Topping a pyramid
Spheres are stacked as shown, with a regular rectangular pattern for the base. You could encase the arrangement in a tightly fitting pyramid, or you could place another identical sphere directly on top of the highest one. Which will be higher, the top of the pyramid or the center of the sphere?
Details: The spheres are identical. The enclosing pyramid has each of its 4 faces tangent to 15 spheres. The additional sphere on top is placed so that it is tangent to the top sphere and its center is directly above the center of the top sphere. That means it does not continue the pattern in any way.
Image credit: Wikipedia
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1 solution
The pyramid encasing the stack is a top half of an octahedron. In an octahedron, the relationship between the inscribed sphere radius r and the circumscribed sphere radius R is R = r 3 . So the height of the top vertex of the pyramid above the top sphere is ( 3 − 1 ) r = 0 . 7 3 r . That is less than r , which is the height of the center of the additional sphere above the top of the original stack.