6. Number Play
Answers to all Figure it Out and in-text questions on parity, magic squares, Virahāṅka–Fibonacci numbers and cryptarithms. CBSE / NCERT Class 7 Ganita Prakash.
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6.1 Numbers Tell us Things
From the first child to the last: 0, 0, 1, 0, 3, 0, 3.
1. Draw a height arrangement for each sequence. (Heights are shown as 1 = shortest, 7 = tallest, from the front of the line.)
(a) 0, 1, 1, 2, 4, 1, 5 → heights 7, 3, 5, 4, 1, 6, 2
(b) 0, 0, 0, 0, 0, 0, 0 → 1, 2, 3, 4, 5, 6, 7 (each child is taller than everyone in front)
(c) 0, 1, 2, 3, 4, 5, 6 → 7, 6, 5, 4, 3, 2, 1 (each child is shorter than everyone in front)
(d) 0, 1, 0, 1, 0, 1, 0 → 2, 1, 4, 3, 6, 5, 7
(e) 0, 1, 1, 1, 1, 1, 1 → 7, 1, 2, 3, 4, 5, 6
(f) 0, 0, 0, 3, 3, 3, 3 → 5, 6, 7, 1, 2, 3, 4
2. Always True, Only Sometimes True, or Never True?
(a) Only sometimes true. The first person always says 0, even if short.
(b) Always true. Nobody is taller than the tallest person.
(c) Always true. There is nobody in front of the first person.
(d) Only sometimes true. A person in the middle says 0 if everyone in front is shorter.
(e) Only sometimes true. If the shortest person stands first, they say 0, and someone else calls the largest number.
(f) 7. The last person can have at most 7 people in front, all taller.
6.2 Picking Parity
(a) Even — the odd ones pair up.
(b) Odd — one odd one is left over.
(c) Even.
No. Their ages are consecutive numbers. One is even and the other is odd, so their sum is odd. 112 is even.
1. Find the parity of the sums.
(a) even + even + odd + odd = even
(b) 2 odd + 3 even = even
(c) 5 even numbers = even
(d) 8 odd numbers = even
2. Lakpa got ₹205. Did he make a mistake?
Yes. An odd number of ₹1 coins gives an odd amount. An odd number of ₹5 coins gives an odd amount. ₹10 coins always give an even amount.
odd + odd + even = even. So, the total must be even, but ₹205 is odd.
3. Find the parity.
(d) even − even = even
(e) odd − odd = even
(f) even − odd = odd
(g) odd − even = odd
Yes. If both sides are odd, the number of squares is odd. If at least one side is even, it is even.
(a) 27 × 13 → odd
(b) 42 × 78 → even
(c) 135 × 654 → even
Always even: 2n, 4k + 2, 10m.
Always odd: 2n + 1, 4k + 3, 6m − 1.
Either parity: n + 1, 5n, 3n + 4.
All even numbers: 2n (n = 1, 2, 3, …). All odd numbers: 2n − 1.
The nth even number is 2n. The 100th odd number is 2 × 100 − 1 = 199.
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6.3 Some Explorations in Grids
Grid 1 (rows 13, 14, 18; columns 24, 9, 12):
| 9 | 1 | 3 |
| 8 | 2 | 4 |
| 7 | 6 | 5 |
Grid 2 (rows 24, 15, 6; columns 12, 16, 17):
| 7 | 8 | 9 |
| 4 | 6 | 5 |
| 1 | 2 | 3 |
(Grid 2 has one more answer: 7, 9, 8 / 4, 5, 6 / 1, 2, 3.)
Together, the three rows contain every number from 1 to 9 exactly once.
So, the row sums add up to 1 + 2 + … + 9 = 45. The same is true for the columns.
Only 5 can be at the centre. 1, 2, 3, 4, 6, 7, 8 and 9 cannot.
Reason: the 4 lines through the centre use the other 8 numbers in 4 pairs. Each pair must make 15 − (centre). For 9 or 8, some pair would need a number below 1; for 1 or 2, a pair would need a number above 9. Checking all, only 5 works.
A corner number lies on 3 lines (a row, a column and a diagonal). With 1, only two lines are possible: 1 + 5 + 9 and 1 + 6 + 8. So, 1 cannot be in a corner. Similarly, 9 has only 9 + 5 + 1 and 9 + 4 + 2.
1 and 9 must be in the middle of opposite sides, with 5 between them.
| 8 | 1 | 6 |
| 3 | 5 | 7 |
| 4 | 9 | 2 |
Each row, column and diagonal adds up to 15.
1. How many different magic squares can be made using 1 – 9?
Only one, if we do not count turning or flipping it. With turns and flips, there are 8 squares.
2. Create a magic square using 2 – 10.
Add 1 to every number of the 1 – 9 square. The magic sum is 18 and the centre is 6.
| 9 | 2 | 7 |
| 4 | 6 | 8 |
| 5 | 10 | 3 |
3. (a) Increase each number by 1. (b) Double each number.
(a) Yes, it is still a magic square. The magic sum increases by 3 (15 → 18).
(b) Yes, it is still a magic square. The magic sum doubles (15 → 30).
4. What other operations give another magic square?
Adding or subtracting the same number to every number, multiplying or dividing every number by the same number, and turning or flipping the square.
