Sphere Packing and Wasted Space

A common interview question: how much of a box is wasted space if you fill it with spheres instead of cubes? Most candidates guess something small, like 10-20%, because spheres "look like" they fill a container pretty well. The real answer for the best possible packing (spheres stacked so each one nestles into the dimples of the layer below) is that spheres occupy only about 74% of the volume — a fixed number, proven mathematically, that doesn't improve no matter how cleverly you arrange them.

The intuition: every sphere touching its neighbors still leaves curved gaps between them that no amount of clever arranging removes, unlike cubes, which tile space with zero waste. Looser packings — spheres dropped in randomly rather than stacked carefully — do much worse, filling only around 64% of the space.

Worked example. A cubic crate is 1 meter on each side (1 m³) and is filled with the densest possible packing of identical balls. Total volume of balls: $1 \times 0.74 = 0.74$ m³. If instead the balls were poured in randomly rather than arranged, expect closer to $1 \times 0.64 = 0.64$ m³ of actual ball volume — a full 10 percentage points of "wasted" space lost purely to sloppy arrangement.