On the Sprague–Grundy function of extensions of proper Nim

Abstract

We consider the game of proper Nim, in which two players alternately move by taking stones from n piles. In one move a player chooses a proper subset (at least one and at most $n-1$) of the piles and takes some positive number of stones from each pile of the subset. The player who cannot move is the loser. Jenkyns and Mayberry (Int J Game Theory 9(1):51–63, 1980) described the Sprague–Grundy function of these games. In this paper we consider the so-called selective compound of proper Nim games with certain other games, and obtain a closed formula for the Sprague–Grundy functions of the compound games, when $n\ge 3$. Surprisingly, the case of $n=2$ is much more complicated. For this case we obtain several partial results and propose some conjectures.

Appendix: Slow Nim

In this section, we consider a variant of Nim, so called slow Nim. A move in a Hypergraph Nim game Nim$_\mathcal{H}$ is called slow if each pile is reduced by at most one token. Let us restrict both players by their slow moves, then the obtained game is called slow hypergraph Nim. We study SG functions and losing positions of slow Moore Nim and slow exact Nim, where they respectively correspond to hypergraphs $\mathcal{H}= \{H \subseteq I \mid 1 \le |H| \le k\}$ and $\mathcal{H}=\{H \subseteq I \mid |H|=k\}$ for some $k \le n$. We provide closed formulas for the SG functions of both games when $n = k = 2$ and $n = k+1 = 3$, where we remak that the SG function for slow exact Nim when $n = k = 2$ is trivial. We also characterize losing positions for slow Moore Nim if either $n \le k+2$ or $n = k+3 \le 6$ holds.

Here we only present the results, where all the proofs can be found in the preprint by Gurvich and Ho (2019).

Given a position $x = (x_1, \ldots , x_n) \in {\mathbb {Z}}_+^I$, we will always assume that its coordinates are nondecreasing $x_1 \le \cdots \le x_n$. The parity vector p(x) is defined as the vector $p(x) = (p(x_1), \ldots , p(x_n)) \in \{0,1\}^I$ such that $p(x_i) = 0$ if $x_i$ is even, and $p(x_i) = 1$ if $x_i$ is odd. It appears that the status of a position x in the slow Moore Nim in the cases below is defined by p(x) .

Proposition 2

The SG function ${\mathcal {G}}$ for slow Moore Nim  for $n = k = 2$ and $n = 3, k = 2$ are uniquely defined by p(x) as follows:

• (i) For $n = k = 2$,

  $$\begin{aligned} {\mathcal {G}}(x) = {\left\{ \begin{array}{ll} 0, \quad \text { if } p(x) = (0,0) \\ 1, \quad \text { if } p(x) = (0,1) \\ 2, \quad \text { if } p(x) = (1,1) \\ 3, \quad \text { if } p(x) = (1,0). \end{array}\right. } \end{aligned}$$

• (ii) For $n = 3$ and $k = 2$,

  $$\begin{aligned} {\mathcal {G}}(x) = {\left\{ \begin{array}{ll} 0 \quad \text { if } p(x) \in \{(0,0,0), (1,1,1)\} \\ 1 \quad \text { if } p(x) \in \{(0,0,1), (1,1,0)\} \\ 2 \quad \text { if } p(x) \in \{(0,1,1), (1,0,0)\}\\ 3 \quad \text { if } p(x) \in \{(0,1,0), (1,0,1)\}. \end{array}\right. } \end{aligned}$$

We next consider losing positions of slow Moore Nim.

Proposition 3

Consider a slow Moore Nim when $n \le k+2$ or $n = k+3 \le 6$. Then for a position $x \in {\mathbb {Z}}_+^I$, we have the following five cases.

• (1) for $n=k$, x is losing if and only if $p(x) = (0, 0, \ldots , 0)$.

• (2) for $n=k+1$, x is losing if and only if $p(x) \in \{(0, 0, \ldots , 0), (1, 1, \ldots , 1)\}$.

• (3) for $n=k+2$, x is losing if and only if $p(x) \in \{(0, 0, \ldots , 0), (0,1, \ldots , 1)$.

• (4) for $n=5$ and $k=2$, x is losing if and only if

  $$\begin{aligned} p(x) \in \{(0,0,0,0,0), (0,0,1,1,1), (1,1,0,0,1), (1,1,1,1,0)\}; \end{aligned}$$

• (5) for $n=6$ and $k=3$, x is losing if and only if

  $$\begin{aligned} p(x) \in \{(0,0,0,0,0,0), (0,0,1,1,1,1), (1,1,0,0,1,1), (1,1,1,1,0,0)\}. \end{aligned}$$

We note that Moore Nim games satisfy that $1\le k \le n$, and the case in which $n=4$ and $k=1$ is a standard 4-pile Nim.

We finally consider slow exact Nim. Note that this game is trivial when $k=1$ or $k=n$. We show that the game is tractable if $n=3$ and $k=2$. Again the parity vector plays an important role, although it does not define the SG function uniquely.

Define six sets of positions $x \in {\mathbb {Z}}_+^3$:

$$\begin{aligned} A&= \{(2a, 2b-1, 2(b+i)) \mid 0 \le a< b, 0 \le i< a, (a+i) \bmod {2} = 1\} \\ B&= \{(2a,2b,2(b+i)+1) \mid 0 \le a \le b, 0 \le i< a, (a+i) \bmod {2} = 1\} \\ C_0&= \{(2a-1, 2b-1, 2(b+i)-1) \mid 0 \le a \le b, 0 \le i< a, (a+i) \bmod {2} = 0\} \\ C_1&= \{(2a-1, 2b-1, 2(b+i)-1) \mid 0 \le a \le b, 0 \le i< a, (a+i) \bmod {2} = 1\} \\ D_0&= \{(2a-1, 2b, 2(b+i)) \mid 0 \le a< b, 0 \le i< a, (a+i) \bmod {2} = 1\} \\ D_1&= \{(2a-1, 2b, 2(b+i)) \mid 0 \le a< b, 0 \le i < a, (a+i) \bmod {2} = 0\}, \end{aligned}$$

and let $C = C_1 \cup C_2$, $D = D_1 \cup D_2$.

Proposition 4

For a slow exact Nim with $n=3$ and $k=2$, the SG function ${\mathcal {G}}$ can be represented by

$$\begin{aligned} {\mathcal {G}}(x) = {\left\{ \begin{array}{ll} 0 \quad \text { if } x \in (\{(2a,2b,c) \mid 2a \le 2b \le c\} {\setminus } B) \cup A \cup C_0 \cup D_0 \\ 1 \quad \text { if } x\in (\{(2a,2b+1,c) \mid 2a \le 2b+1 \le c\} {\setminus } A) \cup B \cup C_1 \cup D_1 \\ 2 \quad \text { if } x \in \{(2a+1,2b+1,c) \mid 2a+1 \le 2b+1 \le c\} {\setminus } C \\ 3 \quad \text { if } x \in \{(2a+1,2b,c) \mid 2a+1 \le 2b \le c\} {\setminus } D. \end{array}\right. } \end{aligned}$$