NCERT Solutions Class 7 Maths Chapter 6: Number Play (Hinglish)
Ganita Prakash, Class 7 Maths ki kitaab Part 1 (Reprint 2026-27) ke Chapter 6, Number Play, ke har sawaal ke jawaab: text ke sawaal, grid aur magic square ki paheliyan, rhythm aur cryptarithm ke sawaal aur paanchon Figure it Out sets. Page numbers kitaab ke hisaab se hain (pages 127–145). Jahan kitaab khud kuch aazmaane ko kehti hai, wahan ek sambhav jawaab diya hai. Chhote notes: Chapter 6 ke notes.
Numbers Tell us Things (pages 127–128)
Numbers kya batate hain? Inka kya matlab ho sakta hai? Line mein khade bachche batate hain ki unke aage kitne bachche unse lambe hain. Chitron mein line ka aage ka hissa baayein taraf hai. Pehli line mein numbers 0, 1, 0, 2, 2, 2, 1 hain, aur bachchon ke jagah badalne ke baad 0, 0, 1, 2, 2, 5, 6; dono lines mein har number is niyam se mel khaata hai.
Teesri line mein har bachche ka number: baayein se daayein, 0, 0, 1, 0, 3, 0, 3. Lambaai ko 1 (sabse chhota) se 7 (sabse lamba) tak likho to line 1, 5, 3, 6, 2, 7, 4 hai: jaise, paanchven bachche (lambaai 2) ke aage 5, 3 aur 6 lambaai waale bachche usse lambe hain, isliye uska number 3 hai.
Figure it Out (page 128)
1. Lambaai ke hisaab se line (lambaai 1, sabse chhota, se 7, sabse lamba, tak; line ka aage waala pehle):
- (a) 0, 1, 1, 2, 4, 1, 5: 7, 3, 5, 4, 1, 6, 2.
- (b) 0, 0, 0, 0, 0, 0, 0: 1, 2, 3, 4, 5, 6, 7, sabse chhota aage, har bachcha apne aage waalon se lamba.
- (c) 0, 1, 2, 3, 4, 5, 6: 7, 6, 5, 4, 3, 2, 1, sabse lamba aage.
- (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, pehle sabse lamba, phir chhote se badhte hue.
- (f) 0, 0, 0, 3, 3, 3, 3: 5, 6, 7, 1, 2, 3, 4.
Saat alag lambaaiyon ke saath har sequence ke liye bas yahi ek line chalti hai.
2. Hamesha, kabhi-kabhi ya kabhi nahi?
- (a) Sirf kabhi-kabhi sach. Pehle bachche ka number hamesha 0 hota hai chaahe sabse zyaada lambaai uski na ho, aur jo bhi apne aage waale sabse lambe hon unka number 0 hota hai.
- (b) Hamesha sach. Sabse lambe bachche se lamba koi nahi, isliye uski ginti 0 hai.
- (c) Hamesha sach. Pehle bachche ke aage koi nahi hai.
- (d) Sirf kabhi-kabhi sach. Beech ke kisi bachche ka number 0 hota hai jab uski lambaai aage ke sab bachchon se zyaada ho, jaise line (b) mein.
- (e) Sirf kabhi-kabhi sach. Line 1, 3, 2 (lambaai) mein lambaai 2 waale bachche ka number, 1, sabse bada hai, par aage khade sabse chhote bachche ka number 0 hai.
- (f) 7. Aakhir mein khade sabse chhote ke aage baaki saaton hain, sab usse lambe.
Picking Parity (pages 129–131)
Kishor ke cards. Sambhav nahi. Har card (13, 9, 1, 7, 11, 5, 3) odd hai. Paanch odd numbers mein do jode bante hain, har jode ka jod even, aur ek odd bachta hai, isliye total odd hai; 30 even hai. Koi case jaanchne ki zaroorat nahi.
Even numbers jodna: jod hamesha even hai, chaahe kitne bhi even numbers jodo, kyunki saari jodiyan jodiyan hi rehti hain.
