Discover Total Number Of Spheres

Discussion in 'Online Games & Puzzles' started by Thyagarajan, Apr 18, 2022.

  1. Thyagarajan

    Thyagarajan IL Hall of Fame

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    Discover
    Total number of
    Spheres
    In
    This
    A R R A N G E M E N T




    upload_2022-4-18_14-6-14.jpeg
     
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  2. Viswamitra

    Viswamitra IL Hall of Fame

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  3. vidhyalakshmid

    vidhyalakshmid Platinum IL'ite

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  4. Thyagarajan

    Thyagarajan IL Hall of Fame

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    In discovering the answer to the problem , if Sum of square of n natural numbers formula crossed your mind, you pat yourself on your back.
    S = product of n x (n+1) x (2n+1) divided by 6
    But what is value of n ?
    Let me wait for few more responses..
     
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  5. jayasala42

    jayasala42 IL Hall of Fame

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    The basal packing contains 81 pieces of 24+20+16+12+8+1=81 spheres.
    Then it gets narrowed to 57,37,21,9,and 1=125
    So over all spheres may be 206 .
    No formula used.
    Vathsala Jayaraman
     
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  6. Gauri03

    Gauri03 Moderator Staff Member IL Hall of Fame

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    The arrangement of spheres is in a formation called a square pyramid. Each layer of the pyramid is a square made up of spheres. There are 7 layers so n = 7
    The top layer has 1 sphere,
    Layer 2 has 2^2 (2 squared) spheres = 4
    Layer 3 has 3^2 = 9
    Layer 4 has 4^2 = 16
    Layer 5 has 5^2 = 25
    Layer 6 has 6^2 = 36
    Layer 7 has 7^2 = 49

    1+4+9+16+25+36+49 = 140

    As you can see the answer is the sum of the squares of the first n natural numbers where n is the number of layers in the pyramid. These kind of numbers are called square pyramidal numbers and can be computed directly using the formula you've given in your reply!
     
  7. Viswamitra

    Viswamitra IL Hall of Fame

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    I misspoke and I got 85 in my revised calculation.
     
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  8. Thyagarajan

    Thyagarajan IL Hall of Fame

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    :hello:Let me wait for some more answers/ participants.
     
  9. sarvantaryamini

    sarvantaryamini Gold IL'ite

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    140 same as gauri and vijayalakshmi answers. This is formula-
    Sum of Squares Formula
     
    Last edited: Apr 20, 2022
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  10. Thyagarajan

    Thyagarajan IL Hall of Fame

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