Question

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?

Answer 1

Number of balls left = 520 balls.

Step-by-step 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 =
`=(n(n+1)(2n+1))/(6)`

Here n = 10

Number of balls
`=(10*(10+1)*(2*10+1))/(6)=385`

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