On most discrete samples for the 1/4 or 1/2 case, the probability of choosing any two points in the subset may be uniform, but the location of the permitted points is definitely not. If you add the requirement that all discrete point samples should be equidistant, then you fix yourself on 1/3 and there is really no way to break that conclusion.

Sorry, in this case "uniform" is ambiguous and I didn't mean the usual definition related to probability, but rather to point spacing.