1. Chapter at a Glance
Numbers hide many secrets. Is the sum of five odd numbers odd or even? Can you fill a 3 × 3 grid so that every row, column and diagonal has the same sum? Why do the Virahānka numbers alternate between odd and even in a fixed rhythm? This chapter explores such questions through parity (odd and even), number grids, magic squares, the Virahānka–Fibonacci sequence and digit puzzles. The aim is to reason and explain — not just to calculate.
- Parity rules for sums, differences and products.
- Using algebra (2n and 2n + 1) to explain parity.
- Grids with given row and column sums; 3 × 3 magic squares.
- The Virahānka sequence and its parity pattern.
- Cryptarithms — puzzles where letters hide digits.
2. Picking Parity: Odd and Even
Every whole number is either even (can be split into pairs with nothing left) or odd (one is always left over). Whether a number is odd or even is called its parity.
- Even numbers can be written as 2n; odd numbers as 2n + 1, where n is a whole number.
- The last digit decides parity: 0, 2, 4, 6, 8 → even; 1, 3, 5, 7, 9 → odd.
| Operation | Result | Reason |
|---|---|---|
| even + even | even | pairs + pairs = pairs |
| odd + odd | even | the two leftover ones make a pair |
| even + odd | odd | one leftover remains |
| even × any number | even | contains a factor 2 |
| odd × odd | odd | (2a + 1)(2b + 1) = 2(…) + 1 |
A sum of several numbers is odd exactly when it contains an odd number of odd terms. The even terms never change the parity.
Without adding, is 23 + 48 + 115 + 9 + 76 odd or even?
Solution: The odd terms are 23, 115 and 9 — three of them. Three is odd, so the sum is odd. (Check: 271.)
Can five odd numbers add up to 100?
Solution: Five odd terms → an odd number of odd terms → the sum is odd. 100 is even, so it is impossible.
Parity in arrangements
Parity can prove that some arrangements are impossible. For example, if a group of people shake hands, the number of people who shake an odd number of hands is always even — because every handshake involves two hands, so the total count of hands shaken is even.
3. Explorations in Grids
Fill a 3 × 3 grid with the numbers 1 to 9 (each once) so that the row and column sums match given targets. Such puzzles use reasoning about totals:
- The total of 1 to 9 is 45. So the three row sums together must be 45, and the three column sums together must also be 45.
- If someone asks for row sums 12, 15, 20, check first: 12 + 15 + 20 = 47 ≠ 45, so it is impossible.
Magic squares
A magic square is a grid in which every row, every column and both diagonals have the same sum, called the magic sum.
- Using 1 to 9, the magic sum is 45 ÷ 3 = 15.
- The centre must be 5: the middle row, middle column and two diagonals all pass through the centre and together use every number once plus the centre three extra times. So 4 × 15 = 45 + 3 × centre, giving centre = 5.
- Even numbers go in the corners, odd numbers on the edges.
- If you add the same number to every entry, or multiply every entry by the same number, the square stays magic.
Magic squares have a long history in India. A famous 4 × 4 magic square is carved at the Parshvanath temple in Khajuraho, and the mathematician Narayana Pandita (14th century) wrote about methods for making them.
4. Nature's Favourite Sequence: Virahānka Numbers
The Virahānka numbers are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … — each number is the sum of the previous two. They appear in nature: the numbers of petals of many flowers (3, 5, 8, 13), the spirals of sunflower seeds and pine cones.
Parity pattern
Write O for odd and E for even: 1 (O), 2 (E), 3 (O), 5 (O), 8 (E), 13 (O), 21 (O), 34 (E), … From 2 onwards the pattern is E, O, O, E, O, O, … — every third number is even. Parity rules explain why: E + O = O, O + O = E, O + E = O, and then the cycle repeats.
Is the 30th Virahānka number (counting 1 as the 1st and 2 as the 2nd) odd or even?
Solution: The even terms are the 2nd, 5th, 8th, … (every 3rd term starting from the 2nd). These positions leave remainder 2 when divided by 3. 30 ÷ 3 leaves remainder 0, so the 30th term is odd.
5. Digits in Disguise: Cryptarithms
In a cryptarithm, letters stand for digits (different letters for different digits, and a number does not start with 0). We use place value and carrying to find them.
Solve: AA + AA = 1B2, where AA is a two-digit number with both digits A, and 1B2 is a three-digit number.
Solution: AA + AA = 2 × AA = 2 × 11A = 22A. We need a three-digit result 1B2, so 22A ends in 2 → A = 6 (22 × 6 = 132). So A = 6, B = 3: 66 + 66 = 132.
Find the digits: K + K + K = 1K (a single digit K added three times gives the two-digit number with tens digit 1 and ones digit K).
Solution: 3K = 10 + K, so 2K = 10 and K = 5. Check: 5 + 5 + 5 = 15 ✔.
6. Solved Problems (NCERT Style)
Is 37 × 51 × 93 odd or even? And 37 × 51 × 94?
Solution: All three factors in the first are odd, so the product is odd. The second has an even factor (94), so it is even.
Show that the sum of any two consecutive numbers is odd.
Solution: One of two consecutive numbers is even and the other odd; even + odd = odd. With algebra: n + (n + 1) = 2n + 1, which is odd.
Add 10 to every entry of the 1–9 magic square. What is the new magic sum?
Solution: Each line has three entries, each increased by 10, so each sum rises by 30. New magic sum = 15 + 30 = 45.
Seven coins lie heads up. In each move you must flip exactly two coins. Can you make all seven coins tails up?
Solution: Track the number of heads. Each move changes it by −2, 0 or +2, so its parity never changes. It starts at 7 (odd) and we want 0 (even). Parity can't change, so it is impossible.
