# Discover Total Number Of Spheres

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

1. ### ThyagarajanIL Hall of Fame

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2. ### ViswamitraIL Hall of Fame

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3. ### vidhyalakshmidPlatinum IL'ite

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4. ### ThyagarajanIL 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. ### jayasala42IL 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. ### Gauri03ModeratorStaff MemberIL 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. ### ViswamitraIL Hall of Fame

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I misspoke and I got 85 in my revised calculation.

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8. ### ThyagarajanIL Hall of Fame

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9. ### sarvantaryaminiGold 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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