Additive sink subtraction

TL;DR

The paper addresses nimber periodicity for additive subtraction games under a sink termination rule. It introduces sink subtraction and proves that for additive rulesets with $S(m,\delta)=\{m,m+\delta,2m+\delta\}$ the nim-sequence is purely periodic, with an explicit period $p(m,\delta)$ depending on $d=\delta\bmod 2m$. The proof splits into two cases: when $0\le d\le m$ a linear-period construction yields $p(m,\delta)=3m+2\delta-d$, and when $m<d<2m$ a block-structure analysis with $B$- and $C$-blocks gives a quadratic period length $|P(m,k)|=\frac{m(4m+3k)}{\gcd(m,k)}$ for $\delta=m+k$. Generalizations to $\delta=2nm+k$ show period growth by $4m^2/\gcd(m,\delta)$, and the work links these findings to broader conjectures on subtraction games and a proposed duality with wall subtraction.

Abstract

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.

Figures (4)

• Figure 1: sink and wall subtraction, respectively. Suppose the ruleset is $S=\{1,5\}$. To the Left, the boy can choose to move into the sink or he can play to the tip of the jetty, and let daddy win. To the Right, the boy has only one move, because the larger move is prohibited by the wall. Thus the wall-game is determined already by parity.
• Figure 2: Nimber periods for $m=19$ and $0<k <19$, starting from $x=1$. The colors correspond to the nimbers; purple (0), blue (1), green (2) and yellow (3). Zoom in for details.
• Figure 3: Nimber periods for $m=5,6,7$ and $0<k <m$, respectively, starting at $x=1$. The rows have different scaling as to fit the full periods in each case. In the case of $m=6$, and rows $k=2,3,4$ the smallest period is a divisor of the full row length. Hence in these cases there are more than one $\zeta$-factors. In the case of $m=7$, we have indicated with red rectangles, for $k=1$ the blocks $B(0)$ and $B(4)$; for $k=2$ the block $C(3)$; for $k=3$ the block $B(6)$; for $k=4$ the factor $\zeta(6)$; for $k=5$, the blocks $B(3)$ and $C(5)$; for $k=6$ the block $C(1)$.
• Figure 4: Nimbers for $m=3$, $\delta\equiv 4\pmod 6$ (left) and $m=4$ with $\delta\equiv 7\pmod 8$ (right).

Theorems & Definitions (8)

• Remark 1
• Theorem 2: Folklore golomb1966mathematical
• proof
• Theorem 3: Main Theorem
• Lemma 5
• proof
• Lemma 6
• proof