given a geometric problem formula_34, we define the "configuration space", formula_3, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) additionally, we consider a "test space" formula_36 and a map formula_37 where formula_38 is a solution to a problem if and only if formula_39. finally, it is usual to consider natural symmetries in a discrete problem by some group formula_2 that acts on formula_3 and formula_42 so that formula_5 is equivariant under these actions. the problem is solved if we can show the nonexistence of an equivariant map formula_44.