NCERT Solutions for Class 7th Maths Chapter 3 Puzzle Time
Updated on2026-09-19
Q1.
You might have noticed and wondered about these different circle designs around the page numbers on each page! The picture below shows all the designs for the numbers from 1 to 100. Try to decode the colour scheme for each number. There are several interesting patterns here. Share your observations with your classmates.
The ring design printed around each page number, shown here for every number from 1 to 100.
Answer
The ring around a number is its prime factorisation, drawn as coloured arcs.
The ring is cut into as many equal arcs as the number has prime factors, counted with repeats.
Each arc is coloured by which prime it stands for.
Prime
Colour of the arc
2
green
3
magenta / pink
5
orange
7
blue
bigger primes
red — with the prime printed in tiny type below the ring
1
a single plain grey ring (no prime factors at all)
2 = 2 → one green arc
4 = 2 × 2 → two green arcs
6 = 2 × 3 → one green, one magenta
8 = 2 × 2 × 2 → three green arcs
12 = 2 × 2 × 3 → two green, one magenta
15 = 3 × 5 → one magenta, one orange
22 = 2 × 11 → one green, one red marked 11
30 = 2 × 3 × 5 → green, magenta, orange
100 = 2 × 2 × 5 × 5 → two green, two orange
Patterns to notice
Primes are whole rings of one colour — 2, 3, 5, 7 in their own colours, and every prime above 7 in solid red.
Powers of 2 (2, 4, 8, 16, 32, 64) are all-green rings cut into 1, 2, 3, 4, 5 and 6 arcs — the number of arcs grows very slowly as the numbers grow.
Every even number has at least one green arc; every multiple of 3 has at least one magenta arc.
Perfect squares such as 4, 9, 25, 36, 49, 100 always have an even number of arcs, and the colours pair up.
Numbers with many arcs are the ones built from small primes — 64 = 2⁶ has six arcs while the much larger 97 has only one.
Two numbers share a colour exactly when they share a prime factor, so their HCF is bigger than 1.
Why it works: every number has exactly one prime factorisation, so every number gets exactly one design. That is why no two numbers from 1 to 100 have the same ring, and why the ring alone tells you the number's factors.
Q2.
Extending this scheme, colour the page numbers from 101 – 110.
Answer
Factorise each number, then draw one arc per prime factor.
Tip: to test whether 101, 103, 107 and 109 are prime you only need to try dividing by primes up to 10, since 11 × 11 = 121 is already past 110. None of 2, 3, 5 or 7 divides them, so all four are prime. That is why this stretch of the page numbers has so many plain red rings.