Odd numbers jodna: ye unki ginti par nirbhar hai. Har odd number ek bacha dot laata hai. Odd numbers ki ginti even ho to bache dots jodiyan bana lete hain aur jod even hai; ginti odd ho to ek dot akela bachta hai aur jod odd hai.
Kya odd number jodiyon ke samooh se ek kam bhi maana ja sakta hai? Haan. Jaise, 9 = 5 jodiyan (10 dots) jinmein ek dot kam hai.
4, 5 aur 6 odd numbers ka jod: (a) even, (b) odd, (c) even.
Martin aur Maria. Sambhav nahi. Unki umar lagaataar numbers hain, aur do lagaataar numbers mein ek odd aur doosra even hota hai, isliye unka jod odd hai (51 + 52 = 103, 55 + 56 = 111, 56 + 57 = 113). 112 even hai.
Figure it Out (page 131)
1. Jodon ki parity: (a) 2 even + 2 odd: even; (b) 2 odd + 3 even: even; (c) 5 even: even; (d) 8 odd: even. Har case mein odd numbers ki ginti even hai.
2. Lakpa ke sikke. Haan, usne galti ki. ₹1 ke sikkon ki odd ginti odd rakam banaati hai; ₹5 ke sikkon ki odd ginti bhi odd rakam banaati hai, kyunki odd × odd = odd; aur ₹10 ke kitne bhi sikke even rakam banaate hain. Odd + odd + even even hai, isliye total kabhi ₹205, yaani odd rakam, nahi ho sakta.
3. Ghataav ki parity: (d) even − even = even; (e) odd − odd = even; (f) even − odd = odd; (g) odd − even = odd.
Small Squares in Grids (pages 131–132)
Bina guna kiye parity batana. Chhote squares ki ginti rows × columns hai. Agar rows ya columns mein se koi bhi even ho, to grid squares ki jodiyon mein bant jaata hai aur total even hai. Sirf jab dono odd hon tab total odd hai.
- (a) 27 × 13: dono odd, isliye odd (351).
- (b) 42 × 78: even.
- (c) 135 × 654: 654 even hai, isliye even.
Parity of Expressions (pages 132–133)
Hamesha even expressions: 2n, 6k + 4, 10m − 2, 100p, 48w − 2; har value mein 2 ek factor hai.
Hamesha odd expressions: 2n + 1, 4n + 3, 10m + 5, 6k − 1; har ek, even number mein odd number jodkar ya ghataakar bana hai.
Kuch bhi ho sakne waale: n + 5, 5n + 2, 7n + 1, bilkul 3n + 4 ki tarah; inki parity n ke saath badalti hai (n + 5 mein n = 1 par 6, n = 2 par 7).
Saare even numbers: expression 2n, n = 1, 2, 3, … ke liye 2, 4, 6, 8, … deta hai, aur yahi 2 ke multiples ka nth term bhi hai. Saare odd numbers: 2n − 1 se 1, 3, 5, 7, … milte hain.
100th even number 2 × 100 = 200 hai, isliye 100vi jagah ka odd number usse ek kam, 199, hai, aur nth odd number 2n − 1 hai.
Some Explorations in Grids (pages 133–134)
Gol ghere numbers kya dikhaate hain? Rows ke jod (daayein) aur columns ke jod (neeche).
Grids bharna.
- Upar baayein 9 aur neeche daayein 5 waala grid (rows 13, 14, 18; columns 24, 9, 12): 9 1 3, 8 2 4, 7 6 5. Ye ek hi jawaab hai.
- Beech baayein 4 aur neeche daayein 3 waala grid (rows 24, 15, 6; columns 12, 16, 17): 7 8 9, 4 6 5, 1 2 3, ya 7 9 8, 4 5 6, 1 2 3.
