Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p

Write a≡3⋅2−1a≡3⋅2−1 and b≡3⋅2−2(modp) where pp is an odd prime. Let cc be a value that is congruent (modpmodp) to either aa or bb. For any xx from Zp∖{0}Zp∖{0}, evaluate each of xx and cx(modp) within the interval (0,p)(0,p). Then consider the quantity μc∗(x)=min(cx−x,x−cx) where the differences are evaluated (modp−1, not modp) in the interval (0,p−1)(0,p−1), and the quantity μc∧(x)=min(cx−x,x−cx) where the differences are evaluated (modp+1modp+1) in the interval (0,p+1)(0,p+1). As xx varies over Zp∖{0}Zp∖{0}, the values of each of μc∗(x) and μc∧(x) give exactly two occurrences of nearly every member of 1,2,…,(p−1)/21,2,…,(p−1)/2. This fact enables aa and bb to be used in constructing some terraces for Zp−1Zp−1 and Zp+1Zp+1 from segments of elements that are themselves initially evaluated in ZpZp.