Protein folding and inverse protein folding problems are examined for the extremely simplified model of short self-avoiding square lattice walks involving only two or three residue types. Simple interresidue contact free energy functions are given and are used to determine which sequences fold uniquely to which conformations. Contrary to general theories of protein folding, this model system shows little correlation between free energy and conformational distance from the native, nor is there any marked energy gap between the native and the best non-native structures. Furthermore, even the given free energy function sometimes fails to identify which sequences fold to a particular target structure. If current ideas about protein folding and structure/sequence compatibility fail in this model system, it is unclear why they should be valid for real proteins.