n-pile Dynamic Nim with Various Endings

1 Introduction. In this paper, two players alternate removing a positive number of counters from one of n piles of counters, and the choice of which pile he removes from can change on each move. On his initial move, the player moving first can remove from one pile of his choice at most t counters. On each subsequent move, a player can remove from one pile of his choice at most f (x) counters, where x is the number of counters removed by his opponent on the proceeding move. The game ends when the total number of counters remaining does not exceed k, k being specified in the beginning, and the winner is the player who moves last. We initially studied the strategy for k arbitrary but fixed, f (x) = x and n = 2. We then generalized the strategy arising from this (k, f, n) = (k, x, 2) game in a straightforward way to arrive at what we call the " ideal " theorem. This ideal theorem specifies the strategy for an arbitrary triple (k, f, n). Of course, this ideal theorem is not always true, and this led us to pose the problem of finding all triples (k, f, n) for which the conclusion of the ideal theorem is true. This paper will give the complete solution to this problem. In [4] we solved the single pile game. Notation. Z is the set of all integers, Z + is the set of positive integers, and