NCERT Solutions Ganita Prakash Chapter 1 Figure it Out — Patterns in Shapes
Book page 11 Updated on2026-09-05
Q1.
Can you recognise the pattern in each of the sequences in Table 3?
Answer
Yes. Each shape sequence follows one clear rule:
Shape sequence
Rule for going to the next shape
Related number sequence
Regular Polygons
Add one more side (and one more corner) each time — triangle, quadrilateral, pentagon, hexagon, …
3, 4, 5, 6, 7, 8, 9, 10 …
Complete Graphs Kn
Add one more dot, then join it to every dot already there.
1, 3, 6, 10, 15 … (triangular)
Stacked Squares
Add one more row and one more column of small squares.
1, 4, 9, 16, 25 … (squares)
Stacked Triangles
Add one more row of small triangles at the bottom.
1, 4, 9, 16, 25 … (squares)
Koch Snowflake
Replace every straight line segment by a “speed bump” of 4 smaller segments.
3, 12, 48, 192 … (3 × powers of 4)
Stacked Squares: 1, 4, 9, 16, 25 small squares.Stacked Triangles: 1, 4, 9, 16, 25 small triangles (upward-pointing green and downward-pointing yellow together).
Q2.
Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Answer
Yes for four of them, and “only in principle” for the fifth.
Regular Polygons — the next shape after the decagon is an 11-sided regular polygon (hendecagon). Easy to draw: keep all sides equal and all corners alike.
Complete Graphs — the next is K7: mark 7 dots on a circle and join every pair. It has 21 lines, so the drawing gets crowded but it can certainly be done.
Stacked Squares — the next has 6 rows of 6 small squares = 36.
Stacked Triangles — the next has 6 rows, 36 small triangles in all.
Koch Snowflake — the next stage can be drawn, but each stage has 4 times as many line segments (3 → 12 → 48 → 192 → 768 …) and each is 3 times shorter. After a few stages the bumps become too tiny to draw by hand or even to see. So the rule is perfectly clear, but the picture soon goes beyond what a pencil can manage.
The rule in one line each: polygons — add a side; complete graphs — add a dot and join it to all the others; stacked squares/triangles — add a row; Koch snowflake — replace every segment by a bump made of 4 segments.