Patterns in Mathematics

1. Chapter at a Glance

Mathematics is often called the study of patterns. A pattern is anything that repeats or grows according to a rule. When we notice a pattern, we can predict what comes next, explain why it happens and use it to solve problems quickly. Patterns appear everywhere — in the numbers on a calendar, the tiles on a floor, the petals of a flower and even the way a cricket score grows over by over.

In this chapter we look at two kinds of patterns: patterns in numbers (number sequences) and patterns in shapes (shape sequences). We also learn how to draw number patterns as dots so that we can see why they behave the way they do. Finally, we discover surprising links between different sequences — for example, that adding odd numbers always gives a square number.

2. What is Mathematics?

Mathematics is a way of thinking. Mathematicians look for patterns, then try to explain them. The explanation is what turns a guess into knowledge. Once a pattern is understood, it can be used again and again in new situations.

• In science and technology: the movement of planets, the timing of tides, the working of mobile phones and computers all depend on mathematical patterns.
• In daily life: we use patterns to plan budgets, read timetables, measure land, cook with the right quantities and play games.
• In art and design: rangoli, kolam, block prints and temple carvings use repeating shapes and symmetry.
• For its own sake: many patterns are studied simply because they are beautiful and interesting. Often, years later, they turn out to be useful too.

India has a long history of mathematics. Ancient scholars studied number patterns in poetry, astronomy and architecture. One famous example — the Virahānka numbers — came from counting rhythms in Sanskrit and Prakrit poetry, as we will see below.

3. Patterns in Numbers: Number Sequences

A number sequence is a list of numbers written in a definite order, following a rule. Each number in the list is called a term. Here are the most important sequences of this chapter.

How to find the rule of a sequence

1. Find the difference between consecutive terms. If it is the same every time, the rule is "add that number" (for example, odd numbers go up by 2).
2. If the differences themselves grow (2, 3, 4, 5, …), the sequence may be triangular or square-like.
3. If each term is a fixed number of times the previous one, the rule is "multiply" (powers of 2 or 3).
4. If each term is the sum of the two before it, it is a Virahānka-type sequence.

Virahānka numbers — a pattern from poetry

In Sanskrit and Prakrit poetry, syllables are either short (1 beat) or long (2 beats). The scholar Virahānka (around 700 CE) asked: in how many ways can a rhythm of n beats be made using short and long syllables? The answers form the sequence 1, 2, 3, 5, 8, 13, 21, 34, …, where each number is the sum of the two before it. For example, a 4-beat rhythm can be made in 5 ways: SSSS, SSL, SLS, LSS, LL. This same sequence was later studied in Europe and is known there as the Fibonacci sequence, but Indian scholars described it centuries earlier.

4. Visualising Number Sequences

Many sequences become easy to understand when we draw them with dots. The picture shows the rule that the numbers alone hide.

• Triangular numbers (1, 3, 6, 10, 15, …) are the number of dots that can be arranged in a triangle. To get the next one, add a new row that is one dot longer than the previous bottom row.
• Square numbers (1, 4, 9, 16, 25, …) are the number of dots in a square arrangement. The nth square number is n × n.
• Cube numbers (1, 8, 27, 64, …) are the number of small cubes needed to build a bigger cube. The nth cube number is n × n × n.
• Odd numbers can be drawn as L-shapes (1 dot, then 3 dots in an L, then 5 dots in a bigger L …). Even numbers can be drawn as two equal rows of dots.

5. Relations among Number Sequences

The most exciting part of this chapter is discovering that different sequences are connected. Here are the key relations, each with a picture-based reason.

(a) Adding odd numbers gives square numbers

Why? Draw a square of dots and split it into L-shaped layers starting from one corner. The layers have 1, 3, 5, 7, … dots — the odd numbers. So the sum of the first n odd numbers is n × n.

(b) Counting up and down gives square numbers

Why? Tilt a square of dots so that it stands on a corner (like a diamond). Reading the rows from top to bottom gives 1, 2, 3, …, n, …, 3, 2, 1. So the sum is n × n.

(c) Two consecutive triangular numbers make a square

Why? A square of dots can be cut along a diagonal into two triangles, one slightly smaller than the other. These two triangles are consecutive triangular numbers.

(d) Adding counting numbers gives triangular numbers

This is simply the triangle picture read row by row. A quick trick: the sum 1 + 2 + … + n = n × (n + 1) ÷ 2. For example, 1 + 2 + … + 10 = 10 × 11 ÷ 2 = 55.

(e) Adding powers of 2

The sum is always one less than the next power of 2. For example, 1 + 2 + 4 + 8 = 15 = 16 − 1. Picture it with a strip: half of a bar + a quarter + an eighth … keeps getting closer to the whole bar.

(f) Adding counting numbers and the cubes

Another beautiful relation: 1³ = 1, 1³ + 2³ = 9 = 3², 1³ + 2³ + 3³ = 36 = 6², 1³ + 2³ + 3³ + 4³ = 100 = 10². The sum of the first few cube numbers is the square of a triangular number (1, 3, 6, 10, …).

6. Patterns in Shapes

A shape sequence is a list of figures that change according to a rule. Important examples:

• Regular polygons: triangle (3 sides), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10). All sides are equal and all angles are equal. The number of sides gives the counting numbers starting from 3.
• Complete graphs: place 1, 2, 3, 4, 5, … points on a circle and join every pair of points with a line. The number of lines is 0, 1, 3, 6, 10, … — the triangular numbers!
• Stacked squares: squares of 1×1, 2×2, 3×3 arrangements — the numbers of small squares are 1, 4, 9, 16, … (square numbers).
• Stacked triangles: big triangles made of small triangles; the number of small triangles is 1, 4, 9, 16, … (also square numbers).
• Koch snowflake: start with a triangle. On each side, replace the middle third by two sides of a smaller triangle pointing outwards. Repeat. The number of sides is 3, 12, 48, 192, … — each time it is multiplied by 4.

7. Relation to Number Sequences

Shape sequences and number sequences are closely linked. Counting something in each shape (sides, lines, small squares, corners) usually gives a familiar number sequence.

8. Master Formula Table

9. Master Summary Table

Growth chart of five sequences (first 6 terms)

Notice how multiplying sequences (powers, cubes) grow much faster than adding sequences (counting, triangular). This difference in growth is an important idea you will meet again in higher classes.

10. Mind Map

11. Tips and Tricks

12. Important One-liners

• Mathematics is the search for patterns and for explanations of why they occur.
• A number sequence is an ordered list of numbers that follows a rule.
• The sum of the first n odd numbers is n × n.
• 1 + 2 + 3 + … + n + … + 3 + 2 + 1 = n × n.
• The sum of two consecutive triangular numbers is always a square number.
• Triangular numbers are 1, 3, 6, 10, 15, 21, 28, …
• Square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
• Cube numbers are 1, 8, 27, 64, 125, 216, …
• Virahānka numbers: each term is the sum of the previous two terms.
• Virahānka found these numbers while counting rhythms of short and long syllables in poetry.
• 1 + 2 + 4 + 8 + … is always one less than the next power of 2.
• The number of lines in complete graphs gives the triangular numbers.
• The Koch snowflake's number of sides is multiplied by 4 at each step.
• A regular polygon has all sides equal and all angles equal.
• The nth odd number is 2n − 1 and the nth even number is 2n.

13. Exam-pattern Questions with Detailed Answers

These questions follow the pattern of school tests and the NCERT exercises. Try each one before opening the answer.

14. Quick Revision Summary