Multi-player Last Nim with Passes

We introduce a class of impartial combinatorial games, Multi-player Last Nim with Passes, denoted by MLNim $$^{(s)}(N,n)$$ ( s ) ( N , n ) : there are N piles of counters which are linearly ordered. In turn, each of n players either removes any positive integer of counters from the last pile, or makes a choice ‘pass’. Once a ‘pass’ option is used, the total number s of passes decreases by 1. When all s passes are used, no player may ever ‘pass’ again. A pass option can be used at any time, up to the penultimate move, but cannot be used at the end of the game. The player who cannot make a move wins the game. The aim is to determine the game values of the positions of MLNim $$^{(s)}(N,n)$$ ( s ) ( N , n ) for all integers $$N\ge 1$$ N ≥ 1 and $$n\ge 3$$ n ≥ 3 and $$s\ge 1$$ s ≥ 1 . For $$n>N+1$$ n > N + 1 or $$n=N+1\ge 3$$ n = N + 1 ≥ 3 , the game values are completely determined for any $$s\ge 1$$ s ≥ 1 . For $$3\le n\le N$$ 3 ≤ n ≤ N , the game values are determined for infinitely many triplets (N, n, s). We also present a possible explanation why determining the game values becomes more complicated if $$n\le N$$ n ≤ N .