What does this research mean for the field?

A reproducible computational package using exhaustive finite-enumeration establishes that a 10x10 grid subset with distinct undirected slopes has at most 10 points, and the maximum base-block orbits for cyclic triple packings by 5-subsets of Z/31Z under reflection invariance is 12. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Resolve exact bounds for two extremal combinatorics problems involving slope-distinct subsets on a finite grid and reflection-invariant cyclic triple packings using reproducible exhaustive enumeration.

September 6, 2026Open Access

Reproducible exhaustive searches for slope-distinct grid sets and reflection-invariant cyclic packings

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BYByungwoong Yoo

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Implication

Computational study demonstrates extremal bounds for grid sets and cyclic packings, highlighting reproducible finite-enumeration workflows.

Key Points

Resolve exact bounds for two extremal combinatorics problems involving slope-distinct subsets on a finite grid and reflection-invariant cyclic triple packings using reproducible exhaustive enumeration.
Formulated finite-enumeration algorithms and mathematical reductions to perform exact computational searches on a 10-by-10 grid.
Conducted exhaustive searches and witness generation for cyclic triple packings by five-subsets of Z/31Z under reflection invariance.
Employed an AI-assisted computational pipeline involving code replay, historical table checks, and independent algorithmic audits.
Exact searches confirmed that any subset of a 10-by-10 grid with distinct undirected slopes contains at most 10 points.
Exhaustive search established that cyclic triple packings by five-subsets of Z/31Z have an exact maximum of 12 base-block orbits under reflection invariance, leaving the unrestricted maximum between 12 and 13.

Cite This Study

Byungwoong Yoo (2026) studied this question.

synapsesocial.com/papers/6a9d1e3328139818eab20efd https://doi.org/10.5281/zenodo.22311019

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Implication

Computational study demonstrates extremal bounds for grid sets and cyclic packings, highlighting reproducible finite-enumeration workflows.

Reproducible exhaustive searches for slope-distinct grid sets and reflection-invariant cyclic packings

Discussion

Member takes

Implication

Key Points

Cite This Study

Reproducible exhaustive searches for slope-distinct grid sets and reflection-invariant cyclic packings

Discussion

Member takes

Implication

Key Points

Cite This Study

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