Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp (1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in \emph{Winning Ways} (1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard wall convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention.

翻译：减法游戏是组合博弈论中的一个经典课题。Golomb (1966) 的结果表明，任何具有有限移动集的减法游戏都拥有最终周期性的尼姆序列，但已知的证明仅能给出周期长度的指数级上界。Flammenkamp (1997) 提出了一个关于三移动减法游戏的惊人分类猜想：非加法规则集表现出形式为“两个移动之和”的线性周期长度，其中具体选择哪两个移动呈现出分形般的行为；而加法集合 $S=\{a,b,a+b\}$ 则具有纯周期性的博弈结果，其周期长度为线性或二次型。尽管在《制胜之道》(1982) 中已受到早期关注，但一般的加法情形仍未解决。我们引入并分析了一种对偶的获胜约定，称之为 {\sc 汇点减法}。与标准的墙约定（禁止移动到负位置）不同，汇点约定规定玩家在移动到非正位置时即获胜。我们证明了 {\sc 加法汇点减法} 存在一个完整的解：其尼姆序列是纯周期的，并具有显式的线性或二次周期公式。我们进一步猜想加法汇点减法与经典的墙减法之间存在一种对偶关系。 关键词：加法减法游戏，尼姆数，周期性，汇点约定。