Apna sawaal banao. Jaise, chupke se 1 8 3, 9 4 6, 2 7 5 bharo aur dost ko sirf rows ke jod 12, 19, 14, columns ke jod 12, 19, 14 aur do numbers suraag ke roop mein do.
Teesra grid namumkin kyun hai? 1 se 9 ke teen alag numbers ka jod kam se kam 1 + 2 + 3 = 6 aur zyaada se zyaada 9 + 8 + 7 = 24 hota hai, par is grid ko row ka jod 5 aur column ka jod 26 chahiye.
Kishor ka 90 aur Vidya ka 45. Upar ke har grid mein rows ke jod ka total 45, columns ka 45, aur chhahon golon ka 90 hai. Rows ke jod milkar 1 se 9 tak ka har number ek baar ginte hain, isliye hamesha 1 + 2 + … + 9 = 45 hote hain; columns ke jod ke saath bhi yahi.
Magic Squares (pages 134–136)
1. Magic sum kya ho sakta hai? Sirf 15: teeno barabar rows ke jod ka total 45 hona chahiye, isliye har ek 45 ÷ 3.
2. Kaunse numbers beech mein nahi aa sakte? Beech ka khaana baaki aathon khaanon ke saath kisi line mein hai. Beech mein 8 ho to 9 uske saath ek line mein aayega aur 8 + 9 = 17 > 15. 7 ho to 9 waali line ko 7 + 9 + (−1) chahiye. 6 ho to 9 waali line ko 0 chahiye. 2 ho to 1 waali line ko 12 chahiye; 3 ho to 11; 4 ho to 10. Isliye 1, 2, 3, 4, 6, 7, 8 aur 9 beech mein nahi aa sakte; sirf 5 aa sakta hai.
Kya 1 kone mein aa sakta hai? Nahi. Kona teen lines (ek row, ek column aur ek diagonal) par hota hai, isliye 1 ko 14 banaane waale teen alag jode chahiye, par sirf 5 + 9 aur 6 + 8 hain. Kya 9 kone mein aa sakta hai? Nahi: use 6 banaane waale teen jode chahiye, aur sirf 1 + 5 aur 2 + 4 hain.
1 aur 9 ki doosri jagahen: ve do aamne-saamne ki edges ke beech mein, 5 ke saath ek line mein baithte hain: 1 baayein aur 9 daayein, ya ulta, ya 1 upar aur 9 neeche, ya ulta.
Poora karna. Beech ki row mein 1 5 9 ho to 1 waale column ko 14 banaane waale do numbers (6 aur 8), 9 waale column ko 6 banaane waale do (2 aur 4), aur 5 se guzarte har diagonal ko 10 banaane waala joda chahiye. Ek nateeja: 8 3 4, 1 5 9, 6 7 2.
Figure it Out (page 136)
1. 1 se 9 se 8 magic squares ban sakte hain, par ve sab ek hi square (jaise 8 1 6, 3 5 7, 4 9 2) ko chauthaai ghumaakar ya palatkar bante hain, isliye asal mein ek hi hai.
2. 2 se 10 ke saath. 1–9 magic square ke har number mein 1 jodo: 9 2 7, 4 6 8, 5 10 3. Jagahon ka pattern wahi hai, beech mein 5 ki jagah 6 hai, aur magic sum 15 ki jagah 18.
3. (a) Har number mein 1 jodne se phir magic square milta hai, magic sum 15 + 3 = 18, kyunki har line mein 1 teen baar judta hai. (b) Har number doguna karne se bhi magic square milta hai, magic sum 2 × 15 = 30.
4. Doosre operations jo naya magic square dete hain: har khaane se ek hi number ghataana; har khaane ko ek hi non-zero number se guna ya bhaag dena; square ko ghumaana ya palatna; aur do magic squares ko khaane-dar-khaane jodna.
