Nim Fractals

TL;DR

This paper explores the enumeration of P-positions in Nim through two sequences, reveals a cellular automaton perspective, and connects Nim sequences to known automata, introducing 10 new sequences.

Contribution

It introduces a novel cellular automaton view of Nim and links Nim sequences to known automata, along with the creation of 10 new sequences.

Findings

Nim P-positions can be enumerated by maximum pile size and total counters.

The three-pile Nim sequence by total counters is a toothpick sequence based on Ulam-Warburton automaton.

The paper introduces 10 new sequences related to Nim P-positions.

Abstract

We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters. We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences.