# P2694: Largest cyclic winning margin for six disjoint six-sided dice
- ID: `P2694`
- Reference: `six-dice-cyclic-margin`
- Page: https://theoremdb.org/statements/P2694
- Record maturity: Reviewed problem with recorded work
## Problem
Partition the labels \(1,\ldots,36\) into six six-element dice \(D_0,\ldots,D_5\). Determine the largest possible value of \(\min_i\Pr(D_i>D_{i+1})\), where indices are modulo \(6\) and each die is rolled uniformly.
### Remarks
- **Remark.** Every label appears on exactly one face, so ties have probability zero.
- **Remark.** Relabeling the dice by a cyclic shift and reversing all labels with the cycle orientation reversed preserve the objective.
### What counts as a solution
- Give a partition attaining the optimal minimum win count and a complete branch, integer-programming, or SAT certificate excluding the next count.
## Status
An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins. [2](#reference-2)
## Work
### Evidence for the current status
**Computation 1 (The optimal cyclic win count is 24 or 25).** An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.
Write
\[
M=36\max_{D_0,\ldots,D_5}\min_i\Pr(D_i>D_{i+1}).
\]
The certified interval is
\[
\boxed{24\leq M\leq25},
\qquad
\boxed{\frac23\leq\max\min_i\Pr(D_i>D_{i+1})\leq\frac{25}{36}}.
\]
The lower bound is attained by
\[
\begin{aligned}
D_0&=\{3,4,5,32,33,34\},&
D_1&=\{1,2,28,29,30,31\},\\
D_2&=\{22,23,24,25,26,27\},&
D_3&=\{16,17,18,19,20,21\},\\
D_4&=\{10,11,12,13,14,15\},&
D_5&=\{6,7,8,9,35,36\}.
\end{aligned}
\]
Its cyclic win counts are \([24,24,36,36,24,24]\).
Komisarski proves that every cycle of six independent random variables with pairwise tie probability zero has some cyclic winning probability strictly below
\[
1-\frac{1}{4\cos^2(\pi/8)}=\frac{1}{\sqrt2}.
\]
The theorem applies directly to fair rolls of these dice. A count of 26 would give probability \(26/36=13/18\), and
\[
\left(\frac{13}{18}\right)^2=\frac{169}{324}>\frac12.
\]
Thus \(13/18>1/\sqrt2\), so six counts of at least 26 are impossible. Integrality gives \(M\leq25\).
The remaining question is whether a partition with all six counts at least 25 exists. No such partition or nonexistence certificate was produced in this research pass.
### Background and intake notes
The current certified interval is 24/36 through 35/36. The lower endpoint comes from an explicit six-die cycle.
- Original intake status: Novelty remains unverified. Nontransitive dice have a broad literature, and no primary-source status audit was completed for this exact six-die partition problem.
- Use integer win counts out of 36. Branching can update only the two cyclic comparisons incident to a moved label.
- Optimizing the average of the six win counts can hide one weak edge. The objective is the minimum count, with the sum useful only as a secondary bound.
- Order-preserving relabeling has already fixed the face labels, so value symmetries are much smaller than arbitrary permutations of 1,...,36.
- Recorded example: Take D_0={3,4,5,32,33,34}, D_1={1,2,28,29,30,31}, D_2={22,23,24,25,26,27}, D_3={16,17,18,19,20,21}, D_4={10,11,12,13,14,15}, D_5={6,7,8,9,35,36}.
### Prior approaches
- **Route 1** (supported): Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search. [2](#reference-2) [1](#reference-1) [3](#reference-3) [4](#reference-4)
### Runnable artifacts
- **Artifact 1** (reproduced): Standard-library Python checks the partition, evaluates every adjacent face pair, and checks the arithmetic that excludes count 26 under the published universal bound.
### Computational notes
- Exact comparison of all 216 ordered face pairs on adjacent dice gave cyclic win counts [24,24,36,36,24,24] for the displayed partition. Seeded label-swap searches also reached minimum count 24 but did not certify optimality.
### Working on this
Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `six-dice-cyclic-margin`, 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.
## References
1. Joshua Rooney, “Winning Probabilities of Balanced and Nontransitive n-tuples of Dice”. arXiv:2505.21950 (2025). 2025 preprint, statement and use of the sharp cycle threshold https://arxiv.org/abs/2505.21950
- preprint; reference source; arXiv:2505.21950, version checked 2026-07-25; checked 2026-07-25
- Source use: citation_only
- For Largest cyclic winning margin for six disjoint six-sided dice: The universal cycle bound leaves one integer case. Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.
2. Andrzej Komisarski, “Nontransitive Random Variables and Nontransitive Dice”. The American Mathematical Monthly 128(5) (2021), 423-434. DOI 10.1080/00029890.2021.1889921. Andrzej Komisarski, Nontransitive Random Variables and Nontransitive Dice, American Mathematical Monthly 128 (2021), 423-434, sharp max-min bound for cycles of n independent random variables; exact witness replay in sdcm-artifact-witness-verifier; Focused searches for six-die cyclic winning margin, balanced nontransitive dice, max-min stochastic precedence cycles, and SAT searches for nonstandard dice, completed 2026-07-25 https://doi.org/10.1080/00029890.2021.1889921
- Also cited at American Mathematical Monthly 128 (2021), 423-434
- Also cited at Andrzej Komisarski, Nontransitive Random Variables and Nontransitive Dice, American Mathematical Monthly 128 (2021), 423-434, sharp max-min bound for cycles of n independent random variables; exact witness replay in sdcm-artifact-witness-verifier
- Also cited at Focused searches for six-die cyclic winning margin, balanced nontransitive dice, max-min stochastic precedence cycles, and SAT searches for nonstandard dice, completed 2026-07-25
- scholarly_publication; reference source; version of record; checked 2026-08-01
- Source use: citation_only
- For Largest cyclic winning margin for six disjoint six-sided dice: The optimal cyclic win count is 24 or 25. An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins. The universal cycle bound leaves one integer case. Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.
- Source named by the research packet.
3. Michael Purcell, “Using a SAT Solver to Find Interesting Sets of Nonstandard Dice”. The American Mathematical Monthly 130(5) (2023), 421-436. DOI 10.1080/00029890.2023.2178218. American Mathematical Monthly 130 (2023), finite SAT encodings for dice https://doi.org/10.1080/00029890.2023.2178218
- scholarly_publication; reference source; version of record; checked 2026-08-01
- Source use: citation_only
- For Largest cyclic winning margin for six disjoint six-sided dice: The universal cycle bound leaves one integer case. Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.
4. James Grime, “The Bizarre World of Nontransitive Dice: Games for Two or More Players”. The College Mathematics Journal 48(1) (2017), 2-9. DOI 10.4169/college.math.j.48.1.2. College Mathematics Journal 48 (2017), 2-9 https://doi.org/10.4169/college.math.j.48.1.2
- scholarly_publication; reference source; version of record; checked 2026-08-01
- Source use: citation_only
- For Largest cyclic winning margin for six disjoint six-sided dice: The universal cycle bound leaves one integer case. Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.