5. Koi bhi nau lagaataar numbers. 1–9 square se shuru karo aur har khaane mein (pehla number − 1) jodo. 3 se 11 ke liye 2 jodo: 10 3 8, 5 7 9, 6 11 4 (jod 21); 9 se 17 ke liye 8 jodo: 16 9 14, 11 13 15, 12 17 10 (jod 39). Nau numbers mein beech waala number beech mein jaata hai, aur magic sum uska teen guna hota hai.
Generalising a 3 × 3 Magic Square (pages 136–137)
Har khaane ko beech ke m se jodna: 8 1 6, 3 5 7, 4 9 2 mein beech 5 = m hai, isliye square m + 3, m − 4, m + 1; m − 2, m, m + 2; m − 1, m + 4, m − 3 hai. Lagaataar numbers se bane har square par yahi roop laagu hota hai.
Figure it Out (page 137)
1. Beech mein 25: 28 21 26, 23 25 27, 24 29 22 (har line ka jod 75).
2. Har row, column aur diagonal ka jod 3m hai; jaise (m + 3) + (m − 4) + (m + 1) = 3m.
3. (a) Har term mein 1 jodne par: m + 4, m − 3, m + 2; m − 1, m + 1, m + 3; m, m + 5, m − 2, har line ka jod 3m + 3. (b) Har term doguna karne par: 2m + 6, 2m − 8, 2m + 2; 2m − 4, 2m, 2m + 4; 2m − 2, 2m + 8, 2m − 6, har line ka jod 6m.
4. Magic sum 60: 3m = 60, isliye m = 20: 23 16 21, 18 20 22, 19 24 17.
5. Haan. 1–9 square ke har number ko 3 se guna karo: 24 3 18, 9 15 21, 12 27 6 (jod 45). Ya 1 se 17 tak ke odd numbers lo: 15 1 11, 5 9 13, 7 17 3 (jod 27).
The First-ever 4 × 4 Magic Square (page 137)
“Chautisa” kyun? Chautis ka matlab 34 hai, aur yantra ki har row, column aur diagonal ka jod 34 hai. 34 banaane waale chaar-chaar ke doosre pattern: chaaron kone (7 + 14 + 9 + 4); beech ka square (13 + 8 + 3 + 10); har kone ka 2 × 2 block, jaise 7 + 12 + 2 + 13; upar aur neeche ki rows ke beech ke do khaane (12 + 1 + 6 + 15); side ke columns ke beech ke do khaane (2 + 16 + 11 + 5); aur toote diagonals, jaise 12 + 8 + 5 + 9 aur 1 + 11 + 16 + 6. Sach mein har 2 × 2 block, edges ke paar lipatne waala bhi, 34 jodta hai.
Magic Squares in History and Culture (pages 138–139)
Har graha ka alag magic sum. Navagraha Yantra ki kisi bhi row ko jodne par: Surya 15, Chandra 18, Mangal 21, Budh 24, Brihaspati 27, Shukra 30, Shani 33, Rahu 36 aur Ketu 39. Har ek apne beech ke number ka teen guna hai. Kubera Yantra (27 20 25, 22 24 26, 23 28 21) ka magic sum 72 = 3 × 24 hai.
The Virahanka–Fibonacci Numbers (pages 139–142)
8 beats ki aur rhythms (S = chhota, L = lamba): S S L L L (1 + 1 + 2 + 2 + 2), L S S L L (2 + 1 + 1 + 2 + 2), S L S L L (1 + 2 + 1 + 2 + 2), L L L S S (2 + 2 + 2 + 1 + 1), S S S S L L (1 + 1 + 1 + 1 + 2 + 2) aur L S S S S L (2 + 1 + 1 + 1 + 1 + 2).
5 ko 1 aur 2 ke jod ke roop mein (8 tareeke): 1 + 1 + 1 + 1 + 1; 1 + 1 + 1 + 2; 1 + 1 + 2 + 1; 1 + 2 + 1 + 1; 1 + 2 + 2; 2 + 1 + 1 + 1; 2 + 1 + 2; 2 + 2 + 1. n = 8 ke liye 34 tareeke hain, bina list banaaye, neeche ke niyam se.
