Chapter 1 — Patterns in Mathematics

Chapter 1 of the Class 6 Maths NCERT textbook (Ganita Prakash), "Patterns in Mathematics", introduces patterns in mathematics — exploring number sequences like triangular numbers, square numbers, cube numbers, Virahānka numbers, and powers of 2, as well as shape sequences like regular polygons, complete graphs, and the Koch snowflake.

• Maths as the search for patterns — The chapter frames maths as finding patterns and explaining why they hold. Number theory studies patterns in whole numbers and geometry studies patterns in shapes, so mathematicians treat the subject as both an art and a science.
• Number sequences and their hidden links — Ten key sequences — counting, odd, even, triangular, square, cube, Virahānka, powers of 2 and 3 — connect in surprising ways. Adding the first n odd numbers gives n², and pairs of triangular numbers add up to squares.
• Shape sequences meet number sequences — Growing shapes like regular polygons, complete graphs, stacked triangles and the Koch snowflake generate their own number patterns. For instance, the count of lines in complete graphs follows the triangular number sequence exactly.

1. 01Mathematics is defined as the search for patterns and for explanations of why those patterns exist; mathematicians view it as both an art and a science.
2. 02Number theory is the branch of mathematics that studies patterns in whole numbers; geometry studies patterns in shapes.
3. 03Table 1 lists 10 number sequences: All 1's, Counting numbers, Odd numbers, Even numbers, Triangular numbers, Squares, Cubes, Virahānka numbers, Powers of 2, Powers of 3.
4. 04Triangular numbers (1, 3, 6, 10, 15, 21, 28, ...) are formed by summing consecutive counting numbers.
5. 05The sum of the first n odd numbers always equals n² — e.g., 1+3+5+7+9+11=36=6².
6. 06Adding counting numbers up and down (1+2+1, 1+2+3+2+1, ...) also gives square numbers.
7. 07Adding pairs of consecutive triangular numbers (1+3, 3+6, 6+10, ...) gives square numbers.
8. 08Cumulative sums of hexagonal numbers (1, 1+7, 1+7+19, ...) give cube numbers: 1, 8, 27, 64, ...
9. 09The number 36 is both a triangular number (8th) and a square number (6²), showing the same number can belong to multiple sequences.
10. 10The Koch Snowflake line segment counts follow 3 times powers of 4: 3, 12, 48, 192, ... — a shape sequence linked to a number sequence not in Table 1.