Counting of Figures
Counting of Figures is a well-known non-verbal reasoning topic of the ADRE 3.0 Grade IV paper. In this type of question a composite figure made of several lines is given, and the candidate must count the number of triangles, squares, rectangles or straight lines present in it. The figures are simple to draw but tricky to count, because the smaller shapes combine to form larger ones, and a careless eye either misses some or counts the same shape twice. The topic tests orderly thinking and a methodical approach rather than any arithmetic talent. The candidate who follows a fixed order and counts the shapes size by size rarely makes a mistake, while the candidate who counts at random is almost certain to go wrong.
Meaning of a Counting Figures Question
In a counting figures question the figure is built up from a small number of base shapes. A triangle may be divided into several smaller triangles by lines drawn from its corners, or a square may be divided into a grid by lines drawn across it. The question asks for the total number of a particular shape in the whole figure, and this total includes both the small base shapes and the larger shapes formed by joining two or more of them. The candidate must count all of them without repetition and without omission. There is often a known formula that gives the answer at once for standard figures.
Useful Rules and Formulas
- Number of triangles from one vertex divided by n lines to the base is n(n+1)/2.
- Number of squares in an n by n grid is the sum of the squares of 1, 2 up to n.
- Number of rectangles in an m by n grid is m(m+1)/2 multiplied by n(n+1)/2.
- Number of triangles formed by n non-parallel lines used three at a time is n(n-1)(n-2)/6.
- Number of straight lines drawn through n points, taken two at a time, is n(n-1)/2.
- Counting always starts with the smallest base shapes and goes on to larger ones.
- Each larger shape must be counted only once, so mark it as it is counted.
- Symmetry of the figure can be used to reduce the work of counting.
- A divided square of side two contains five squares in all.
- A figure with three lines crossing at one point gives six angles but more triangles.
Types of Questions
The commonest type asks for the number of triangles in a figure that has been divided by lines from one or more vertices. Another type asks for the number of squares or rectangles in a grid. A third type asks for the number of straight lines in a figure made of overlapping segments. Some questions combine two shapes, such as triangles and rectangles in the same figure, and ask for the total of one of them. A few questions ask for the number of circles or curved shapes, though these are less common in the Grade IV paper.
How to Solve a Counting Figures Question
- Identify the base shape from which the figure is built.
- Count the smallest shapes of the required kind first.
- Count the shapes formed by joining two base shapes, then three, and so on.
- Use a known formula if the figure matches a standard pattern.
- Add all the counts and check once more for any missed shape.
Worked Examples
If a triangle has its two sides divided by three lines drawn from the apex to the base, the number of triangles is three times four divided by two, which is six. If a square is divided by one line across and one line down, it becomes four small squares and the whole figure contains five squares in all. If a rectangle is divided into two columns and three rows, the number of small rectangles is six and, using the formula, the total number of rectangles is three times four divided by two, which is six, multiplied by four times five divided by two, which is ten, giving sixty. If five points are joined, the number of straight lines is ten.
How This Topic Appears in the Examination
Counting of figures is usually asked as one or two objective questions carrying one mark each. The figure is small enough to be drawn on the question paper, and the candidate may mark it with a pencil if rough work is allowed. Because the answer is a single number, a small slip in counting spoils the whole question, so the order of counting matters more than speed. With practice a candidate can finish a standard counting question in well under a minute and use the saved time for the longer parts of the paper.
Memory Tips
Always count from the smallest shape to the largest, because this order prevents both repetition and omission. Learn the three standard formulas for triangles from one vertex, for squares in a grid and for rectangles in a grid, since these cover most questions. When the figure is complicated, count one row or one column and multiply, taking care of the extra shapes formed across rows. Mark each shape lightly with a pencil as you count it. If two counts differ, count again slowly rather than accepting the average.
Common Mistakes to Avoid
- Counting the small shapes but forgetting the larger combined shapes.
- Counting the same shape more than once.
- Applying a formula to a figure that does not match the standard pattern.
- Miscounting when lines cross each other at a point.
- Confusing the number of lines with the number of segments.
How to Prepare This Topic
- Learn the standard formulas for triangles, squares and rectangles.
- Redraw each figure on rough paper and count in a fixed order.
- Practise at least fifteen counting questions every day.
- Check answers by counting once again in the reverse order.
- Time yourself so that each question takes less than a minute.
Additional Solved Examples
If a square is divided into four equal squares by a cross, the total number of squares is four small plus one large, that is five. If the same grid has three lines across and three lines down, making a three by three grid, the number of squares is one plus four plus nine, that is fourteen. If a triangle is divided by two lines from the apex and also by one horizontal line, the triangles must be counted in two sets, one above the horizontal line and one below, and the two sets added. If a figure consists of two overlapping squares, the number of squares may be more than two because the overlap creates a third square.
Advanced Tips
When a figure has several lines meeting at one point, count the triangles in groups that share that common vertex, because each group can be counted by the standard formula and the results added. For grids, count the squares by size, one by one, rather than by position, since counting by size avoids skipping any square. If a rectangle is divided unequally, first count the number of vertical segments and the number of horizontal segments and then multiply the two totals. Remember that every large square in a grid is counted once, and that a shape formed by joining two small squares is a rectangle, not a square, unless the small squares are arranged in a square block. Finally, when the question seems too large to count, look for the hidden formula, because the paper usually chooses a figure that matches one.
Related Non-Verbal Topics
Counting of figures is closely related to Embedded Figures, Figure Classification and Figure Series. Embedded figures asks you to find one shape hidden in a complex drawing, and counting figures asks you to find how many such shapes are hidden, so the two topics share the same scanning skill. Classification compares figures for a common property, which also needs a clear look at the parts of a figure. Series uses the idea of a regular change, which appears in counting figures when a growing pattern is given. Practising these topics together makes the eye quick at separating and recognising shapes even in a busy drawing.
Quick Revision Checklist
- Count from the smallest shape to the largest.
- Use the formula n(n+1)/2 for triangles from one vertex.
- Use the sum of squares for squares in a grid.
- Multiply the two rectangle formulas for a grid.
- Mark every shape once as it is counted.
- Check the total by a second count in reverse order.
Frequently Asked Questions
What is a counting figures question?
It gives a composite figure and asks you to count the total number of a particular shape, such as triangles or squares, present in it.
Why are counting figures difficult?
They are difficult because the small shapes combine to form larger ones, so a careless count misses some shapes or counts the same shape twice.
How many squares are there in a two by two grid?
A two by two grid contains four small squares and one large square, that is five squares in all, which follows the sum of squares rule.
What is the formula for triangles from one vertex?
If lines divide the base into n parts from a common vertex, the number of triangles is n(n+1)/2. For example, three parts give six triangles.
How should I count to avoid mistakes?
Count from the smallest shapes to the largest, mark each shape once, and check the total with a second count in the reverse order.