Since 2 is a primitive root of 3^m for each positive integer m, the set of points { (n, 2ⁿ mod 3^m) : n ≥ 0}, viewed as a subset of Z_{≥ 0} × Z_{≥ 0} is bi-periodic, with minimal periods ϕ(3^m) (horizontally) and 3^m (vertically). We show that if one considers the classes of n modulo 6, one obtains a finer structural classification. This result is presented within the context of the question of strong normality of Stoneham numbers.