Re: Q6
I'll give one concrete numerical example to hopefully set you on the right path.
Lets say I am considering N = 3 and pick points {1, 7, 11}. Can I find a way to set up two intervals to get all possible 2^3 = 8 possible dichotomies? By your short hand I'm assuming o represents +1 and x represents 1.
If I want the arrangment xox, that means 1 and 11 have to be outside my choices for intervals and 7 has to be inside. So one particular choice of intervals would be [5,9] and [99,100]. Another would be [5,8] and [6,10]. You can go easily verify that all possible 2^3 arrangements are possible. The only trick here is that you're restricted by the geometry of an interval, meaning it starts at one particular value, ends at another, and contains every point in between.
