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,
Seems as though it should have 650.
If one side of the bottom has 12, then it should be 12x12 = 144
Then the 2nd layer 11 x 11 = 121
Third layer 10x10 = 100
Continuing like that until the 2nd from the top is 2x2 = 4 and the top is 1x1 = 1.
Adding those up, 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 = 650
If one side of the bottom has 12, then it should be 12x12 = 144
Then the 2nd layer 11 x 11 = 121
Third layer 10x10 = 100
Continuing like that until the 2nd from the top is 2x2 = 4 and the top is 1x1 = 1.
Adding those up, 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 = 650
Googaholic are looking at this as a four-sided pyramid, not as the three-sided one as suggested?
The answer tells you that under the top ball (1) there are 3 balls supporting it, not four.
The answer tells you that under the top ball (1) there are 3 balls supporting it, not four.
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