Evil Twins in Sums of Wildflowers

TL;DR

This paper investigates the evil twin property in sums of wildflowers and mutant flowers in combinatorial game theory, extending existing theories and proving computational hardness results.

Contribution

It extends the evil twin property to sums of wildflowers and mutant flowers, generalizes misère genus theory, and proves NP-hardness of outcome computation.

Findings

A large set of wildflower sums has the evil twin property.

Many mutant flower sums also exhibit the evil twin property.

Computing outcome classes of mutant flower sums is NP-hard.

Abstract

A game $G$ is said to have the evil twin property if there exists $G^* \in \{G,G+*\}$ such that $o^+(G) = o^-(G^*)$ and $o^+(G^*) = o^-(G)$. We study sums of wildflowers, games of form $G:H$. We find that a large closed set of sums of wildflowers has the evil twin property, extending work of McKay--Milley--Nowakowski and Lo. Our argument partially generalizes the mis\`ere genus theory of Conway to partizan games, and requires proving several general theorems on ways to extend sets with the evil twin property. Many sums of mutant flowers of the form $\{*x_1,\dots,*x_n\}:a$, where $a$ is a number, also have the evil twin property. We also prove that this set of mutant flowers is the largest such closed set with the evil twin property, and that it is $\mathsf{NP\text{-}hard}$ to compute the outcome class of a sum of mutant flowers under either play convention via a reduction from \textsc{3-Sat}. Previous work on this topic was done by McKay, Milley, and Nowakowski, and later Lo.