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The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games

Balaji R. Kadam, Matthieu Dufour, Silvia Heubach

Abstract

We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN($n,A$) and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN($n,k$) with $n=7, k=3$ and $n=8,k=3$.

The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games

Abstract

We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN() and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN() with and .

Paper Structure

This paper contains 13 sections, 12 theorems, 18 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and Game Reductions
  3. Invariance and $\mathcal{P}$-positions
  4. Finding the $\mathcal{P}$-position patterns using invariant vectors
  5. Proving the $\mathcal{P}$-position pattern using invariant vectors
  6. Existence of a move from $\bm{p} \notin \mathcal{P}$ to $\bm{p}' \in \mathcal{P}$
  7. There is no move from $\bm{p} \in \mathcal{P}$ to $\bm{p}' \in \mathcal{P}$
  8. Efficient Play with Invariant Vectors
  9. $\mathcal{P}$-positions of PathNim and Game $H$
  10. The $\mathcal{P}$-positions of CN$\left(7,3\right)$
  11. The $\mathcal{P}$-positions of CN$\left(8,3\right)$
  12. Detour into Simplicial Complexes
  13. Conclusion and Future Work

Key Result

Lemma 2.5

Let $\bm{p}$ be a position of a game $G$ and $\tilde{\bm{p}}$ the associated position in the merge-reduced game $\tilde{G}^C_{v^*}$. Then the Grundy values $g(\bm{p})$ and $g(\tilde{\bm{p}})$ are the same. In particular, $\bm{p}$ is a $\mathcal{P}$-position of $G$ if and only if $\tilde{\bm{p}}$ is

Figures (5)

  • Figure 1: Using invariant vectors to find an option $\bm{p}' \in \mathcal{P}$ of $\bm{p} \notin \mathcal{P}$ that can be reached via legal move $\bm{m}$.
  • Figure 2: Visualization of the Invariance Reduction Process. Boxed numbers reference the steps of the Invariance Reduction Process.
  • Figure 3: Quantities involved in the move to $\bm{p}' \in \mathcal{P}_{n,k}$ in case 3 of the proof of Theorem \ref{['thm:P-pos path']}.
  • Figure 4: The steps of the Invariance Reduction Process for a position $\bm{p} \notin \mathcal{P}_H$ with $\min(\bm{p})=a$, which reduces to game PN$\left(5,3\right)$ (case 1 in the proof of Theorem \ref{['thm:H']}), using the position of Example \ref{['ex:irp']}.
  • Figure 5: Generic labeling of positions of CN$\left(8,3\right)$.

Theorems & Definitions (50)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2: Zero reduction
  • Remark 2.3
  • Definition 2.4: Merge reduction
  • Lemma 2.5
  • proof
  • Example 2.6
  • ...and 40 more