Saari 13 chhe-beat rhythms (har 5-beat rhythm ke aage “1 +”, phir har 4-beat rhythm ke aage “2 +”): 1 + 1 + 1 + 1 + 1 + 1; 1 + 1 + 1 + 1 + 2; 1 + 1 + 1 + 2 + 1; 1 + 1 + 2 + 1 + 1; 1 + 1 + 2 + 2; 1 + 2 + 1 + 1 + 1; 1 + 2 + 1 + 2; 1 + 2 + 2 + 1; 2 + 1 + 1 + 1 + 1; 2 + 1 + 1 + 2; 2 + 1 + 2 + 1; 2 + 2 + 1 + 1; 2 + 2 + 2.
55 ke baad: 34 + 55 = 89. Agle teen: 144, 233, 377. Ek aur: ye even hoga, kyunki 233 aur 377 dono odd hain aur odd + odd = even (ye 610 hai).
Parities: 1 odd, 2 even, 3 odd, 5 odd, 8 even, 13 odd, 21 odd, 34 even, … Block odd, even, odd dohraata hai, kyunki odd + even = odd, even + odd = odd aur odd + odd = even. Doosre se shuru karke har teesra number even hai.
Daisy phoolon ki pankhudiyan: 13, 21 aur 34, sab Virahanka numbers.
Digits in Disguise (pages 142–143)
T + T + T = UT. Teen barabar digits ka jod usi digit par khatam hota hai. T = 2 se 6 aur T = 3 se 9 milta hai, jo do-digit number nahi; sirf T = 5 chalta hai: 5 + 5 + 5 = 15, isliye U = 1.
K2 + K2 = HMM. Ikaai ka digit 2 + 2 = 4 hai, isliye M = 4. Phir K + K ko 4 par khatam hokar 1 se shuru hone waala number banaana hai: K + K = 14, isliye K = 7 aur H = 1: 72 + 72 = 144.
Chaar aur:
- YY + Z = ZOO: do-digit number aur ek digit ka jod zyaada se zyaada 99 + 9 = 108 hai, isliye Z = 1 aur YY = 99: 99 + 1 = 100 (Y = 9, Z = 1, O = 0).
- B5 + 3D = ED5: 5 + D ka ant 5 hai, isliye D = 0; B + 3 do-digit number E0 banaata hai, yaani 10, isliye B = 7: 75 + 30 = 105 (B = 7, D = 0, E = 1).
- KP + KP = PRR: do-digit number ka doguna 200 se kam hai, isliye P = 1; phir R = 1 + 1 = 2, aur K + K = 12, isliye K = 6: 61 + 61 = 122.
- C1 + C = 1FF: jod kam se kam 100 hai, isliye C = 9: 91 + 9 = 100 (F = 0).
Figure it Out (pages 143–144)
1. Dorjee ka bulb. Off. Har baar dabaane par haalat badalti hai, isliye even baar ke baad bulb phir on hota hai. 76 baar ke baad vo on hai, aur 77vi baar, kul milaakar odd ginti, use off kar deti hai.
2. Liswini ke khule panne. Nahi, jod 6000 nahi ho sakta. Har panne ki ek taraf odd page number aur doosri taraf agla even number hota hai, jaise 7 aur 8, isliye har panne ka jod odd hai, aur 50 panne even total dete hain. Isliye sirf parity 6000 ko rad nahi karti. Thoda aur dhyaan se dekhne par faisla ho jaata hai: ek panne ke pages 2k − 1 aur 2k hain, jo milkar 4k − 1 bante hain, yaani 4 ke multiple se ek kam. Pachaas panne milkar 4 × (saare k ka jod) − 50 bante hain. Ise 6000 hone ke liye 4 × (saare k ka jod) = 6050 chahiye, par 6050 ÷ 4 = 1512.5 poora number nahi hai.
