Consider a triangular pyramid of balls, constructed by placing each ball (except those on the bottom) in the hollow formed by the three balls under it. If such a pyramid has 12 balls on the side of the bottom level, how many balls in total does it contain?

For each level n we are looking at a triangle n balls on a side. The number of balls is given by the following equation:

  n
  ∑i = n(n+1)/2
i=1

for a pyramid n levels high, we are looking for:

  n
  ∑[i(i+1)/2] = n(n+1)(n+2)/6
i=1

When n= 12, we get 12(13)(14)/6 = 364

Posted by TomM on 2002-08-02 20:06:57