2-pile Nim with a Restricted Number of Move-size Imitations

TL;DR

This paper investigates a constrained version of 2-pile Nim where players are limited in how many consecutive moves they can imitate, revealing strategic similarities to a modified Wythoff Nim game with blocking moves.

Contribution

It introduces a novel variation of 2-pile Nim with move imitation restrictions and establishes its strategic connection to a Wythoff Nim variant with blocking maneuvers.

Findings

Game strategy closely resembles a Wythoff Nim variant with diagonal blocking.

Imitation constraints influence optimal move choices.

Theoretical analysis generalizes the imitation concept.

Abstract

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a variant with a blocking manoeuvre on $p-1$ diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is.