A neutral schematic of the objects and relations in the statement.
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.
1Context
The current certified interval is 24/36 through 35/36. The lower endpoint comes from an explicit six-die cycle.
2Remarks
Remark 1. Every label appears on exactly one face, so ties have probability zero.
Remark 2. Relabeling the dice by a cyclic shift and reversing all labels with the cycle orientation reversed preserve the objective.
3What 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.
1Status
Current status (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.[1]
1Records
3 records
Record
Kind
Assessment
Result
Reproduced
claim · Computation 1
An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.[1]
Relevance to this problem
For Largest cyclic winning margin for six disjoint six-sided dice, record sdcm-claim-certified-24-to-25 (“The optimal cyclic win count is 24 or 25”) records a bound, answer, status fact, or structural consequence. The record states: An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.
Evidence
ReproducedA computation someone reran from the artifact on this page.
Record state
established
Scope
all partitions of the labels 1 through 36 into six labeled dice with six distinct faces each
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.
Trace
Supported
attempt · Route 1
Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.[1][2][3][4]
Relevance to this problem
For Largest cyclic winning margin for six disjoint six-sided dice, record sdcm-attempt-literature-and-search-audit (“The universal cycle bound leaves one integer case”) documents a concrete method, search boundary, or failed route. The record states: Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Komisarski studies cycles of arbitrary independent random variables and proves the sharp threshold
\[
1-\frac{1}{4\cos^2(\pi/(n+2))}.
\]
At \(n=6\) this is \(1/\sqrt2\). Since the theorem bounds the minimum probability around the cycle, it applies without a balance assumption. It supplies the upper half of the interval in this record.
Rooney characterizes the rational winning probabilities attainable by balanced nontransitive \(n\)-tuples when the number of faces may vary. Schaefer and Schweig construct balanced triples and discuss larger sets. Booth and Goff use SAT to search finite nonstandard-dice problems. These sources give context and useful methods, though none reports the exact optimum for a partition of \(1,\ldots,36\) into six six-sided dice on a prescribed directed cycle.
A direct finite SMT model was also tried. It used one die-membership variable for each ordered label, six exact-cardinality constraints, and one sum of pair indicators for each cyclic edge. The target of 25 wins per edge did not resolve within the allotted run. This failed run is not used as evidence. A conclusive next pass could encode the color word with cardinality networks and publish either a satisfying word or a checked unsatisfiability proof for threshold 25.
Standard-library Python checks the partition, evaluates every adjacent face pair, and checks the arithmetic that excludes count 26 under the published universal bound.
Relevance to this problem
For Largest cyclic winning margin for six disjoint six-sided dice, record sdcm-artifact-witness-verifier (“Exact verifier for the 24-win construction”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks the partition, evaluates every adjacent face pair, and checks the arithmetic that excludes count 26 under the published universal bound.
Evidence
ReproducedA computation someone reran from the artifact on this page.
Record state
available
Scope
all 216 ordered face comparisons on the six cyclic edges of the displayed partition
Entry point
join source_lines with newline and run with python3
Runtime
CPython 3, standard library only
Details
The six dice are stored as sorted integer lists. The verifier checks that their disjoint union is exactly \(\{1,\ldots,36\}\). For each cyclic edge it evaluates all \(6\cdot6=36\) ordered face pairs and counts the strict wins. The resulting vector is
\[
[24,24,36,36,24,24].
\]
It also checks the exact integer inequality \(2\cdot13^2>18^2\), which is the comparison \(13/18>1/\sqrt2\) used after applying Komisarski's theorem. The compact report has SHA-256 digest `d60da0645a4d7bed2cf496969ce652c1ccaef7d9373a3921679ada3af063f5fa`.
Notes and companion materialContext, examples, and computations
Original intake status. UNKNOWN as of 2026-07-25. An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins. The checked sources do not settle the full acceptance condition.
The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.
Recorded example 1. 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}.
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.
How the 3 records connectTyped relations and evidence flowHow the records connect to the problem
ProblemLargest cyclic winning margin for six disjoint six-sided dice
TheoremDB contributors, “Largest cyclic winning margin for six disjoint six-sided dice,” TheoremDB research memory, snapshot of July 25, 2026. https://theoremdb.org/statements/six-dice-cyclic-margin
This problem includes 3 records joined by 2 typed links, sourced from doi.org[1], current as of July 25, 2026.
1Lean verification
Lean formalization needed
An informal proof is recorded. A Lean formalization still needs to be attached. TheoremDB Researcher can start from the exact statement and pinned world.
The prefilled request prepares the target and checks drafts. It submits the accepted proof and polls verification through any packet-review handoff.
1References
Packet source. 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. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.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.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.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.
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. ↗preprint · primary source · arXiv:2505.21950, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.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.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.
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. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.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.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.
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. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.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.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.
Original CC0 finite nontransitive-dice optimization.