Introduction
Maths becomes much more interesting when we start noticing patterns. A number pattern is simply a sequence of numbers that follows a specific rule. Once we understand the rule, we can easily find missing numbers and predict what comes next.
Take this example:
2, 4, 6, 8, 10, __
Each number increases by 2, so the missing number is 12. This simple idea of recognising patterns also applies to square numbers, which follow their own special rules.
What Are Square Numbers?
A square number is formed when a number is multiplied by itself.
Examples:
3 × 3 = 9
9 × 9 = 81
Both 9 and 81 are square numbers. These numbers are called “square” because they can be arranged into perfect squares with equal rows and columns.
The sequence of square numbers is:
1, 4, 9, 16, 25, …
These numbers form the foundation of many square number patterns used in school mathematics.
Understanding Square Number Patterns
There are several interesting square number patterns that help students see relationships between numbers. Learning these patterns improves number sense and makes problem-solving faster. This is why such topics are often emphasised in structured programmes like the best psle tuition in singapore, where students are trained to recognise patterns quickly and accurately.
Below are some common and useful square number patterns.
1. Sum of Odd Numbers
The sum of the first n odd numbers is always equal to n².
Examples:
1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 7²
This pattern shows how square numbers grow using odd numbers.
2. Difference Between Consecutive Square Numbers
The difference between two consecutive square numbers is always an odd number.
General rule:
(n+1)2−n2=2n+1(n + 1)² − n² = 2n + 1
Example:
4² − 3² = 16 − 9 = 7
2 × 3 + 1 = 7
3. Adding Triangular Numbers
Triangular numbers are formed by adding natural numbers in sequence:
1, 3, 6, 10, 15, 21, …
A useful pattern is that the sum of two consecutive triangular numbers always results in a square number.
Examples:
3 + 6 = 9 = 3²
15 + 21 = 36 = 6²
4. Numbers Between Two Square Numbers
Between two consecutive square numbers n² and (n + 1)², there are exactly 2n non-perfect square numbers.
Example:
8² = 64 and 9² = 81
Number of non-square numbers between them = 2 × 8 = 16
5. Product of Consecutive Even or Odd Numbers
The product of two consecutive even or odd numbers can be expressed using square numbers.
General form:
(a−1)(a+1)=a2−1(a − 1)(a + 1) = a² − 1
Example:
17 × 19 = 323
18² − 1 = 324 − 1 = 323
6. Squaring Numbers Made Up of Ones
Numbers that contain only the digit 1 follow a beautiful and simple square pattern.
| Number | Square |
|---|---|
| 1 | 1 |
| 11 | 121 |
| 111 | 12321 |
| 1111 | 1234321 |
To square these numbers, count the digits, write numbers from 1 up to that number, then write them back down to 1.
Helpful Facts About Square Numbers
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If a square number ends in 1, its square root ends in 1 or 9
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If a square number ends in 6, its square root ends in 6
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If a square number ends in 5, its square root also ends in 5
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The square of an odd number is always odd
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The square of an even number is always even
Worked Examples
1. How many numbers are there between 289 and 324 without counting individually?
289 = 17²
324 = 18²
Non-square numbers = 2 × 17 = 34
2. Find the square of 1111111
The number has 7 digits. Write numbers from 1 to 7 and then back to 1:
1234567654321
3. How many non-square numbers are there between 100 and 121?
100 = 10²
121 = 11²
Non-square numbers = 2 × 10 = 20
Conclusion
Square numbers follow clear and logical rules that make maths easier to understand. By mastering these square number patterns, students gain confidence and improve their problem-solving skills. With consistent practice and guidance—such as that provided by the best psle tuition in singapore—students can apply these concepts effectively in exams and everyday maths.
