Mis\`ere-play Hackenbush Sprigs

Abstract

A Hackenbush Sprig is a Hackenbush String with the ground edge colored green and the remaining edges either red or blue. We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in mis\`ere-play, as well as in normal-play. We find the outcome of a disjunctive sum of Sprigs in mis\`ere-play and show that it is the same as the outcome of that sum plus star in normal-play. Along the way it is shown that the sum of a Sprig and its negative is equivalent to 0 in the universe of mis\`ere-play dicotic games, answering a question of Allen.