On the $\mathcal{P}$-positions of some infinite families of Slow $A$-Nim

Abstract

We introduce the game Slow $A$-Nim which generalizes a number of recently studied games. Slow $A$-Nim is played on $n$ stacks of tokens, and the set $A$ indicates the number of stacks a player can play on. Once a player has decided on the number $a$ of stacks, s/he will select any $a$ stacks and then remove one token from each stack. The last player to move wins. We give results on the $\mathcal{P}$-positions of Slow $A$-Nim for several infinite families. The results for $A=\{n-1\}$, which is the game Slow Exact $k$-Nim for $k=n-1$ extend recent results for small values of $n$. The other two families, $A=\{n-1,n\}$ and $A=\{1,n\}$ have not been previously studied. The $\mathcal{P}$-positions for $A=\{n-1\}$ and $A=\{n-1,n\}$ are closely related and have a very elegant description in terms of reduced positions, that is, positions for which unplayable tokens are disregarded. We also provide some general results that will be useful in the study of other sets $A$.

Title changed to reflect results are for general sets $A$ instead of just singletons $\{k\}$. Paper was reorganized, results for $A=\{1,k\}$ added, some figures improved, and redundancies removed. Some proofs rewritten (direct proofs instead of contradiction). Since the number of changes are numerous, the new version should be read as a new paper