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 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 {\em 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经典主题。Golomb (1966)的一个结果表明，所有具有有限移动集合的减法游戏最终都具有周期性的nim序列，但已知的证明仅给出了周期的指数级上界。Flammenkamp (1997)对三步减法游戏提出了一项引人注目的分类猜想：非加法规则集表现出线性周期长度，其形式为“两个移动之和”，其中两个移动的选择呈现出分形行为；而加法集合 $S=\{a,b,a+b\}$ 则具有纯周期结果，周期长度为线性或二次形式。尽管《Winning Ways》(1982)中早有提及，但一般的加法情况仍未解决。我们引入并分析了一种对偶的获胜规则，称之为{\sc 消去减法}。与禁止移动到负位置的经典{\em 壁}规则不同，消去规则宣布移动到非正位置的玩家获胜。我们证明{\sc 加法消去减法}存在完整解：其nim序列是纯周期的，具有显式的线性或二次周期公式，并猜想加法消去减法与经典的壁减法之间存在对偶性。关键词：加法减法游戏；Nim数；周期性；消去规则。