Additive Sink Subtraction Analysis

• Additive sink subtraction is a combinatorial game-theoretic framework that defines terminal sink positions under additive move-set rules, leading to explicit nim-sequence periodicity.
• It employs novel arithmetic characterizations and a duality with the wall convention to classify nim-sequence behavior into distinct linear and quadratic cases.
• The explicit periodic formulas and structural proofs provide actionable insights for analyzing combinatorial strategies and decision-making processes.

Additive sink subtraction is a combinatorial game-theoretic construct that expands on the classical theory of subtraction games by introducing a modified winning condition—called the sink convention—within the context of additive rulesets. The periodicity and exact structure of nim-sequences in this framework admit an explicit and comprehensive classification, distinguishing it sharply from the classical wall convention. The nomenclature arises from a duality between movement constraints (wall) and special terminal positions (sink), and the associated arithmetic characterizations exploit the additive algebraic structure of the allowed move sets (Bhagat et al., 26 Jan 2026).

1. Foundations: Subtraction Games and Conventions

Subtraction games are impartial combinatorial games defined over non-negative integers, typically viewed as heap sizes. For a given finite move-set $S \subset \mathbb{N}$, play from position $x > 0$ consists of subtracting any $s \in S$ (with $x - s$ as the next position). The classical restriction is the wall convention, forbidding moves into negative integers: only $x - s \geq 0$ positions are legal.

In the sink subtraction convention, all positions $x \leq 0$ are treated as "sink" states—terminal P-positions. Any move $x \to x - s$ is legal for $s \in S$; if $x - s \leq 0$, the next player cannot move, and under normal play, the participant who enters such a sink wins. The nimber (Sprague-Grundy value) at $x$ is denoted $x > 0$0, with boundary condition $x > 0$1 for all $x > 0$2 and the mex rule:

$x > 0$3

2. Additive Move-Sets and Periodicity Phenomena

The central focus is on additive subtraction games where the move-set is $x > 0$4 for parameters $x > 0$5. These rulesets are distinguished by their inclusion of the sum $x > 0$6 whenever $x > 0$7 are present, an algebraic closure property leading to unique periodic structures in the nim-sequence.

Flammenkamp conjectured—in the wall context—that non-additive 3-move sets typically exhibit period length equal to the sum of two moves, with intricate fractal selection of these pairs, while additive sets may display either linear or quadratic period lengths. In the sink convention, additive sets admit a tractable, explicit, and purely periodic nim-sequence with period length given by a closed formula, in contrast to the exponential upper bound known from Golomb's result for general finite sets.

3. Main Structural Theorem: Explicit Periodicity

Let $x > 0$8, $x > 0$9. The periodicity of the nim-sequence $s \in S$0 for additive sink subtraction with $s \in S$1 is exactly:

$s \in S$2

The sequence is purely periodic of period $s \in S$3. The explicit construction of a period depends on whether $s \in S$4 falls in the linear or quadratic regime, with modular congruence controlling the combinatorial pattern.

4. Explicit Pattern Construction—Linear and Quadratic Cases

Linear Case ($s \in S$5):

Let $s \in S$6 so that $s \in S$7. One period is the concatenation:

$s \in S$8

where $s \in S$9 (resp. $x - s$0, etc.) is a contiguous block of $x - s$1 consecutive positions with nim-value 1 (resp. 2, etc.). Each factor contributes to the periodic structure, wrapping around modulo $x - s$2.

Quadratic Case ($x - s$3):

Write $x - s$4 ($x - s$5, $x - s$6). The period is derived through $x - s$7-block decompositions with modular index shifts:

• Define $x - s$8, $x - s$9, $x - s \geq 0$0 positive representative of $x - s \geq 0$1, $x - s \geq 0$2.
• Blocks are:
  • $x - s \geq 0$3-block (if $x - s \geq 0$4): $x - s \geq 0$5.
  • $x - s \geq 0$6-block (if $x - s \geq 0$7): $x - s \geq 0$8.
  • $x - s \geq 0$9.
  • Insert $x \leq 0$0 between block $x \leq 0$1 and block $x \leq 0$2 whenever $x \leq 0$3.

The complete period is $x \leq 0$4, a concatenation of these blocks, total length:

$x \leq 0$5

5. Proof Strategy and Periodicity Validation

Periodicity is established using Golomb’s window argument: each nimber depends only on the nimbers of its $x \leq 0$6 predecessors, implying the sequence of length $x \leq 0$7 slides ("window"), so only finitely many windows and thus eventual periodicity.

For additive sink subtraction, the precise period is conjectured and then validated inductively. Each position within one period is examined for consistency with the mex rule and move set. For the quadratic case, crucial modular identities (e.g., $x \leq 0$8 with $x \leq 0$9) and structural inequalities ($x \to x - s$0 vs. $x \to x - s$1) classify the block type and ensure exact repetition with period $x \to x - s$2 (Bhagat et al., 26 Jan 2026).

6. Illustrative Examples

Examples provide concrete instantiation of the general pattern:

$x \to x - s$3	$x \to x - s$4	$x \to x - s$5	Case	Period Length $x \to x - s$6	Period
2	1	$x \to x - s$7	Linear	7	(1,1,2,2,3,0,0)
3	2	$x \to x - s$8	Linear	11	(1,1,1,2,2,2,3,3,0,0,0)
3	4	$x \to x - s$9	Quadratic	45	Explicit $s \in S$0-block concatenation

Each period can be verified directly or by computer-assisted check to satisfy the mex rule at every step, with no shorter period possible.

7. Duality Conjecture and Mathematical Significance

A duality between additive sink subtraction and the classical wall subtraction is conjectured, suggesting deeper algebraic and combinatorial interconnections between winning conventions and periodicity characteristics.

The classification achieved for additive sink subtraction contrasts with the intractability of the corresponding wall subtraction case for general additive moves. The purely periodic and explicit nature of the solution for the sink convention provides constructive tools for analyzing broader classes of subtraction games, informing both combinatorial theory and algorithmic applications in related domains such as strategic decision processes and symbolic dynamics (Bhagat et al., 26 Jan 2026).