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Additive Subtraction Games

Urban Larsson, Hikaru Manabe

Abstract

We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the P-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Miklós and Post (2024) established outcome-periodicity, but without reference to that closed formula. The primitive quadratic case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction with Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical P-positions.

Additive Subtraction Games

Abstract

We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the P-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Miklós and Post (2024) established outcome-periodicity, but without reference to that closed formula. The primitive quadratic case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction with Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical P-positions.
Paper Structure (11 sections, 12 theorems, 65 equations, 2 figures)

This paper contains 11 sections, 12 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Anti-collision of candidate $\mathscr{P}$-positions
  3. Higher nim-values
  4. The arithmetic structure of the $\delta$-collisions
  5. Gaps and residue classes
  6. Collision windows and the parameters $L$ and $R$
  7. Three residue regions and the distribution of $c(r)$
  8. Proof of the $\delta$-collision counting theorem
  9. Some numerical examples
  10. Game theory conclusion
  11. Open problems and discussion

Key Result

Theorem 1

Assume $a<\delta<2a$ and $\gcd(a,\delta)=1$, with $b=a+\delta$. Then $\mathcal{W}_0$ is the set of $\mathscr{P}$-positions of the additive subtraction game $\{a,b,a+b\}$, and the set of nim-value one positions is $\mathcal{W}_0+a$. The set of nim-value two positions is $\mathcal{W}_0-b$, and the set

Figures (2)

  • Figure 1: Consider $a=4$ and $\delta=7$. The gaps along the number line have sizes $1,1,1,5,1,12$ where $12>\delta$. The central gap of size $a+1=5$ is flanked by $L=3$ consecutive one-gaps on the left and $R=1$ on the right; the large dashed gap $>\delta$ prevents extending collision windows further right. The blue and red braces show the two possible collision windows of length $\delta=7$.
  • Figure 2: Each column $r$ contains exactly $c(r)$ admissible lattice points. Thus the total number of points equals the number of lattice points $(r,x)$ satisfying $r-a\leqslant x\leqslant r-1$ inside the rectangle $1\leqslant r\leqslant\delta-1$, $0\leqslant x\leqslant d-1$.

Theorems & Definitions (27)

  • Theorem 1: Main Theorem
  • Remark 2
  • Lemma 3: Anti-collision
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 17 more