Heap games, numeration systems and sequences

TL;DR

This paper introduces a new family of heap games and a novel sequence-based strategy, utilizing exotic numeration systems to analyze the complexity and efficiency of the strategies involved.

Contribution

It presents a new two-parameter family of heap games and links their strategies to a novel class of sequences and exotic numeration systems, providing insights into their computational complexity.

Findings

The strategy based on the new sequences is computationally hard to implement.

Exotic numeration systems can determine the existence of efficient strategies.

A new class of sequences is introduced and shown to be equivalent to the sequences used in game strategies.

Abstract

We propose and analyse a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences, and demonstrate its equivalence with the class of sequences defined for the strategy of our games.