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Number pyramid aur grid wali paheliyaan

Short mein jawab

Number pyramid mein har dibba apne theek neeche ke do dibbon ka jod hota hai. Khaali dibbe jodkar ya ghataakar milte hain, ya unhe letter-number dekar aur hal karke. Teen pankti ke pyramid ka top a + 2b + c hai. Calendar aur algebra grid mein letter-numbers jod ko equation bana dete hain, jo chhupe number bata deti hai.

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Number pyramid ka niyam

Number pyramid dibbon ka ek dher hai. Neeche ki pankti mein koi bhi number ho sakte hain, aur baaki har dibbe mein uske theek neeche ke do dibbon ka jod hota hai. Neeche 2, 5 aur 3 hon, to upar ki pankti 7 aur 8 hai, aur top 15. Upar jaate hue jodte hain; neeche aate hue ghataate hain. Agar top 20 hai aur uske neeche baayein 8, to 8 ke bagal wala dibba 20 − 8 = 12 hoga. Agar 8 ke neeche 3 hai, to uska padosi 8 − 3 = 5, aur aakhri dibba 12 − 5 = 7.

Number pyramid: jodkar upar jaana aur ghataakar neeche aana
Number pyramid: upar banaaiye ya neeche aaiye

Jab ghataana shuru hi na ho sake

Kabhi koi bhi dibba apne padosiyon se nahi milta. Maan lijiye teen pankti ke pyramid mein top 50 hai aur neeche ki pankti ke kinaaron par 10 aur 6. Beech ki pankti ke khaali dibbon ko a aur b, aur neeche ke khaali dibbe ko c kahiye. Niyam se teen equations milti hain: a = 10 + c, b = c + 6 aur a + b = 50. Pehli do ko teesri mein rakhne par (10 + c) + (c + 6) = 50, yaani 16 + 2c = 50, 2c = 34 aur c = 17. Phir a = 27 aur b = 23, aur sach mein 27 + 23 = 50.

Pyramid mein letter-numbers: a, b aur c ki equations aur bhara hua pyramid
Jab koi dibba seedha na mile

Neeche ki pankti se top ka number

Algebra poora pyramid banaaye bina top bhi bata deta hai. Neeche a, b hon to top a + b hai. a, b, c hon to beech ki pankti a + b aur b + c hai, isliye top a + 2b + c: beech ka number do baar ginta hai kyonki wo upar ke dono dibbon ke neeche hai. Neeche a, b, c, d wale chaar pankti ke pyramid ka top a + 3b + 3c + d hai. Gunak 1, 1, phir 1, 2, 1, phir 1, 3, 3, 1 padosiyon ko jodkar badhte hain, us tribhuj ki panktiyon ki tarah jise Bharat mein Meru Prastara kehte hain. Jaise neeche 2, 6, 5 ho to top 2 + 12 + 5 = 19 hai.

Chaar pankti ke pyramid ka top letter-numbers mein, aur 2, 3 aur 4 pankti ke gunak
Neeche ki pankti se top ka number

Neeche ki pankti mein pehle kuch Virahanka–Fibonacci numbers rakhiye to ek mazedaar baat hoti hai: har dibbe mein isi kram ka koi number aata hai, kyonki kram ke do padosi jodne par aage ka number banta hai, aur har nayi pankti top ko kram mein do jagah aage le jaati hai.

1, 2, 3 aur 5 par bana pyramid, jiska har dibba Virahanka–Fibonacci number hai
Pyramid mein Virahanka–Fibonacci numbers

Calendar ka jaadu

Mahine ke calendar mein kisi bhi date ke daayein wali date 1 zyada aur neeche wali 7 zyada hoti hai. Isliye agar 2 × 2 square ki upar-baayein date a hai, to square mein a, a + 1, a + 7 aur a + 8 hain, aur unka jod 4a + 16 hai. Dost jod 56 batayein to 4a = 40 aur a = 10, yaani square 10, 11, 17 aur 18 hai.

Calendar par 2 × 2 square: a, a + 1, a + 7 aur a + 8
Calendar ka jaadu: 2 × 2 square

Isi soch se naye trick bante hain. 3 × 3 square ka jod beech ki date ka 9 guna hota hai, kyonki uski nau dates beech ke aas-paas jode banaati hain. Ek column ki teen dates a, a + 7 aur a + 14 hain, jinka jod 3a + 21 hai.

Calendar ke do aur trick: 3 × 3 square aur ek column ki teen dates
Apna calendar trick banaaiye

Algebra grid

Algebra grid mein har aakaar ek number ki jagah hai, aur har pankti ka aakhri dibba us pankti ka jod batata hai. Teen square milkar 24 hon to ek square 8 hai. Do circle aur ek square milkar 20 hon to do circle milkar 12, yaani har circle 6. Jab har pankti mein aakaar mile-jule hon, to do panktiyaan jodne par har aakaar barabar sankhya mein aa sakta hai, jisse pata chalta hai ki har tarah ka ek-ek milkar kitna hai.

Squares aur circles wala algebra grid, pankti dar pankti hal kiya hua
Algebra grid: aakaar numbers ki jagah

Aur padhein: number socho waale trick, divisibility ke trick aur sabse bada guna, chapter ke notes aur NCERT solutions.

Practice karo

  1. Ek pyramid ki neeche ki pankti 3, 7, 2 hai. Top number hai:

    1. (a) 12
    2. (b) 19
    3. (c) 21
    4. (d) 10
    Jawab dekho

    Jawab: (b) 19

  2. Dates ka ek 2 × 2 square jodne par 76 deta hai. Uski upar-baayein wali date hai:

    1. (a) 13
    2. (b) 15
    3. (c) 17
    4. (d) 19
    Jawab dekho

    Jawab: (b) 15

  3. Teen square milkar 33 aur do circle aur ek square milkar 21 hain. Ek circle kis number ki jagah hai?

    1. (a) 4
    2. (b) 5
    3. (c) 6
    4. (d) 11
    Jawab dekho

    Jawab: (b) 5

  4. Neeche ki pankti 1, 1, 1, 1 wale chaar pankti ke pyramid ka top kya hai?

    Jawab dekho

    Jawab: 1 + 3 + 3 + 1 = 8. Banaakar bhi yahi milta hai: 2, 2, 2, phir 4, 4, phir 8.

  5. Dates ka ek 3 × 3 square jodne par 117 deta hai. Beech ki date kya hai?

    Jawab dekho

    Jawab: Nau dates ka jod beech ki date ka 9 guna hota hai, isliye beech ki date 117 ÷ 9 = 13 hai.

Students ke sawal

Pyramid ko upar se kaise bharein?

Aisa khaali dibba dhoondhiye jiska usi pankti ka saathi aur un dono ke upar wala dibba pata ho. Ghataakar use nikaaliye, phir ek-ek ghataav karke neeche badhte jaaiye.

Agar ghataakar koi dibba na mile to?

Khaali dibbon ko letter-numbers dijiye, har dibbe ki equation likhiye aur unhe milaaiye. Teen pankti ke pyramid mein top 50 aur neeche ke kinaaron par 10 aur 6 hon, to beech ka dibba c se 16 + 2c = 50 banta hai, yaani c = 17.

Dates ka 2 × 2 square 4a + 16 kyon deta hai?

Calendar mein daayein wali date 1 zyada aur neeche wali 7 zyada hoti hai, isliye chaar dates a, a + 1, a + 7 aur a + 8 hain. Inka jod 4a + 16 hai.

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