3. 2 × 3 grid. Ek tareeka: upar ki row 1, 2, 4 (jod 7, odd) aur neeche ki row 3, 6, 5 (jod 14, even); columns ke jod 4, 8 aur 9 hain, yaani even, even aur odd. Doosra tareeka: 2, 1, 4 ke neeche 6, 3, 5.
4. Magic sum 0. 1–9 magic square ke har number se 5 ghataao: 3, −4, 1; −2, 0, 2; −1, 4, −3. Har row, column aur diagonal ka jod 0 hai (ye m = 0 waala general roop hai).
5. (a) even; (b) even; (c) even; (d) odd.
6. 1 se 100. Even. 1 se 100 tak 50 odd numbers hain (1, 3, …, 99), yaani even ginti, isliye unka jod even hai, aur even numbers parity nahi badalte. (Sach mein 1 + 2 + … + 100 = 5050.)
7. 987 aur 1597 ke baad: 987 + 1597 = 2584, phir 1597 + 2584 = 4181. Unse pehle: 1597 − 987 = 610, phir 987 − 610 = 377.
8. Angaan ki seedhi. 34 tareeke. Kisi bhi seedhi par uska aakhri kadam 1 ya 2 seedhi ka hota hai, isliye kisi seedhi tak pahunchne ke tareeke = ek neeche waali ke tareeke + do neeche waali ke tareeke: 1, 2, 3, 5, 8, 13, 21, 34.
9. 20th term even hai. Parities odd, even, odd dohraati hain, isliye 2nd, 5th, 8th, 11th, 14th, 17th aur 20th terms even hain (20th term 10946 hai).
10. Kaunse statements sach hain? (m, j, p, q aur f ko 1, 2, 3, … maano.)
- (a) Sach. 4m even hai, isliye 4m − 1 hamesha odd hai (3, 7, 11, …), haalaanki ye har odd number nahi deta.
- (b) Galat. 6j − 4 se 2, 8, 14, … milte hain aur 4 aur 6 jaise even numbers chhoot jaate hain.
- (c) Galat. 2q − 1 har odd number deta hai, par 2p + 1 se 3, 5, 7, … milte hain aur 1 chhoot jaata hai.
- (d) Galat. 2f even hai, isliye 2f + 3 hamesha odd hai.
11. UT + TA = TAT. Do do-digit numbers ka jod 200 se kam hai, isliye T = 1. Ikaai mein T + A ka ant T hai, yaani A + 1 ka ant 1, isliye A = 0. Phir U1 + 10 = 101, isliye U1 = 91 aur U = 9: 91 + 10 = 101.
Shaadi ki ginti (page 145)
Kya ye even number nahi hona chahiye? Haan. Har shaadi do logon ko jodti hai, isliye saal bhar mein shaadi karne waale log jodiyon mein aate hain aur unka total even hona chahiye. 14,70,369 jaisa odd aankda batata hai ki ginti mein kahin galti hai.
Chhote notes aur practice topic pages pe hain: parity: odd aur even, Virahanka numbers kya hain aur magic square kya hai.
Practice karo
-
1 + 2 + 3 + … + 10 odd hai ya even?
Jawab dekho
Jawab: (a) Odd
-
3 × 3 magic square mein 4 se 12 tak ke nau lagaataar numbers hain. Beech mein kya hai?
Jawab dekho
Jawab: (b) 8
-
Do lagaataar Virahanka numbers 377 aur 610 hain. Agla kya hai?
Jawab dekho
Jawab: (b) 987
-
Cryptarithm A + A = 1B mein A, 5 se 9 tak ka koi digit ho sakta hai. Agar B = 4 ho, to A kya hai?
Jawab dekho
Jawab: A = 7, kyunki 7 + 7 = 14.
-
Bina guna kiye batao, 25 × 49 grid ke chhote squares ki ginti odd hai ya even?
Jawab dekho
Jawab: Odd, kyunki 25 aur 49 dono odd hain, aur guna tabhi odd hota hai jab har factor odd ho.
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