Embedded figures and figure counting for AFCAT: find the hidden shape, count without missing any

Two question types share this page because they need the same skill: seeing smaller shapes inside a busy figure without being distracted by the lines around them.

In an embedded figure question, you are given a simple shape and four complex figures, and you pick the one that contains the simple shape. In a counting question, you are given one figure and asked how many triangles, squares or rectangles it contains. The first is about careful searching. The second is about counting in an order that neither misses nor repeats anything.

Embedded figures

The rules of the game are strict, and the options are designed around them.

• The hidden shape must appear at the same size as the one given.
• It must be in the same orientation. A rotated or mirrored copy does not count unless the question says so.
• Extra lines may cross it. It must still be traceable along lines that are actually drawn.

How to search

1. Pick the most distinctive part of the simple shape: an unusual angle, a curve, a short side next to a long one.
2. Scan each option for that part only. Most options fail here within a few seconds.
3. Trace the rest of the shape from that anchor in the surviving options, checking every side exists.
4. Check orientation last. A right angle at bottom-left is not the same as one at bottom-right.

Example 1. The simple shape is a square with one diagonal, running from its top-left corner to its bottom-right corner. Each option is a square of the same size with some lines inside.

• (a) The other diagonal, from top-right to bottom-left, plus a horizontal line across the middle.
• (b) A vertical line and a horizontal line through the centre.
• (c) Both diagonals.
• (d) A vertical line through the centre only.

The distinctive part is the diagonal from top-left to bottom-right. Only (c) has it. (a) has a diagonal, but the wrong one, which is exactly the orientation trap. Answer: (c).

Counting triangles

Lines from one corner

Take a triangle and draw n lines from its top corner down to the base. Counting the two sides, there are now n + 2 lines from that corner. Any two of them, together with the base, make a triangle. So the count is the number of ways of choosing 2 lines from n + 2:

Triangles = (n + 1)(n + 2) / 2

Adding lines parallel to the base

If the triangle also has m lines drawn parallel to the base, each one acts as another base for the same set of triangles. Multiply:

Triangles = [(n + 1)(n + 2) / 2] × (m + 1)

For example, with 2 lines from the top corner and 1 line parallel to the base: 6 × 2 = 12 triangles.

The label-and-list method

For figures that no formula fits, label every point where lines meet or end, then list triangles in a fixed order. Group them by size, or go letter by letter: every triangle using A, then every one using B but not A, and so on.

Example 2: a square with both diagonals. Label the corners A (top-left), B (top-right), C (bottom-right) and D (bottom-left), and the centre O.

• Small triangles with a vertex at O: AOB, BOC, COD, DOA. That is 4.
• Half-squares cut by one diagonal: ABC, ABD, ACD, BCD. That is 4.
• Total: 8.

Example 3: add one vertical line. Take Example 2 and add a vertical line from P, the midpoint of the top side AB, down to R, the midpoint of the bottom side CD. It passes through O.

• Smallest triangles: AOP, BOP, COR, DOR. That is 4.
• Quarter triangles: AOB, BOC, COD, DOA. That is 4. (AOB is made of AOP and BOP.)
• Half-squares: ABC, ABD, ACD, BCD. That is 4.
• Total: 12.

Counting by size, smallest first, makes it very hard to miss a group. If you have time to spare, a final check is to run through every set of three labelled points and ask whether all three sides lie on drawn lines.

Counting squares and rectangles

Squares in an n × n grid

An n × n grid contains squares of every size from 1 × 1 up to n × n. There are n² of the smallest, (n − 1)² of the next size, and so on.

Squares = 1² + 2² + … + n² = n(n + 1)(2n + 1) / 6

Squares in an m × n grid (m rows, n columns, m smaller)

Add m × n + (m − 1)(n − 1) + (m − 2)(n − 2) + … until one factor reaches 1.

For a 2 × 3 grid: 2 × 3 + 1 × 2 = 6 + 2 = 8.

Rectangles in an m × n grid

A rectangle is fixed by choosing 2 of the m + 1 horizontal lines and 2 of the n + 1 vertical lines.

Rectangles = [m(m + 1) / 2] × [n(n + 1) / 2], which includes the squares.

For a 2 × 2 grid: 3 × 3 = 9. For a 3 × 3 grid: 6 × 6 = 36.

Practice set

1. A triangle has 4 lines drawn from its top corner to the base. How many triangles does it contain?
2. How many squares are there in a 5 × 5 grid?
3. How many rectangles, including squares, are there in a 2 × 3 grid?
4. A triangle has 3 lines from its top corner to the base and 1 line parallel to the base. How many triangles?
5. How many squares are there in a grid of 3 rows and 4 columns?
6. A square has both diagonals and both midlines (one vertical and one horizontal line through the centre). How many triangles does it contain?
7. The simple shape is a right-angled triangle with its right angle at the bottom-left, the same size as half of the option square. The option is a square with one diagonal, from its top-left to its bottom-right corner. Does the option contain the shape?

Answers:

1. 5 × 6 ÷ 2 = 15.
2. 25 + 16 + 9 + 4 + 1 = 55.
3. 3 × 6 = 18.
4. (4 × 5 ÷ 2) × 2 = 10 × 2 = 20.
5. 12 + 6 + 2 = 20.
6. 16. There are 8 smallest triangles (each quarter square is split by half a diagonal), 4 made of two small ones with a vertex at the centre (such as the top triangle formed by the two top corners and the centre), and 4 half-squares.
7. Yes. The diagonal cuts the square into two triangles. The lower one has corners at top-left, bottom-right and bottom-left, with its right angle at the bottom-left.

What to do next

• Derive the triangle formula once for yourself by listing the triangles when n = 2. You will remember it far better.
• Practise 10 counting questions using the label-and-list method, even when a formula exists, and compare answers.
• Do five embedded-figure questions a day, always naming the distinctive feature before scanning.
• Move on to dot situation and figure completion, then spatial ability.