Lagnever AI English

Magic square kya hai?

Short mein jawab

Magic square aisa grid hai jismein har row, column aur diagonal ka jod ek hi hota hai, jise magic sum kehte hain. 3 × 3 grid mein 1 se 9 tak ke numbers hon to magic sum 15 aur beech ka number 5 hona chahiye; aise 8 squares bante hain, sab ek hi square ko ghumaakar ya palatkar.

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Grid mein jod

3 × 3 grid ko 1 se 9 tak ke numbers se bharo, har ek ek baar, aur har row aur har column ka total bahar golon mein likho. Grid 4 7 5, 6 1 2, 3 9 8 mein rows ka jod 16, 9 aur 20 hai aur columns ka 13, 17 aur 15. Numbers kaise bhi rakho, teeno rows ke total milkar 45 hote hain, aur teeno columns ke bhi, kyunki unmein 1 se 9 tak ka har number theek ek baar aata hai, aur 1 + 2 + … + 9 = 45. Isliye chhahon gole milkar 90 banaate hain.

1 se 9 ka grid jiski rows ka jod 16, 9, 20 aur columns ka jod 13, 17, 15 hai, har set ka total 45
1–9 grid ke rows aur columns ke jod

Jab grid namumkin ho

1 se 9 ke teen alag numbers ka sabse chhota jod 1 + 2 + 3 = 6 hai, aur sabse bada 9 + 8 + 7 = 24. Jo paheli kisi row ya column ka jod 5 ya 26 maange, vo kabhi hal nahi ho sakti, bina ek bhi tareeka aazmaaye.

Do hal ki hui grid paheliyan aur teesra grid jo namumkin hai kyunki vo 5 aur 26 ke jod maangta hai
Grids bharna

Magic square kya hai?

Numbers ka square grid magic square hai agar har row, har column aur dono diagonals ka jod ek hi number ho, jise magic sum kehte hain. Andaaza lagaana bekaar hai: 3 × 3 grid ko 1 se 9 se bharne ke 3,62,880 tareeke hain. Iske bajaye hum ek-ek sawaal par tark karte hain.

1–9 magic square banaana

Magic sum. Teeno rows ke jod ka total 45 hai aur teeno barabar hain, isliye har ek 45 ÷ 3 = 15 hona chahiye.

Beech ka khaana. Beech ka khaana baaki har khaane ke saath kisi line mein hai. Agar 9 beech mein ho, to 8 uske saath ek line mein aayega, aur 9 + 8 pehle hi 17 hai, bahut zyaada. Agar 1 beech mein ho, to 2 uske saath ek line mein aayega, aur 1 + 2 + 12 chahiye, par 12 hai hi nahi. Isi tarah ki jaanch 5 ke alaawa har number ko rad kar deti hai.

Kone. Kona teen lines par hota hai, isliye kone mein 1 ho to use 14 banaane waale teen alag jode chahiye, par sirf 5 + 9 aur 6 + 8 hain. Isi tarah 9 bhi kone mein nahi aa sakta. Isliye 1 aur 9 aamne-saamne ki edges ke beech mein, 5 ke saath ek line mein jaate hain, aur baaki apne aap bhar jaata hai: jaise 8 3 4, 1 5 9, 6 7 2.

1–9 magic square banaana: magic sum 15, beech mein 5, aur 1 aur 9 aamne-saamne ki edges ke beech mein, jisse 8 3 4, 1 5 9, 6 7 2 banta hai
1–9 magic square banaana

1 se 9 ke 8 magic squares hain, sab ek hi square ko ghumaakar ya palatkar.

Magic square badalna

Har khaane mein ek hi number jodne se har line barabar rehti hai aur magic sum us number ke teen guna se badh jaata hai, isliye 1–9 square ke har khaane mein 1 jodne se 2–10 ka square milta hai jiska jod 18 hai. Har khaane ko doguna karne se jod doguna, 30, ho jaata hai, aur har khaane se 5 ghataane par aisa square milta hai jiska magic sum 0 hai aur jismein negative numbers hain.

Jod 15 waala magic square, phir 1 jodkar (jod 18), doguna karke (jod 30) aur 5 ghataakar (jod 0)
Magic square badalna

General roop

Lagaataar numbers se bane kisi bhi 3 × 3 magic square ke beech ke number ko letter-number m likho. Tab khaane m + 3, m − 4, m + 1; m − 2, m, m + 2; m − 1, m + 4, m − 3 bante hain, aur har row, column aur diagonal ka jod 3m hota hai. m = 25 par magic sum 75 hai, aur magic sum 60 ke liye m = 20 lete hain.

