I think you're all making this far too difficult?.
As I understood the question, he wanted either a gain of 10, or an attenuation of 10, and fully variable between the two options - which is what I provided, either inverting or not, with one opamp.
The schemes for changing from inverting to not-inverting are flawed, because what happens when you get to the balance point - no output at all!.
Perhaps 'petesmc' should clarify the question further - presumably it's a school question?.