Ultimately bipartite subtraction games

We introduce the notion of ultimately bipartite impartial games. These are games that are ultimately periodic in the simplest possible manner. We examine ultimately bipartite subtraction games and demonstrate a curious feature: for ‘large’ games it is clear who has the winning position and how the game should be strategically played, but during play, as the game eventually becomes small, it is no longer so easy to know what the strategic moves are. We give examples which indicate that ultimately bipartite subtraction games are quite common.