A pyramid is made of metal balls. The lowest layer consists of 4*4 balls, the next layers of 3*3 and 2*2 , and one ball at the top. Every two balls that touch each other are connected with a screw. How many screws are used to build this pyramid?
Answers
Given : A pyramid is made of metal balls. The lowest layer consists of 4*4 balls, the next layers of 3*3 and 2*2 , and one ball at the top.
Every two balls that touch each other are connected with a screw.
To Find : How many screws are used to build this pyramid
Solution:
First we count balls touching each other in same layer :
Lowest layer has 4 * 4 balls
where inner 4 balls touches other 4 balls
4 corner balls touches other 2 balls
remaining 8 balls touches other 3 balls
=> 4 * 4 + 4 * 2 + 8 * 3 = 48 ( two balls needed for one screw)
Hence 24 Screws in lowest layer ( 1st )
in 2nd layer 3 * 3 balls
1 balls touches other 4 balls
4 corner balls touches other 2 balls
remaining 8 balls touches other 3 balls
=> 1 * 4 + 4 * 2 + 4 * 3 = 24 ( two balls needed for one screw)
Hence 12 Screws
in 3rd layer 2 * 2 balls
4 corner balls touches other 2 balls
two balls needed for one screw)
Hence 4 Screws
Screws in same layers = 24 + 12 + 4 = 40
Now balls touching other layer balls
Each ball of 2nd layer ( 3 * 3) is touching 4 balls of 1st layer
Hence 9 * 4 = 36 screws
Each ball of 3rd layer ( 2 * 2) is touching 4 balls of 2nd layer
Hence 4 * 4 = 16 screws
1 Ball of 4th layer is touching 4 balls of 3rd layer
Hence 1 * 4 = 4 screws
Screws in different layers = 36 + 16 + 4 = 56
Total Screws : 40 + 56 = 96
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