7. Master Formula Table
| Idea | Rule |
|---|---|
| Even number | 2n |
| Odd number | 2n + 1 |
| Sum parity | odd ⇔ odd number of odd terms |
| Product parity | odd ⇔ all factors odd |
| Magic sum (1 to 9) | 45 ÷ 3 = 15 |
| Centre of 3 × 3 magic square | magic sum ÷ 3 |
| Virahānka rule | next = sum of previous two |
| Virahānka parity | every 3rd term even (E, O, O repeating) |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Parity | Odd/even rules | Predict without calculating |
| Impossibility proofs | Parity cannot change | Explain "why not" |
| Grids | Row totals = grand total | Check feasibility |
| Magic squares | Equal sums; centre 5 | Complete a square |
| Virahānka numbers | Sum of previous two | Terms, parity |
| Cryptarithms | Place value and carries | Find digits |
9. Mind Map
- 2n and 2n + 1
- Sum and product rules
- Impossibility
- Row and column sums
- Magic squares
- Virahānka numbers
- E, O, O pattern
- Cryptarithms
- Coin flips
10. Tips and Tricks
- To find the parity of a long sum, just count the odd terms.
- In 3 × 3 magic squares with numbers in an equal-step sequence, the middle number goes in the centre.
- In cryptarithms, start with the ones column and think about carries.
- When a puzzle seems impossible, look for something that never changes (an invariant), such as parity.
- Thinking odd + odd is odd.
- Forgetting that 0 is even.
- Giving two letters the same digit in a cryptarithm.
11. Important One-liners
- Parity means whether a number is odd or even.
- Odd + odd = even; even + odd = odd.
- A product is even if any factor is even.
- A sum is odd only if it has an odd number of odd terms.
- The magic sum of a 3 × 3 square using 1–9 is 15.
- The centre of that magic square is always 5.
- Every third Virahānka number is even.
- Virahānka numbers appear in petals and seed spirals.
- A cryptarithm hides digits behind letters.
12. Exam-pattern Questions with Detailed Answers
The sum of 4 odd numbers and 3 even numbers is always:
- odd
- even
- a multiple of 7
- cannot say
Show answer
There are 4 odd terms — an even number of them — so the sum is even.
Answer: (B) even
Write the next three Virahānka numbers after 34, 55.
Show answer
34 + 55 = 89; 55 + 89 = 144; 89 + 144 = 233.
Answer: 89, 144, 233
Is 1,357 × 2,468 odd or even?
Show answer
2,468 is even, so the product is even.
Answer: Even
Complete the magic square (magic sum 15): top row 8, _, 6; middle row _, 5, _; bottom row 4, _, 2.
Show answer
Top: 8 + x + 6 = 15 → x = 1. Bottom: 4 + y + 2 = 15 → y = 9. Left column: 8 + a + 4 = 15 → a = 3. Right column: 6 + b + 2 = 15 → b = 7.
Square: 8 1 6 / 3 5 7 / 4 9 2. Check middle row 3 + 5 + 7 = 15 ✔ and diagonals 8 + 5 + 2 = 15, 6 + 5 + 4 = 15 ✔.
Answer: 8 1 6 / 3 5 7 / 4 9 2
Can the sum of three consecutive odd numbers be 100? Explain.
Show answer
Three odd terms give an odd sum. 100 is even, so it is impossible.
Answer: No
A 3 × 3 grid must use 1–9 with row sums 10, 15 and 21. Is this possible?
- Yes
- No
Show answer
The three row sums must add to 1 + 2 + … + 9 = 45. Here 10 + 15 + 21 = 46 ≠ 45.
Answer: (B) No
Using algebra, show that the product of two odd numbers is odd.
Show answer
(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which has the form 2n + 1.
Answer: The product has the form 2n + 1, so it is odd
Solve the cryptarithm: A B + B A = 1 2 1 (A and B are different digits, and AB is a two-digit number).
Show answer
AB + BA = (10A + B) + (10B + A) = 11(A + B) = 121, so A + B = 11. Many pairs work, e.g. A = 2, B = 9: 29 + 92 = 121 ✔.
Answer: Any digits with A + B = 11, e.g. 29 + 92
In a 3 × 3 magic square made from 3, 4, 5, …, 11, what is the magic sum and the centre number?
Show answer
Total = 3 + 4 + … + 11 = 63. Magic sum = 63 ÷ 3 = 21. Centre = 21 ÷ 3 = 7.
Answer: Magic sum 21; centre 7
A dot moves on a straight track, one step left or right each second. It starts at 0. Can it be at position 0 after exactly 15 seconds?
Show answer
Each step changes the position by 1, flipping its parity. After 15 steps (odd), the position must be odd. 0 is even.
Answer: No
How many even numbers are among the first 25 Virahānka numbers (1, 2, 3, 5, …)?
Show answer
Even terms are at positions 2, 5, 8, 11, 14, 17, 20, 23 (every 3rd from the 2nd). That is 8 positions up to 25.
Answer: 8
In a party of 9 people, could each person shake hands with exactly 3 others?
Show answer
Total of everyone's handshake counts = 9 × 3 = 27. But each handshake is counted twice (once by each person), so this total must be even. 27 is odd.
Answer: No, it is impossible
13. Quick Revision Summary
- Even = 2n, odd = 2n + 1; a sum is odd iff it has an odd number of odd terms; a product is odd iff all factors are odd.
- Parity can prove that something is impossible.
- 3 × 3 magic square with 1–9: magic sum 15, centre 5.
- Virahānka numbers: sum of the previous two; every third one is even.
- Cryptarithms are solved with place value and carrying.