Chapter 1 — Patterns in Mathematics

Chapter 1 of Ganita Prakash (Class 6 Maths) 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.

This chapter frames mathematics as the search for patterns and their explanations. It introduces number sequences (counting, odd, even, triangular, square, cube, Virahānka, powers of 2 and 3) and shows how they relate to each other — for example, the sum of the first n odd numbers always equals n², and adding consecutive triangular numbers gives square numbers. It then extends to shape sequences (regular polygons, complete graphs, stacked squares, stacked triangles, Koch snowflake) and reveals surprising links between shape sequences and number sequences, such as the number of lines in complete graphs following the triangular number sequence.

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.