Counting of Figures for the Assam Police Constable (AB & UB) Exam
Counting of Figures is a non-verbal reasoning topic of the SLPRB Assam Police Constable (AB & UB) examination in which a single composite figure is shown and the candidate must count the exact number of triangles, squares, rectangles or straight lines hidden inside it. The topic tests careful observation and a systematic method rather than any knowledge of formulae. Most candidates lose marks here not because they cannot see the figures but because they count in a random order, miss the large figures that are formed by joining several small ones, or count the same figure twice. This page explains the types of figures that are asked, the counting method, worked examples, shortcuts and the traps that cost marks. The material follows the standard reasoning reference books used for Assam recruitment examinations and is reviewed before it is published.
What the Topic Means
In a counting question a big figure is drawn with several lines crossing one another, and it contains many small and large figures within it. The candidate is asked to find how many triangles, squares, rectangles, straight lines or other shapes are present in the whole figure. The small figures are easy to see, but the larger ones, which are made by joining two or more small figures, are usually missed. A triangle may be formed by combining a triangle with a trapezium, a rectangle may be formed by two squares placed together, and a straight line may be made of several small segments lying in a row. The answer must include every possible figure and not only the obvious ones, so the whole task depends on a fixed order of counting.
The Shapes That Are Counted
- Triangles inside a large triangle, a square or a star.
- Squares and rectangles inside a grid of horizontal and vertical lines.
- Straight lines in a figure made of many segments.
- Circles, semicircles or quadrilaterals in a combined figure.
- A mixture of two shapes, such as both triangles and squares in one figure.
Counting Triangles in a Network of Lines
Triangles are the most common shape in this topic. The usual figure is a large triangle with lines drawn from its corners to a point inside, or lines drawn parallel to its base. The safe method is to count the smallest triangles first and then the triangles made by joining two, three or more of them. In a triangle with three lines drawn from the three corners to one interior point there are three small triangles and one large triangle, giving four in all. In a triangle with one line parallel to the base there is the small triangle at the top, the whole triangle, and the triangles formed with the middle region, which must all be listed one by one. The candidate should mark each triangle with a number so that no triangle is counted twice.
Counting Squares and Rectangles
When a rectangle is cut by several horizontal and vertical lines, small squares, big squares and rectangles of many sizes are formed. The number of squares in a grid of n rows and n columns equals the sum of the squares of the numbers from n down to one, and the number of rectangles is found by choosing two horizontal and two vertical lines. For a figure of plain lines it is easier to count row by row and column by column and then add the combinations. Care must be taken because a square is also a rectangle; if the question asks for rectangles the squares are usually counted as well, but if it asks only for squares then the rectangles must not be counted.
Counting Straight Lines
When the question asks for the number of straight lines, the candidate must look for long lines that pass through several intersections and must not count them as separate small segments. A single line drawn across the figure may look like two or three short lines, but it must be counted once. The method is to trace each long line from one end to the other and to tick it off before moving to the next.
A Step by Step Method
- Look at the whole figure and note the shape that has to be counted.
- Mark the smallest figures of that shape with a number so that none is counted twice.
- Count the figures made by joining two small figures.
- Count the figures made by joining three, four and more small figures.
- Add the count for every size and write the total.
- Check the total once again from a different starting point.
Worked Examples
Example one: a large triangle is divided by three lines drawn from the three corners to a point inside; there are three small triangles and one large triangle, so the total is four. Example two: a square is divided into four small squares by one horizontal and one vertical line through the middle; the four small squares, the original square and the rectangles formed by two squares each must all be added. Example three: in a triangle with one line drawn parallel to the base there is the small triangle at the top, the trapezium below it and the whole triangle, and the triangles formed with the middle region give a larger total. Example four: a grid of two rows and two columns of lines forms one large square, four small squares and four rectangles of two squares each. Example five: a star made of two overlapping triangles contains several small and large triangles and must be counted corner by corner.
How This Topic Appears in the Examination
The topic usually appears as two to four objective questions of one mark each. The common forms are to count the number of triangles in a given figure, to count the number of squares or rectangles in a grid, to count the number of straight lines, and to count a particular shape inside a mixed figure. The options are close numbers such as six, seven and eight, so a single missed figure changes the answer.
Shortcuts and Speed Methods
- Count the smallest figures first and give each a number.
- Then count the bigger figures that are formed by joining them.
- Use a formula for a regular grid, such as the sum of squares for the number of squares.
- In a triangle with lines from one corner to the opposite side, count the regions and use them to find the triangles.
- Recount from a different corner to cross-check the total.
Common Mistakes to Avoid
- Counting only the small figures and forgetting the large combined figures.
- Counting the same figure twice because it is seen from two sides.
- Losing count in the middle and starting again in a disorganised way.
- Forgetting that a square is also a rectangle and counting it wrongly.
- Not checking the total a second time.
Preparation Plan
- Learn to count triangles, squares and lines with a fixed order.
- Practise ten counting figures a day with a pencil in hand.
- Write the numbers inside the figure so that nothing is missed.
- Check each answer by counting from a different starting point.
- Revise the standard figures such as the triangle with internal lines and the square grid once a week.
More Practice Questions
- Count the triangles in a square with both diagonals drawn.
- Count the squares in a figure of three rows and three columns.
- Count the straight lines in a figure made of many small segments.
- Count the rectangles in a figure of overlapping squares.
- Count the triangles in a star made of two overlapping triangles.
Quick Revision Checklist
- Identify the shape that has to be counted.
- Number the smallest figures first.
- Add the larger combined figures.
- Write every size separately and add the totals.
- Cross-check once from a different corner.
Counting of Figures and Observation Skills
Counting of figures trains the eye to see the whole as well as the parts, and the same skill helps in embedded figures and pattern completion. A candidate who can number the small figures and build up the larger ones will also solve embedded-figure questions quickly, because both depend on recognising a shape inside a bigger shape. The topic rewards patience more than cleverness, and a slow, ordered count is always better than a fast, guessed one. With regular practice the standard figures become familiar and the total can be read almost at a glance.
Frequently Asked Questions
What is counting of figures?
It is a non-verbal reasoning topic in which a single composite figure is given and the candidate must count the exact number of triangles, squares, rectangles or straight lines that it contains.
Why do I miss the answer?
Most misses happen because the large figures formed by joining several small figures are forgotten, or because the same figure is counted twice; a numbered, orderly count removes both problems.
Is there a formula for squares in a grid?
Yes. In an n by n grid the number of squares is the sum of the squares of the numbers from n down to one, so a three by three grid contains fourteen squares.
How do I count large figures?
Count the smallest figures first and then build larger figures by joining them two at a time, then three at a time, until no larger figure is left.
Is counting of figures scoring in Assam Police?
Yes. The figure is fixed and the count is certain once the method is followed, so a trained candidate can gain sure marks in a short time.