  We present a graphical approach to understanding the degeneracy, density of
states, and cumulative state number for some simple quantum systems. By taking
advantage of basic computing operations we define a straightforward procedure
for determining the relationship between discrete quantum energy levels and the
corresponding density of states and cumulative level number. The density of
states for a particle in a rigid box of various shapes and dimensions is
examined and graphed. It is seen that the dimension of the box, rather than its
shape, is the most important feature. In addition, we look at the density of
states for a multi-particle system of identical bosons built on the
single-particle spectra of those boxes. A simple model is used to explain how
the $N$-particle density of states arises from the single particle system it is
based on.
