  We compare to different extensions of the ancient game of nim: Moore's
nim$(n, \leq k)$ and exact nim$(n, = k)$. Given integers $n$ and $k$ such that
$0 < k \leq n$, we consider $n$ piles of stones. Two players alternate turns.
By one move it is allowed to choose and reduce any (i) at most $k$ or (ii)
exactly $k$ piles of stones in games nim$(n, \leq k)$ and nim$(n, = k)$,
respectively. The player who has to move but cannot is the loser. Both games
coincide with nim when $k=1$. Game nim$(n, \leq k)$ was introduced by Moore
(1910) who characterized its Sprague-Grundy (SG) values 0 (that is,
P-positions) and 1. The first open case is SG values 2 for nim$(4, \leq 2)$.
Game nim$(n, = k)$, was introduced in 2018. An explicit formula for its SG
function was computed for $2k \geq n$. In contrast, case $2k < n$ seems
difficult: even the P-positions are not known already for nim$(5,=2)$. Yet, it
seems that the P-position of games nim$(n+1,=2)$ and nim$(n+1,\leq 2)$ are
closely related. (Note that P-positions of the latter are known.) Here we
provide some theoretical and computational evidence of such a relation for
$n=5$.
