We compare to different extensions of the ancient game of nim: Moore's
nim(n,≤k) and exact nim(n,=k). Given integers n and k such that
0<k≤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,≤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,≤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,≤2).
Game nim(n,=k), was introduced in 2018. An explicit formula for its SG
function was computed for 2k≥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,≤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