NCERT Ganita Prakash Part I • Chapter 6 (pp. 127–145)

Number Play

Discover the hidden rules, patterns, and puzzles inside numbers. Learn how numbers encode arrangements, unlock parity invariants, construct magic squares, generate Virahāṅka–Fibonacci poetic rhythms, and crack cryptarithms!

5 Core Sections 40 Practice Questions 20-Question Final Quiz

Hero Puzzle: What Comes Next?Sequence Challenge

1235813?

What You Will Learn in Chapter 6

Progress: 17%

•1. Interpret information in height sequences

•2. Identify patterns in arrangements

•3. Understand parity as pairwise completeness

•4. Classify numbers as odd or even

•5. Reason about parity of sums and differences

•6. Reason about parity of products

•7. Prove puzzles impossible using parity

•8. Explore number patterns in 3×3 grids

•9. Understand the definition of magic squares

•10. Determine magic sum = 15 for 1–9 square

•11. Understand why 5 must be at the centre

•12. Construct magic squares systematically

•13. Generalise magic squares using centre m

•14. Understand the Virahāṅka–Fibonacci sequence

•15. Link the sequence to Sanskrit poetic rhythms

•16. Formulate recurrence: N(b) = N(b-1) + N(b-2)

•17. Determine higher terms in the sequence

•18. Explore repeating parity patterns (period 3)

•19. Solve digit disguise puzzles (cryptarithms)

•20. Deduce digits from place value and carries

Section 6.1 (pp. 127–128)

Numbers Tell us Things

Height Arrangements • Inversion Sequences

Can numbers encode an entire physical arrangement without anyone disclosing their actual measurements? The textbook opens with seven children standing in a row from left to right. Each child calls out a number based on a simple rule:

The Height Rule:

Each child calls out the number of children standing in front of them (to their left) who are taller than them.

Interactive Height Arrangement Simulator

Swap adjacent children to see how their called-out numbers update dynamically!

Front (Left)───────────────▶Back (Right)

145 cm

138 cm

160 cm

152 cm

130 cm

175 cm

168 cm

Generated Sequence:[0, 1, 0, 1, 4, 0, 1]

Target: [0, 1, 1, 2, 4, 1, 5]

Textbook Activity: Always, Sometimes, or Never True? (p. 128)

For each statement, reason through the logic, make your choice, and compare with the textbook explanation:

If a person says '0', then they are the tallest in the group.

If a person is the tallest in the group, then their number is '0'.

The first person's number is '0'.

If a person is standing in between (neither first nor last), then they cannot say '0'.

The person who calls out the largest number is the shortest.

In a group of 8 people, the largest number possible for anyone to call out is 7.

Exam Pitfalls

Common Misconceptions in Number Play

Thinking Odd + Odd is Always Odd

Common Trap: Since both numbers are odd, students assume their sum must be odd.

Mathematical Truth: Odd + Odd is ALWAYS EVEN. Every odd number has exactly one unpaired leftover unit. When two odd numbers combine, their two single leftovers join to form a complete pair!

Calculating the Full Answer to Test Parity

Common Trap: Thinking you must compute huge numbers like 135 × 654 to know if the result is even or odd.

Mathematical Truth: Parity is an INVARIANT property. Any even number multiplied by anything yields an even number. Since 654 is even, 135 × 654 is automatically even without multiplying!

Believing Numbers in a Magic Square Can be Placed by Guessing

Common Trap: Thinking you can pick any random number for the centre of a 3 × 3 magic square.

Mathematical Truth: The centre cell of a 1–9 magic square MUST BE 5. The centre participates in 4 distinct lines of three numbers (row, column, and 2 diagonals) that must all sum to 15. Only 5 has four complementary pairs.

Assuming Magic Squares Only Work for Numbers 1 to 9

Common Trap: Believing magic squares cannot be made with other numbers like 2–10, negative numbers, or non-consecutive numbers.

Mathematical Truth: Any 9 numbers in arithmetic progression or transformed via addition/scaling form a magic square. With centre m, the magic sum is always 3m. A magic square can even have sum 0 using negative numbers!

Thinking Fibonacci Discovered the Sequence First

Common Trap: Believing Leonardo of Pisa (Fibonacci) invented the sequence in 1202 CE.

Mathematical Truth: The sequence was discovered over 500 years earlier by Indian prosodists (notably Virahāṅka c. 700 CE, and earlier referenced by Piṅgala c. 300 BCE) while analyzing short (1-beat) and long (2-beat) poetic meters.

Confusing Sequence Position (n) with Sequence Value

Common Trap: Saying the 100th even number is 100, or the 100th odd number is 101.

Mathematical Truth: The nth even number is 2n (for n=100, 2 × 100 = 200). The nth odd number is 2n − 1 (for n=100, 200 − 1 = 199).

Guessing Blindly in Cryptarithms

Common Trap: Randomly plugging in digits 0–9 without analyzing constraints.

Mathematical Truth: Cryptarithms are solved through mathematical deduction: inspecting units column parity, looking at maximum possible carries (a sum of two numbers can only carry 1 into the next column), and identifying repeated letters.

Reference Glossary

Key Chapter Terms & Notations

Parity

Even / Odd

The mathematical classification of an integer as either even (divisible by 2) or odd (not divisible by 2).

Invariant

I(x) = C

A mathematical property or quantity that remains completely unchanged after applying transformations or operations.

Magic Square

M_n

An n × n grid of numbers where the sum of each row, column, and both diagonals is equal to the magic sum.

Magic Sum

S = 3m

The constant total of any row, column, or diagonal in a magic square. For a 3 × 3 grid, S = 3 × centre number.

Chautīsā Yantra

Sum = 34

A 10th-century 4 × 4 magic square from Khajuraho, India, where rows, cols, diags, and sub-squares sum to 34.

Virahāṅka Numbers

V_n = V_{n-1} + V_{n-2}

The sequence 1, 2, 3, 5, 8, 13, 21... discovered by Indian prosodists to count combinations of 1-beat and 2-beat poetic syllables.

Cryptarithm / Alphametic

Letters ➔ Digits

A mathematical puzzle where letters replace digits in an arithmetic equation, requiring deductive solving.

Consecutive Integers

n, n + 1

Integers that follow each other in order without gaps. Two consecutive integers always have opposite parities and an odd sum.

Help & Clarifications