New Values for Top Entails
Julian West · 1997
The game of Top Entails introduces the curious theory of entailing moves. In Winning Ways, simple positions are analysed and stacks of one and three coins are shown to be loony positions, since any move leaving such a stack is a losing move, regardless of other stacks that might also be present. We analyse all stacks of up to 600,000 coins and find three more loony stacks: those having 2403, 2505 and 33,243 coins. The game of Top Entails is introduced in Winning Ways [Berlekamp et al. 1982, p. 376]. It is a nim-like, impartial game played with stacks of coins and the following two types of move. A player may either (1) split just one stack into two strictly smaller stacks; or (2) remove just one coin from just one stack; in this case the opponent is entailed to reply with a move (of either type) in the resulting stack. An unentailed stack of zero coins can never really occur in this game, since neither type of move can create one. Nevertheless, we can posit one as a hypothetical object; indeed, it is clear that it has value *0 = 0, since it hardly matters whether there is such a stack on the table or not! What about an entailed stack of zero coins? Such a position can certainly occur, by an entailing (type 2) move from any stack containing just one coin. Since the next player, N, is now forced to move in an empty stack, the game ends instantly with a win for the previous player, P .