On the Sprague-Grundy Function of Tetris Extensions of Proper {\sc Nim}

TL;DR

This paper studies the Sprague-Grundy function of generalized Nim games played on hypergraphs, introducing Tetris extensions and providing formulas for proper Nim when the number of piles is three or more, with partial results for two piles.

Contribution

It introduces Tetris extensions of hypergraph Nim and derives a closed-form formula for the Sprague-Grundy function for proper Nim with three or more piles, extending prior work.

Findings

Closed formula for SG functions of Tetris extensions of proper Nim for n ≥ 3

Partial results and conjectures for the case n=2

Surprising complexity difference between cases with 2 and ≥3 piles

Abstract

Given a hypergraph $\cH \subseteq 2^I \setminus \{\emptyset\}$ on the ground set $I = \{1, \ldots, n\}$, we assign to each $i \in I$ a nonnegative integer $x_i$, that is a pile of $x_i$ tokens, and consider the following generalization of the classical game of {\sc Nim}: Two players alternate turns. In a move a player chooses an arbitrary edge $H \in \cH$ and reduces all piles $i \in H$. The player who is out of moves loses. We call the obtained game hypergraph {\sc Nim}. Such a game is called proper {\sc Nim}, when $\cH=2^I \setminus\{I,\emptyset\}$ is the family of all proper subsets of $I$. Jenkyns and Mayberry \cite{JM80} described the Sprague-Grundy (or SG in short) function of these games. In this paper we introduce Tetris extensions of hypergraph {\sc Nim}, and obtain a closed formula for the SG functions of the extensions of proper {\sc Nim}, when $n\geq 3$. Surprisingly, the case of $n=2$ is much more complicated. For this case we only suggest several partial results and conjectures.