5. Magic squares using 9 consecutive numbers
Start with the 1 – 9 square and add the same number to every entry. For 3 – 11 add 2; for 9 – 17 add 8. The middle number of the set goes in the centre.
| m + 3 | m − 4 | m + 1 |
| m − 2 | m | m + 2 |
| m − 1 | m + 4 | m − 3 |
1. Magic square with centre 25
| 28 | 21 | 26 |
| 23 | 25 | 27 |
| 24 | 29 | 22 |
2. Sum of any row, column or diagonal
3m. Example: (m + 3) + (m − 4) + (m + 1) = 3m.
3. (a) Adding 1 to every term (b) Doubling every term
| m + 4 | m − 3 | m + 2 |
| m − 1 | m + 1 | m + 3 |
| m | m + 5 | m − 2 |
| 2m + 6 | 2m − 8 | 2m + 2 |
| 2m − 4 | 2m | 2m + 4 |
| 2m − 2 | 2m + 8 | 2m − 6 |
4. Magic square with magic sum 60
3m = 60, so m = 20.
| 23 | 16 | 21 |
| 18 | 20 | 22 |
| 19 | 24 | 17 |
5. Can nine non-consecutive numbers make a magic square?
Yes. Multiply the 1 – 9 square by 3 (magic sum 45):
| 24 | 3 | 18 |
| 9 | 15 | 21 |
| 12 | 27 | 6 |
Every row, column and diagonal adds up to 34 (chautīs).
Other patterns: the four corners 7 + 14 + 9 + 4 = 34; the central 2 × 2 square 13 + 8 + 3 + 10 = 34; each corner 2 × 2 square, like 7 + 12 + 2 + 13 = 34.
6.4 Virahāṅka–Fibonacci Numbers
5 has 8 ways: 1+1+1+1+1, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, 1+2+2, 2+1+2, 2+2+1.
6 has 8 + 5 = 13 ways, and 8 has 34 ways.
After 55: 34 + 55 = 89. Next three after 89: 144, 233, 377.
1 (odd), 2 (even), 3 (odd), 5 (odd), 8 (even), 13 (odd), 21 (odd), 34 (even), …
After 1, the pattern even, odd, odd repeats. The number after 377 is 610, which is even.
6.5 Digits in Disguise
T + T + T = UT: T = 5, U = 1 (5 + 5 + 5 = 15).
K2 + K2 = HMM: 2 + 2 = 4, so M = 4. 72 + 72 = 144, so K = 7, H = 1. H cannot be 2 or 3, since the largest sum is 92 + 92 = 184.
YY + Z = ZOO: 99 + 1 = 100, so Y = 9, Z = 1, O = 0.
B5 + 3D = ED5: 75 + 30 = 105, so B = 7, D = 0, E = 1.
KP + KP = PRR: 61 + 61 = 122, so K = 6, P = 1, R = 2.
C1 + C = 1FF: 91 + 9 = 100, so C = 9, F = 0.
1. A bulb is ON. The switch is toggled 77 times. On or off?
Off. Each pair of toggles brings it back to ON. 77 is odd, so after 76 toggles it is ON and the last one turns it OFF.
2. Can the page numbers of 50 loose sheets add up to 6000?
Each sheet has two pages, like 7 and 8, whose sum is odd. 50 odd numbers add up to an even number. So, parity alone does not rule out 6000.
But look closer: a sheet has pages 2k − 1 and 2k, with sum 4k − 1, which is 1 less than a multiple of 4. 50 such sums give a total that is 2 less than a multiple of 4. 6000 is a multiple of 4, so the sum cannot be 6000.
3. Fill the 2 × 3 grid with 3 odd (o) and 3 even (e) numbers.
Rows: o, e, e (sum odd) and o, e, o (sum even). Columns: o + o even, e + e even, e + o odd.
| o | e | e |
| o | e | o |
4. Magic square with magic sum 0 (subtract 5 from every number of the 1 – 9 square)
| 3 | −4 | 1 |
| −2 | 0 | 2 |
| −1 | 4 | −3 |
5. Fill in with odd or even.
(a) even (b) even (c) even (d) odd
6. Parity of 1 + 2 + … + 100
There are 50 odd numbers. 50 odd numbers make an even sum. So, the sum is even (it is 5050).
7. Next and previous numbers of 987, 1597
Next: 987 + 1597 = 2584, 1597 + 2584 = 4181.
Previous: 1597 − 987 = 610, 987 − 610 = 377.
8. Ways to climb 8 steps taking 1 or 2 steps at a time
It is the same as writing 8 as a sum of 1s and 2s: 34 ways.
9. Parity of the 20th term
Even terms come at positions 2, 5, 8, 11, 14, 17, 20, … So, the 20th term is even.
10. Which statements are true?
(a) True. 4m is even, so 4m − 1 is odd.
(b) False. 6j − 4 gives 2, 8, 14, … It misses 4, 6, 10, …
(c) Not fully true. 2q − 1 (q = 1, 2, 3, …) gives all odd numbers, but 2p + 1 misses 1 unless p can be 0.
(d) False. 2f is even, so 2f + 3 is always odd.
11. UT + TA = TAT
The sum of two 2-digit numbers is less than 200, so T = 1. Then 1 + A ends in 1, so A = 0. U + 1 = 10, so U = 9.
U = 9, T = 1, A = 0: 91 + 10 = 101.
If it counts people, it should be even, because each marriage has 2 people. So, the number given is wrong (it may count marriages, not people).