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Two approaches for determining Hilbert functions of fat point subschemes of$\mathbb P^2$ are demonstrated. A complete determination of the Hilbertfunctions which occur for 9 double points is given using the first approach,extending results obtained in a previous paper using the second approach. Inaddition the second approach is used to obtain a complete determination of theHilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the pointsare smooth points of an irreducible plane cubic curve. Additional results areobtained using the first approach for $n\geq 9$ double points when the pointslie on an irreducible cubic (but now are not assumed to be smooth points of thecubic).

Hilbert functions of fat point subschemes of the plane: the two-fold way