Two Dimensional Subtraction -- Transfer Games

Alon Danai; Paul Ellis; Thotsaporn Aek Thanatipanonda

arxiv: 2602.14325 · v2 · pith:TUQ3HG4Cnew · submitted 2026-02-15 · 🧮 math.CO

Two Dimensional Subtraction -- Transfer Games

Alon Danai , Paul Ellis , Thotsaporn Aek Thanatipanonda This is my paper

Pith reviewed 2026-05-25 06:59 UTC · model grok-4.3

classification 🧮 math.CO

keywords nim-valuesperiodicitysubtraction gamestransfer gamesimpartial gamestwo pilesGrundy numberscombinatorial game theory

0 comments

The pith

Nim-values of a large class of two-dimensional subtraction-transfer games are periodic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that nim-values in a broad family of two-pile impartial games become periodic after a finite threshold. Moves allow either removing tokens from one pile or transferring tokens between the two piles. Periodicity lets the entire sequence of values be described by a finite list plus a repeating cycle. The authors compute exact periods for many specific move sets and introduce several new notions of periodicity to capture the observed patterns.

Core claim

The nim-values of a large class of two-dimensional subtraction-transfer games are periodic. These impartial normal-play games have two piles of tokens; a move consists of subtracting a positive number of tokens from one pile or transferring a positive number from one pile to the other. For many concrete subtraction and transfer rules the exact period is calculated, and several new notions of periodicity are developed.

What carries the argument

The nim-value (Grundy number) function on game positions, shown to be eventually periodic for the defined large class of subtraction and transfer move sets.

If this is right

After a finite number of positions the outcome of every larger position is determined by its size modulo the period.
Exact periods are obtained for many individual games, giving complete tables of all nim-values.
New notions of periodicity distinguish different degrees of regularity in the value sequences.
The periodicity supplies a finite description that replaces infinite case-by-case computation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same periodicity technique may apply to three-pile or higher versions with analogous moves.
Automated search over move sets could identify the boundary between periodic and non-periodic cases.
The repeating structure may correspond to linear relations among the allowed subtraction and transfer vectors.

Load-bearing premise

The specific subtraction amounts and transfer rules place the game inside the large class for which nim-values are guaranteed to repeat after some point.

What would settle it

A concrete subtraction-transfer rule set inside the claimed class whose nim-values fail to repeat after any finite threshold.

read the original abstract

We generalize the results and conjectures of Tam\'{a}s Lengyel, showing that the \textsc{nim}-values of a large class of two-dimensional subtraction-transfer games are periodic. These are impartial, normal-play games with two piles of tokens, where players alternate either taking some tokens from a pile or transferring tokens from one pile to the other. In many cases, we calculate the exact period. We also develop several new notions of periodicitiy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Lengyel on nim-value periodicity to more two-pile subtraction-transfer games with some exact period calculations, but the abstract leaves the supporting arguments unshown.

read the letter

The main takeaway is that the authors generalize Lengyel's periodicity results for nim-values to a larger class of impartial two-pile games that allow both subtraction and transfers between piles. They report exact periods in many cases and introduce a few new periodicity notions. That is the concrete advance here. The work stays within the standard framework of normal-play combinatorial game theory and builds directly on the cited prior results without obvious circularity. The exact period calculations are the part that could be immediately useful to someone computing Grundy numbers for similar games. The new notions of periodicity are presented as additional tools rather than a complete overhaul. The soft spot is the lack of any proof outline or sample verification in the abstract, which makes it impossible to check how the induction or case analysis actually proceeds or whether boundary positions are handled cleanly. The restriction to the subclass where periodicity holds is the usual way these statements are made and does not create an extra problem on its own. If the full paper supplies reproducible calculations or machine-checkable steps for the periods, that would strengthen the claim considerably. This is a paper for people already working on subtraction games or Grundy periodicity in impartial games. A reader who knows Lengyel's conjectures would see the extension as a natural next step and might pick up the new periodicity ideas for their own calculations. It is not broad enough to interest people outside combinatorial game theory. I would send it to peer review so the details of the arguments and the exact period claims can be examined.

Referee Report

0 major / 2 minor

Summary. The paper generalizes results and conjectures of Tamás Lengyel by establishing that the nim-values (Grundy numbers) of a large class of two-pile subtraction-transfer games are periodic under normal play. These impartial games allow moves that subtract tokens from one pile or transfer tokens between the two piles. The authors compute exact periods in many cases and introduce several new notions of periodicity.

Significance. If the periodicity claims hold, the work extends the scope of known periodic Grundy-number behavior in combinatorial game theory to include transfer moves, which may enable closed-form or efficient evaluation of positions in infinite families. The new periodicity notions could prove useful beyond this specific setting for analyzing other impartial games.

minor comments (2)

[Abstract] The abstract states the main periodicity result but does not indicate the proof strategy or verification method; a one-sentence outline of the approach would help readers assess the scope immediately.
[Introduction] Clarify the precise definition of the 'large class' of games early in the introduction, including any restrictions on subtraction sets or transfer rules that are required for the periodicity to hold.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of the paper's contributions, and recommendation of minor revision. No specific major comments appear in the report, so we have nothing further to address point by point at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generalizes external results and conjectures from Tamás Lengyel on nim-value periodicity for two-dimensional subtraction-transfer games. The abstract and context reference this prior independent work without any self-citation chains, fitted inputs renamed as predictions, or self-definitional reductions. The restriction to a 'large class' of games is a standard scope definition for stating the periodicity result, not a circular premise. No load-bearing steps reduce to the paper's own inputs by construction; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard Sprague-Grundy theorem for assigning nim-values to impartial positions; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)

standard math Sprague-Grundy theorem assigns nim-values (Grundy numbers) to positions in impartial normal-play games.
Required to define the nim-values whose periodicity is claimed.

pith-pipeline@v0.9.0 · 5596 in / 1082 out tokens · 32241 ms · 2026-05-25T06:59:00.575341+00:00 · methodology

Two Dimensional Subtraction -- Transfer Games

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)