Ordinal sums of impartial games

In an ordinal sum of two combinatorial games G and H, denoted by G:H, a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H. It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze G(G:H) where G and H are impartial forms, observing that the G-values are related to the concept of minimum excluded value of orderk. As a case study, we introduce the ruleset oak, a generalization of green hackenbush. By defining the operation gin sum, it is possible to determine the literal forms of the bases in polynomial time.