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Back-of-the-envelope Physics:
It's all about quantizing electrons in a spherical potential. Inspired by Harold Kroto.

Background:
Consider an element with N protons and N electrons.
The electrons occupy the N lowest states in the potential, which is approximately spherical.
Hence, in order to roughly understand the periodic table, we need to order these electron states.
An electron state depends on a few discrete quantum numbers: n (radial coordinate), j (total angular momentum), m (projection of j), and m_s (projection of spin).
An angular momentum j defines a multiplet of 2j+1 discrete projections, m={-j, -j+1, -j+2, ...,+j-1, +j}.
Similarly, the s=1/2 spin of the electron defines a doublet, i.e. 2s+1=2 states, with a projected spin m_s={-1/2,+1/2}.

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