 2) ... until one side reaches 1.

$S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)$

For a square grid of n × n this becomes $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$.

Example: a chessboard (8 × 8) has $\frac{8 \times 9 \times 17}{6} = 204$ squares. A 4 × 3 grid has 12 + 6 + 2 = 20 squares.

Counting rectangles in a grid

A rectangle is fixed the moment you choose its two vertical sides and two horizontal sides. A grid of m columns has m + 1 vertical lines, and n rows give n + 1 horizontal lines. So

$R = \binom{m+1}{2} \times \binom{n+1}{2} = \frac{m(m+1)}{2} \times \frac{n(n+1)}{2}$

Example: a 3 × 2 grid has 4 vertical and 3 horizontal lines: C(4, 2) × C(3, 2) = 6 × 3 = 18 rectangles.

A single strip of m cells has $\frac{m(m+1)}{2}$ rectangles (choose a start cell and an end cell).

Rectangles that are not squares = R − S.

Tilted squares

Joining the middles of the four sides of a square makes a tilted square (a diamond). Joining the middles again makes another upright square, and so on. Each tilted square counts once. Grid lines passing through a diamond cut it into triangles, not squares, so do not invent extra squares there.

Irregular grids

Plus signs, L-shapes and grids with missing lines break the formula. Then:

1. Count the 1 × 1 cells.
2. Look for each bigger size and check that all four sides are drawn.
3. For rectangles, pick each cell as the top-left corner and count how far you can extend right and down.

Special cases

• A line missing from the middle of a grid kills every rectangle that needs it as a side.
• Cells must be equal for the square formula. If they are not, count by hand.
• The outer boundary counts as one rectangle (and one square if it is square).

Quick revision

• 2 × 2 → 5 squares, 9 rectangles; 3 × 3 → 14 squares, 36 rectangles; 4 × 4 → 30 squares, 100 rectangles.
• Every square is a rectangle.
• Irregular figure: check all four sides of every candidate.