游戏Max-Welter
The game Max-Welter
摘要：
在韦尔特的游戏中，在一个半无限长的方格上，每个方格最多只能有一个硬币，每个方格编号为$0, 1, 2, \ldots$，两个玩家轮流将一枚硬币移到左边的一个空方格。跳过其他硬币是合法的。首先不能移动的玩家输了。我们研究了韦尔特博弈的一个变体，我们称之为Max-Welter，在这个变体中，玩家只允许把硬币移到最右边。我们解决了获胜策略，并描述了Sprague-Grundy值为1的位置。我们提出了两个定理分类的一些特殊情况下，计算的Sprague-Grundy值的大小$k$的位置变得更容易通过考虑另一个位置的大小$k-1$。我们建立了两个结果的周期性的Sprague-Grundy值。然后，我们表明，游戏最大Welter被归类在一个适当的子类，Gurvich呼吁强烈悲惨的驯服游戏。
英文摘要：
On a semi-infinite strip of squares rightward numbered $0, 1, 2, \ldots$ with at most one coin in each square, in Welter's game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The player who first cannot move loses. We examine a variant of Welter's game, that we call Max-Welter, in which players are allowed to move only the coin furthest to the right. We solve the winning strategy and describe the positions of Sprague-Grundy value 1. We propose two theorems classifying some special cases where calculating the Sprague-Grundy value of a position of size $k$ becomes easier by considering another position of size $k-1$. We establish two results on the periodicity of the Sprague-Grundy values. We then show that the game Max-Welter is classified in a proper subclass of tame games that Gurvich calls strongly miserable.
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