Three-Move Subtraction Games

• Three-move subtraction games are impartial combinatorial games defined by a fixed three-element subtraction set, exhibiting rich periodic and algebraic structures.
• Algorithms such as the Sprague–Grundy function and the PTFN procedure efficiently classify winning positions and reveal underlying recurrence relations.
• The analysis uncovers explicit formulas, connections to Beatty sequences, expansion properties, and open problems in multi-dimensional game segmentation.

A three-move subtraction game is an impartial combinatorial game where two players alternately remove a fixed, allowable number of tokens from one or several heaps. When the subtraction set contains exactly three elements, the analysis reveals rich periodic and algebraic structure. The study of these games encompasses single- and multi-dimensional settings, periodicity, explicit formulae for winning positions, expansion properties, and deep connections to Beatty sequences and combinatorial game theory.

1. Formal Definition and Structure

A classic three-move subtraction game on a single heap is played with a fixed subtraction set $S=\{s_1,s_2,s_3\}$, where $1\leq s_1<s_2<s_3$. Players alternate moves; each move consists of subtracting exactly one $s\in S$ from the heap, provided $s\leq n$, for heap size $n\geq0$. The player unable to move wins under normal play convention. Every position $n$ is recursively classified as a $\mathcal{P}$-position (previous player wins with best play) or an $\mathcal{N}$-position (next player wins) via structural recurrences rooted in Zermelo's principle: all terminal positions are $\mathcal{P}$, and any $\mathcal{N}$-position admits a move to a $\mathcal{P}$-position, while all moves from a $\mathcal{P}$-position lead to $\mathcal{N}$-positions (He et al., 2012).

Extending to $d$-dimensional games, a position $\mathbf{x}\in\mathbb{N}_0^d$ allows subtractions by vectors $\mathbf{s}\in S\subset\mathbb{N}_0^d$ with coordinatewise constraint $\mathbf{s}\leq\mathbf{x}$. Normal play, periodicity, and recursive outcome classification generalize accordingly (Larsson et al., 2023).

2. Algorithms for Solving Three-Move Games

Central algorithms for identifying winning positions are the Sprague–Grundy function and the PTFN ("P-position To Find N-position") procedure. The Sprague–Grundy function assigns to each heap $n$ the nim-value

$$G(n) = \mathrm{mex}\{ G(n-s) : s\in S,\ s\leq n \},$$

where $\mathrm{mex}$ denotes the minimum excluded value (Ho, 2012). $\mathcal{P}$-positions are those with $G(n)=0$.

In contrast, the PTFN algorithm employs a $0$–$1$ bit array $X[0\dots n]$ indicating $\mathcal{P}$-positions ($0$) and $\mathcal{N}$-positions ($1$). The recurrence for three-move games (with $a<b<c$) is

$$X[n]= \begin{cases} 0, & n<a,\ 1, & \exists\, s\in \{a,b,c\}:\ X[n-s]=0,\ 0, & \text{otherwise}. \end{cases}$$

Equivalently,

$$X[n]=1 \iff (\neg X[n-a]) \lor (\neg X[n-b]) \lor (\neg X[n-c]), \qquad X[n]=0 \iff X[n-a]=X[n-b]=X[n-c]=1.$$

The algorithm runs in $O(n|S|)$ time and uses $O(n)$ space, faster and more space-efficient than Sprague–Grundy when $|S|=3$ (He et al., 2012).

3. Periodicity and Explicit Formulas

Every fixed finite subtraction set yields an ultimately periodic sequence for both $X[n]$ and $G(n)$. In numerous three-move families, one observes alternating gap patterns among $\mathcal{P}$-positions and closed-form formulas. For instance, in $S=\{1,3,4\}$, $\mathcal{P}$-positions occur at $P=\{0,2,7,9,14,16,21,\dots\}$ with gaps repeating $2,5,2,5,\dots$ and closed form

$p_{2k}=7k, \quad p_{2k+1}=7k+2 \qquad (k\ge0).$

Analogous patterns emerge for various arithmetic and algebraic templates, such as $S(a,b,a+b)$ and $S(1,a,b)$, with the period and pre-period determined by explicit parameter-dependent formulas (Ho, 2012).

Moreover, in two-dimensional settings, periodicity is maintained along rows and columns. Theorem: Every finite two-dimensional set is eventually row- and column-periodic, with outcome-type repeating after some $T$ once past an initial transient $x_0$ or $y_0$ (Larsson et al., 2023).

