23 May 2006

Puzzling Tuesday

Here is an easy one:

You have 2 identical cubic boxes. You completely fill (without going above the top) one of them with iron spheres and the other one with iron cubes. The length of an inner edge of each box is an exact multiple of one edge of a cube and of the diameter of a sphere.

1. Which box will have more items (cubes or spheres) in it?
2. Which filled box will have a smaller density?

(Modified from a question in Ebbing & Gammon, General Chemistry, 6th ed., 1999.)


pascal said...

If I remember my cubic closest packing math correctly, the box with the cubes will more dense, since there will be no intersticial space, but I think the box with the spheres will have more items.


Yes, the box with the cubes will be denser, because the cubes can be stacked against each other with no space in between.

The number of cubes in the cube-filled box will be the same as the number of spheres in the sphere-filled box. If the inner edge of each box is k times the edge of a cube and k times the diameter of a sphere, then there will be kxkxk cubes and kxkxk spheres in the respective boxes. I guess I should have explicitly stated that one edge of each cube is equal in length to the diameter of each sphere.