You have a cubic box full of tennis balls. they are packed so that the bottom layer is 10 x 10; the next layer where the balls are placed in the indents between the balls in the lower layer is 9 x 9. next layer is 10 x 10, etc. if you take all the balls out and then build a pyramid on a 10 x 10 base inside the box, how many balls will you have left over?

 Number of balls left = 520 balls.

Explanation:

First case

 Since the box is cubic, number of layers balls = 10

 Total number of 10 x 10 layer = 5

 Total number of 9 x 9 layer = 5

  Total number of balls = 5 x 10 x 10 + 5 x 9 x 9 = 905

Second case.

  Number of balls in bottom layer = 10 x 10 = 100

  Number of balls in second layer = 9 x 9 = 81

  So, total number of balls = 100 + 81 + 64 + 49 + ...........+ 1

                                            = 1² + 2² + 3² +.... 10²  

 We have sum of squares of first n natural numbers =

  Here n = 10

  Number of balls

 Number of balls left = 905 - 385 = 520 balls.

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