Table of Contents
What is a Number Pattern?
Number patterns refer to the pattern or sequence in a given series of numbers.
It establishes a common relationship between all the numbers in the series. These patterns can be used to determine any missing number in a series.
For example:
What are the missing numbers in the diagram given below?
Upon carefully observing the numbers in the diagram we see a pattern that the next number is 2 less than the number preceding it. The 4th and the last numbers are missing in the series. These numbers can be determined by applying the pattern observed.
So the 4th number will be 10 – 2 = 8, and the last number will be, 6 – 2 = 4.
What is a Number Rule?
A number rule is basically a relation between numbers in a number pattern.
It tells how the numbers are related to each other. It also describes the manner in which the pattern starts and continues.
For example:
In the above diagram every next number is 3 less than the previous one.
The number rule here is, “the next number is obtained by subtracting 3 from the previous number”.
A number rule can be based on any basic math operation such as addition, subtraction, multiplication or division.
For example:
In the above diagram, the number rule applied here is:
In every row, the next number is obtained by multiplying the previous number with 2.
Let us look at another example to clearly understand the number rule.
In the above figure, we see a pattern or a number rule in which the next number is obtained by dividing the previous number by 5.
How do we create a Number Pattern using a given Number Rule?
A number rule can be used to create a number pattern. Once the number rule to be used is identified, apply it on the starting number. Apply the rule again on the second number obtained, then on the third number, and so on. Repeat this process until the number pattern of the required size is obtained.
In this manner we will get a number pattern on a given number rule for 1 row.
If we want to add more rows in the number pattern, we will specify the starting number for that row and apply the given number rule on it to obtain the remaining numbers depending on the size of the pattern.
For example:
Use the rule “subtract 10” to create a number pattern.
The starting number is 100. In addition to this, describe another feature of this number pattern.
So the numbers are 100, 90, 80, 70 and 60. Besides the given pattern, we also observe that the numbers in the series are even and multiples of 10.
Apart from the number rule that defines the number pattern there may be some additional features in a number pattern, such as, numbers in a number pattern may be multiples of 5, or even or odd or prime numbers, and so on.
For example, in the number pattern shown above, the numbers are even and multiples of both 5 and 10.
What is the Shape Pattern?
A shape pattern is a series of shapes, where a group of shapes are repeated more than once. These patterns follow certain rules while being repeated.
How to create a Shape Pattern using a given Rule?
To create a shape pattern, first of all, we will determine the initial shapes given.
Then, we will follow the specified pattern or shape rule to get the next shapes in the series.
How do we find a Shape at a given Position in a Shape Pattern?
In a given shape pattern, a shape at a given position can be determined by applying the pattern rule as given below:
- Identify the shape pattern rule.
- Divide the required position number by the number of shapes in the group of shapes that are repeating.
- If the required position number is divisible by the number of shapes, that is, the remainder is 0, then the shape at the required position will be the last shape in the group.
- If the required position number is not divisible by the number of shapes, that is, the remainder is not 0, then the shape at the required position will be determined by the value of the remainder.
- In case of problems involving multiplication-based patterns, such as, if at 1st position there are 4 balls, at the \(2^{nd}\) position there are 8 balls then how many balls are at the \(7^{th}\) position?
In this, we will simply multiply the number of balls at the 1st place with the required position number, that is, 7, which gives us,
4 × 7 = 8 balls at the \(17^{th}\) position.
Let’s understand this better with the help of an example.
Create a shape pattern by repeating the rule “circle, triangle pentagon.” What is the \(20^{th}\) shape in the series?
Step 1: Create the pattern
Step 2: Since there are 3 shapes we will divide 20 by 3
20 ÷ 3 = 6 , Remainder = 2
So when the pattern repeats 6 times,
The \(18^{th}\) shape is a pentagon.
The \(19^{th}\) shape is a circle.
The \(20^{th}\) shape is a triangle.
Hence, the \(20^{th}\) shape in the series is a triangle.
In the above example, a group of 3 shapes are repeating. Hence, we divide the required position number by 3. 20 is not divisible by 3 as 2 is the remainder. So from 18 we count 2 numbers forward, to get to the shape at the \(20^{th}\) position.
Solved Examples
Example 1:
In a square table of numbers, shade every second number in the first row. What are the numbers, and what pattern do you see in these numbers?
Solution:
The marked numbers are 2, 4, 6, 8, 10
The pattern or number rule applied on the above numbers is that every next number is obtained by adding 2 to the preceding number.
2 First number
2 + 2 = 4
4 + 2 = 6
6 + 2 = 8
8 + 2 = 10
Hence, the numbers are 2, 4, 6, 8 and 10 and the rule is “add 2” to the numbers starting from 2.
Example 2:
Draw the missing shape in the given pattern. Explain the pattern.
Solution:
In the above figure, we observe that the shape pattern is formed by repeating “rhombus, square, triangle”.
Hence the missing figure is a rhombus.
Example 3:
Describe the dot pattern. How many dots are in the \(20^{th}\) figure?
Solution:
In the above dot pattern, figure 1 contains 4 dots, figure contains 8 dots and so on.
Each succeeding figure is a multiple of 4, that is,
Figure 1 = 4 × 1 = 4 dots
Figure 2 = 4 × 2 = 8 dots
Figure 3 = 4 × 3 = 12 dots.
So in figure 20, there are 4 × 20 = 80 dots.
Example 4
What pattern do you notice in the ones digit of multiples of 9?
Solution:
Upon carefully observing the multiples of 9, we notice that, at the ones place the digits decrease by 1 in every next multiple, that is,
9 – 1 = 8
8 – 1 = 7
7 – 1 = 6
6 – 1 = 5
5 – 1 = 4
4 – 1 = 3
3 – 1 = 2
2 – 1 = 1
1 – 1 = 0.
Hence, the number rule observed in the ones digits of the multiples of 9, is “subtract 1”.
Example 5: Use the rule to generate a pattern of 6 numbers.
“Multiply by 3”
Solution:
The size of the pattern given is 6, that is, there will be 6 numbers in the pattern. Let the starting number be 2,
Then the second number will be,
2 x 3 = 6
Similarly, the remaining 4 numbers in the series will be,
6 x 3 = 18
18 x 3 = 54
54 x 3 = 162
162 x 3 = 486
Hence, the required number patterns will be,
2, 6, 18, 54, 162 and 486.
A number pattern is a sequence or pattern observed in a series of numbers in which numbers are related to other numbers by a specific rule, called the number rule.
It is a pattern or sequence involving repeating shapes or dots. Each pattern follows a set rule according to which the shapes are repeated.
In the leap years 2008, 2012, 2016 and 2020 we observe that they are all divisible by 4. So the pattern observed in these years is “divisible by 4”.