On remoteness functions of exact slow NIM, NIM^1 =(4,2) and NIM^1 =(5,2)

Given integer $n$ and $k$ such that $0<k<n$ and $n$ piles of stones, two players make moves alternately, reducing by every move exactly $k$ piles from $n$, by one stone each. The player who must move but cannot is the loser. We call this game exact slow NIM and denote it $\mathrm{NIM}^1(n, k)$. In recent articles [25,26,27,28], a very simple explicit rule was suggested choosing a move in each position of this game. Furthermore, in case $n=k+1$, it was proven that each move chosen by this rule reduces the Smith's remoteness function of the game exactly by one. This result allows us to compute the remoteness function efficiently, thus, solving the game. In the present article, we apply the same rule for two new cases: $k=2, n=4,5$ and show that it still works efficiently: The remoteness function is reduced by one with almost every move. Exceptions are sparse and have a regular pattern. We suggest simple formulas explaining all known exceptions and, thus, solve the two considered new families of games too.