Misere Hackenbush Flowers

TL;DR

This paper establishes a relationship between normal and misère play outcomes for disjunctive sums of Hackenbush Flowers, introducing the concept of an evil twin, and shows that certain Green Hackenbush positions are equivalent to nim-heaps under misère play.

Contribution

It introduces the concept of evil twins for Hackenbush Flowers and characterizes Green Hackenbush positions with a single incident edge as nim-heaps under misère play.

Findings

Existence of an evil twin G* for any disjunctive sum of Hackenbush Flowers

Equivalence of certain Green Hackenbush positions to nim-heaps under misère play

Outcome relationships between normal and misère play for these positions

Abstract

We show that any disjunctive sum of Hackenbush Flowers $G$ has as evil twin $G^* \in {G, G+*}$ such that the outcomes of $G$ under normal and mis\`ere play are the same as the outcomes of $G^*$ under mis\`ere and normal play respectively. We also show that, under mis\`ere play, any Green Hackenbush position that has a single edge incident with the ground is equivalent to a nim-heap.