# P101194: Density of losing positions in the subtract-a-cube game

- ID: `P101194`
- Reference: `subtract-a-cube-density`
- Page: https://theoremdb.org/statements/P101194
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/subtract-a-cube-density?representation=markdown
- Record maturity: Archived record

## The problem

Two players alternately remove a positive perfect cube of tokens from a single pile; the player unable to move loses. Let \(L(N)\) be the number of losing starting positions \(n \le N\). Does \(L(N)/N \to 0\) as \(N \to \infty\)?

### Problem setup

- **Remark.** The allowed moves are 1, 8, 27, 64, 125, and so on.
- **Definition.** A position is losing (a P-position) when its Sprague-Grundy value is 0.

### What counts as a solution

- Prove that the density of losing positions tends to zero, or prove that it does not.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

The losing positions form a cube-difference-free set, so the polynomial Szemerédi theorem forces their natural density to be zero. [1](#reference-1)

**Recorded scope.** the full set of losing positions in the normal-play subtract-a-cube game

[Read this exact record and its supporting evidence](/records/sac-claim-density-zero)


### Resolution argument

Let \(C\) be the set of P-positions. Two elements of \(C\) cannot differ by a positive cube, since the larger position could move to the smaller one. Suppose \(C\) had positive upper density. Bergelson and Leibman's polynomial extension of Szemerédi's theorem, applied to \(p(t)=t^3\), would give \(x,y\in C\) and \(t\geq 1\) with \(x-y=t^3\). This contradicts the defining property of P-positions. Therefore \(\limsup_{N\to\infty}|C\cap[1,N]|/N=0\), and the nonnegative ratios converge to zero.

The argument settles the database question. It applies more generally when the legal moves are the positive values of an integer-valued intersective polynomial with zero constant term.

## Research packet

### Saved solution and evidence

**Proposition 1 (Subtract-a-cube losing positions have density zero).** The losing positions form a cube-difference-free set, so the polynomial Szemerédi theorem forces their natural density to be zero.

Let \(C\) be the set of P-positions. Two elements of \(C\) cannot differ by a positive cube, since the larger position could move to the smaller one. Suppose \(C\) had positive upper density. Bergelson and Leibman's polynomial extension of Szemerédi's theorem, applied to \(p(t)=t^3\), would give \(x,y\in C\) and \(t\geq 1\) with \(x-y=t^3\). This contradicts the defining property of P-positions. Therefore \(\limsup_{N\to\infty}|C\cap[1,N]|/N=0\), and the nonnegative ratios converge to zero.

The argument settles the database question. It applies more generally when the legal moves are the positive values of an integer-valued intersective polynomial with zero constant term.

### Background and intake notes

Screened in the same sweep as the triangular version. The two behave very differently in the computed range, which is the reason both are recorded.

- Original intake status: Status not established. No literature search was performed. Record this as a candidate question, not as a known open problem.
- No literature search was carried out.
- Because cubes have counting function N^(1/3) rather than N^(1/2), any argument that settles the square case need not transfer, and the observed densities differ by an order of magnitude.

- Recorded example: The first losing positions are 2, 4, 6, 9, 11, 13, 15, 18, 20, 22, 24, 34.

### Open directions

- **Route 1** (conjectured): Density zero leaves a quantitative problem: estimate the number of subtract-a-cube P-positions through \(N\). [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (executable material): Exact dynamic programming finds 21,912 positive P-positions through 200,000 and a density of 0.10956 at the endpoint.

### Computational notes

- Exact Sprague-Grundy computation for all n up to 200000. Density of losing positions: 0.16840 up to 10000, 0.13630 up to 50000, 0.13002 up to 100000, 0.10956 up to 200000. Total losing positions up to 200000: 21912. Maximum Grundy value observed: 20.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `subtract-a-cube-density`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## Lean verification

A solution has been submitted and awaits independent review.

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## References

1. <a id="reference-1"></a>Vitaly Bergelson and Alexander Leibman, “Polynomial extensions of van der Waerden’s and Szemerédi’s theorems,” Journal of the American Mathematical Society 9 (1996), 725–753. Section 0.2, p. 726. https://doi.org/10.1090/S0894-0347-96-00194-4
   - Also cited at Bergelson and Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, JAMS 9 (1996), section 0.2, page 726
   - journal_article; primary source; checked 2026-07-28
   - Open copy: https://people.math.osu.edu/leibman.1/preprints/psz.pdf
   - Source use: citation_only
   - The polynomial recurrence theorem implies that every positive-density set has two elements differing by a nonzero cube.
   - Source named by the research packet.
2. <a id="reference-2"></a>Antal Balog, János Pelikán, János Pintz, and Endre Szemerédi, “Difference sets without κ-th powers,” Acta Mathematica Hungarica 65 (1994), 165–187. Quantitative bounds for sets whose difference sets avoid perfect powers. https://doi.org/10.1007/BF01874311
   - Also cited at Quantitative direction informed by Balog, Pelikán, Pintz, and Szemerédi, Difference sets without k-th powers, 1994
   - journal_article; secondary source; checked 2026-07-28
   - Source use: citation_only
   - Provides quantitative context beyond the qualitative density-zero conclusion.