Dynamic One-Pile Blocking Nim

The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the number of counters that can be removed on each successive turn changes during the play of the game. Both the minimum and the maximum number of counters that can be removed is dependent upon the move number. Also, on each move, the opposing player can block some of the moving player’s options. This number of blocks also depends upon the move number. There is great interest in generalizations and modifications of simple, deterministic two-player “take-away-games” — for a nice survey, see chapter 4 of [1]. We discuss here a modification where the player-not-to-move may effect the options of the other player. Modifications of this type have been called Muller twists in the literature. See [4]. In [3], we discuss games in which the number of counters that can be removed depends on the number removed in the previous move. the electronic journal of combinatorics 10 (2003), #N00 1 We begin with some notation. The set of integers is denoted by Z, the positive integers by N and the nonnegative integers by N0. If a, b ∈ Z with a ≤ b, then [a, b] denotes {x ∈ Z : a ≤ x ≤ b}. Rules of the Game: We are given three sequences (ci ∈ N0)i∈N, (mi ∈ N)i∈N and (Mi ∈ N)i∈N which satisfy the following conditions: ∀i ∈ N, ci ≤ ci+1 (1) and ui = Mi −mi − ci for each i ∈ N and ∀i ∈ N, 0 ≤ ui ≤ ui+1. (2) These two conditions imply that (Mi−mi ∈ N0)i∈N is a nondecreasing sequence. There are two players and a pile of counters. These two players alternate removing counters from the single pile according to the following rules: denote by k ∈ N the movecounter and by pk ∈ N0 the pile size before the k-th move. Then the player to make the k-th move must remove from the pile any number of counters x ∈ [mk, Mk] satisfying x ≤ pk. There is also a further restriction set by the other player for the selection of x: before a player makes his k-th move, the opponent can prohibit up to ck of the current options. Therefore a player cannot move if pk is less than the smallest available option. Condition (1) and condition (2) can be interpreted as saying that the number of at most blocked options ci as well as the number of at least available options ui + 1 must be a nondecreasing function of the turn number i. The game ends as soon as one of the two players cannot move, and this player is called the loser. As an example, look at the third move of such a game. Suppose, [m3, M3] = [5, 10], c3 = 2 and p3 = 15. Since c3 = 2, the opponent of the player-to-move, can block at most two of the moving player’s six options. Suppose that he denies the removal of 6 counters and the removal of 10 counters from the available interval [5, 10]. This means the player-to-move can remove from the 15 counter pile either 5, 7, 8 or 9 counters. If we modify the example so that p3 = 6 and the opponent prohibits the removal of 5 or 6 counters, the player-to-move can not move at all and loses this game. Whether the starting player, also called the first player, or the second player will win the game depends therefore only on the pile size at the beginning. The possible pile sizes, numbers in N0, which are a lost position for the player-to-move are called safe positions. The complement is called the unsafe positions, these are the winning positions for the player-to-move. These two sets of pile sizes will be characterized in the following as a set of disjoint intervals of maximal length in N. Theorem 1. The safe positions of the game are ⋃ k∈N0 [Ak, Bk − 1] with