# math problem

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If one side of the bottom layer of a triangular pyramid of bowling balls has 12 balls, how many are there in the whole pyramid?

• math problem -

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The above figure shows the bottom layer of the pyramid, which has
1+2+3+4+5+6+7+8+9+10+11+12=78 balls
The next layer will have one row less, namely
1+2+3...11=66
and so on until the first layer has only one ball.
In fact, the layer with n balls on the side has n(n+1)/2 balls in the layer.

Add up all the sums to get 364 balls in all.

Again, for a pyramid of n layers (where the bottom layers has n balls on each side) the total number of balls is given by n(n+1)(n+2)/6.

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