Beech mein m waala generalised 3 × 3 magic square, aur beech mein 25 aur 20 waale udaaharan
Generalised magic square

Itihaas mein magic squares

Sabse purana darj magic square, Lo Shu square, 2000 saal se pehle ke China se aata hai; kahaani hai ki Lo nadi ki baadh ke dauraan ek kachhua ise apni peeth par laaya. Bhartiya ganitagyon ne magic squares banaane ke aam tareeke nikaale, 5 × 5 aur bade squares samet. Sabse pehla darj 4 × 4 magic square, Chautisa Yantra, Khajuraho ke Parshvanath Jain mandir ke 10th century ke shilalekh mein hai; chautis ka matlab 34, aur iski rows, columns, diagonals, kone aur 2 × 2 blocks sab 34 jodte hain. Tamil Nadu ke Palani mein 8th century ke mandir ke khambhe par 3 × 3 square khuda hai, aur Navagraha aur Kubera Yantra jaise yantra aaj bhi gharon aur dukaanon mein lagte hain.

Chautisa Yantra, jismein rows, columns, diagonals, kone, beech ka square, 2 by 2 blocks aur toote diagonals sab 34 jodte hain
Chautisa Yantra

Ye kyun zaroori hai

Magic squares andaaze ke bajaye total se tark karna sikhaate hain. General roop algebra ke letter-numbers ka istemaal karta hai, aur parity: odd aur even ke vichaar grid paheliyon mein bhi madad karte hain; Bhartiya parampara ka ek aur pattern Virahanka numbers kya hain mein hai. Har sawaal ka jawaab Chapter 6 ke NCERT solutions mein hai; poore Chapter 6 ke notes bhi dekho.

Practice karo

  1. 1 se 9 tak ke numbers ka jod kya hai?

    1. (a) 36
    2. (b) 45
    3. (c) 54
    4. (d) 90
    Jawab dekho

    Jawab: (b) 45

  2. General roop se bane 3 × 3 magic square ke beech mein 25 hai. Uska magic sum kya hai?

    1. (a) 25
    2. (b) 50
    3. (c) 75
    4. (d) 100
    Jawab dekho

    Jawab: (c) 75

  3. Chautisa Yantra ki har row, column aur diagonal ka jod kya hai?

    1. (a) 15
    2. (b) 30
    3. (c) 34
    4. (d) 36
    Jawab dekho

    Jawab: (c) 34

  4. 1–9 waale magic square ke beech mein 9 kyun nahi aa sakta?

    Jawab dekho

    Jawab: Baaki har khaana beech se guzarti kisi line par hai, isliye 8 ko 9 ke saath ek line mein aana padega, aur 9 + 8 = 17 pehle hi magic sum 15 se zyaada hai.

  5. Magic sum 60 waala 3 × 3 magic square banao.

    Jawab dekho

    Jawab: General roop mein beech mein 60 ÷ 3 = 20 lo: 23 16 21, 18 20 22, 19 24 17.

Students ke sawal

Magic square kya hai?

Magic square numbers ka aisa square grid hai jismein har row, har column aur dono diagonals ka total ek hi hota hai, jise magic sum kehte hain. Jaise, 8 1 6, 3 5 7, 4 9 2 mein har line ka jod 15 hai.

1 se 9 tak ke numbers se kitne 3 × 3 magic squares ban sakte hain?

Aath. Ye sab ek hi square hain jise chauthaai-chauthaai ghumaaya ya palta gaya ho, jaise 8 1 6, 3 5 7, 4 9 2 aur uske ghumaaye aur aaine waale roop; is matlab mein 1–9 ka bas ek hi buniyaadi magic square hai.

Chautisa Yantra kya hai?

Ye sabse purana jaana-maana 4 × 4 magic square hai, Khajuraho ke Parshvanath Jain mandir ke 10th century ke ek shilalekh se. Chautis ka matlab 34 hai, aur iski har row, column aur diagonal ka jod 34 hai, aur chaaron kone, beech ka square aur har 2 × 2 block bhi 34 jodte hain.

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