## Round 5: Problem 2

A pile of identical spheres (each of radius 1 meter) are placed in a pile so that they form a triangular pyramid that culminates with one lone sphere at the top of the pile. If the pile of spheres is higher than 2009 centimeters, what is the minimum number of spheres that can be assembled to form the pyramid?

Height of the tetrahedron formed by 4 spheres

= Increase in height by one layer

= sqrt(24)/3.

let h = number of layers.

2 + (h-1) * sqrt(24)/3 > 20.09

h-1 > 11.0778

h > 12.0778

h = 13

We can find number of spheres in layer k by

summing 1 to k, so

there are 455 spheres.

anonymousMay 13, 2009 at 18:23

You forgot to subtract the top and bottom halves of the top and bottom squares. You should be using 19.09 rather than 20.09.

Franklin LeeMay 13, 2009 at 22:04

Sorry, no, I’m wrong. You should be using 18.09, which seems to be what you used.

I’m convinced.

Franklin LeeMay 13, 2009 at 22:24