### Brain Teasers

# Ball Pyramid

Mad Ade has built a pyramid out of old tennis balls. One side of the bottom layer of a triangular pyramid has 12 balls. How many are there in the whole pyramid?

Note that the pyramid is equilateral and solid.

Note that the pyramid is equilateral and solid.

### Answer

There are total 364 balls.As there are 12 balls along one side, it means that there are 12 layers of balls. The top most layer has 1 ball. The second layer has 3 (1+2) balls. The third layer has 6 (1+2+3) balls. The fourth layer has 10 (1+2+3+4) balls. The fifth layer has 15 (1+2+3+4+5) balls. Similarly, there are 21, 28, 36, 45, 55, 66 and 78 balls in the remaining layers.

Hence, the total number of balls are

= 1 + 3 + 6 + 10 + 15 + 21 +28 + 36 + 45 + 55 + 66 + 78

= 364 balls

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## Comments

Nice! Building up a 100-level-pyramid with 171700 balls would be a rather big job.

Some addition can be avoided as any level L has L*(L+1)/2 tennis balls.

Which looks like:

(12*1 + 11*2 + 10*3 + 9*4 + 8*5 + 7*6) * 2

(12*1 + 11*2 + 10*3 + 9*4 + 8*5 + 7*6) * 2

It also can be written (looking at the pattern of two layers at a time) as

2^2 + 4^2 + 6^2 + 8^2 + 10^2 + 12^2 = 364.

Interesting puzzle because of the many number patterns,

2^2 + 4^2 + 6^2 + 8^2 + 10^2 + 12^2 = 364.

Interesting puzzle because of the many number patterns,

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