4. Connections to Beatty Sequences and Generalizations

A significant class of three-move games produces $\mathcal{P}$-positions as complementary pairs of Beatty sequences: $$P=\{(\lfloor n\alpha\rfloor,\lfloor n\beta\rfloor)\}$$, with $\alpha,\beta$ irrational and $1/\alpha+1/\beta=1$. Fraenkel and modified subtraction games use mex-style recurrences

$$a_n = \mathrm{mex}\{ a_k, b_k : k < n \}, \qquad b_n = f(a_n) + b_{n-1} + (a_n - a_{n-1}),$$

where $f$ encodes move constraints, possibly derived directly from Beatty differences (2208.00041).

A necessary and sufficient "Beatty-complementarity inequality" $2\min f-\max f\ge1$ governs which pairs arise. The Relaxed Wythoff family generalizes this further, showing that for any complementary Beatty pair, one can construct a simple three-move game whose $\mathcal{P}$-positions are exactly the pair, providing a solution to the Fraenkel problem and a modification of the Duchêne–Rigo conjecture. Examples include the classical golden ratio case and parameterized $t$-Wythoff families.

5. Expansion Properties and Rigidity

The notion of an expansion set $S^{\text{ex}}$ formalizes which additional moves can be added to $S$ without altering the nim-sequence. The precise characterization:

• $s\in S^{\text{ex}}$ iff $G(n+s)\neq G(n)$ for all $n$ in one period.
• For larger $s$, periodicity governs inclusion: $s\in S^{\text{ex}}$ iff $s-p\in S^{\text{ex}}$, with $p$ the period.

The expansion set can be explicitly computed for families (e.g., $S=\{1,8,11,27\}$ yields $S^{\text{ex}} = \{1,8,11\} \cup \{13,20,27\}^{+19}$) (Ho, 2012).

A central open conjecture asserts that ultimately bipartite games (those with tail nim-sequence $0,1,0,1,\dots$) are non-expandable; no proper superset of $S$ can maintain the same nim-sequence. This conjecture is proven for several families, supported by combinatorial obstructions and explicit counterexamples for expansion by odd moves above $s_k$.

6. Multi-Dimensional Three-Move Games and Segmentation Phenomena

Three-move rulesets in two or more dimensions reveal "mosaic" and "slice" structures in the game-board segmentation. Symmetric-additive and arithmetic-additive families admit complete solutions via segmentwise coloring algorithms (termed "coloring automaton" Editor's term). In the symmetric-additive case $S=\{(a,b),(b,a),(a+b,b+a)\}$, $\mathcal{P}$-positions fill segments defined by rational slopes, recursively closed under specific translation vectors.

Experimentally, for many small three-move sets, the first quadrant decomposes (modulo a finite "corner") into finitely many semi-infinite slices, each bounded by rational slope lines. The coloring automaton precisely identifies $\mathcal{P}$-positions with periodic patterns within each segment (Larsson et al., 2023).

Open conjectures propose that every three-move $d$-dimensional game admits segmentation into finitely many semi-infinite slices solved by such automata, and that $k$-move games attain at most $k+1$ slices. Computational complexity is open: two-move games are solvable in linear time, but the complexity for three moves remains unresolved.

7. Open Problems and Directions

Classification of the periodicity and nim-sequence behavior for all three-move sets is incomplete; several family subcases (e.g., $S(a,b,a+b)$ with even $k$) lack explicit solution. The full structure of expansion sets and rigidity for general sets needs further exploration (Ho, 2012). In higher dimensions, a natural necessary and sufficient condition ensuring solvability by a coloring automaton remains open, as does the problem of computational complexity for decision queries in three-move games.

The inverse construction for Beatty pairs, the periodic mosaic phenomenon, and outcome segmentation remain central research themes, with implications for combinatorial game theory and algebraic combinatorics (2208.00041, Larsson et al., 2023).

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Table 1: Summary of Key Algorithmic and Structural Features

Aspect	Single-Heap $|S|=3$	Multi-Dimensional
Solution Recurrence	PTFN or mex/Grundy	Coloring Automaton, mex
$\mathcal{P}$-position pattern	Ultimately periodic	Row/column periodic, slices
Explicit formulas	Exists for many families	Segment-wise constructions
Expansion set	Computable, conjectural rigidity	Yet to be classified
Beatty connection	Extensive via recurrences	Present in variants

The theory of three-move subtraction games forms a nexus of combinatorial periodicity, algebraic structure, extremal rigidity, and open-ended complexity, with explicit solutions available for key families and deep connections to number theory and game dynamics.