What does this research mean for the field?

The exact maximum number of points in an n×n×n grid with no three collinear points is 1, 8, 16, 28, 40, and 64 for n=1 through 6, and a(p) ≥ p² for every prime p. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine exact values, certified lower bounds, and algebraic constructions for the maximum number of non-collinear points in an n×n×n grid.

August 22, 2026Open Access

Certified exact values for the no-three-in-line problem in the three-dimensional grid

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Authors

AKAleksei KudriashovNational Heritage Institute

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Implication

Computational study determines exact values and bounds for 3D grid point configurations without collinear triples, establishing foundational limits in discrete geometry.

Key Points

To determine exact values, certified lower bounds, and algebraic constructions for the maximum number of non-collinear points in an n×n×n grid.
Encoded grid collinearity constraints into propositional formulas across n³ variables, requiring at most two points on every line with three or more lattice points.
Employed a certified decision procedure with DRAT certificates verified by independent checkers for unsatisfiability and exhaustive triple-checking integer arithmetic for witness lower bounds.
Constructed a general lower bound for prime dimensions using an anisotropic binary quadratic form.
Determined exact maximum non-collinear grid points for sizes one through six: a(1..6) = 1, 8, 16, 28, 40, 64.
Established certified lower bounds of a(7) ≥ 71 and a(8) ≥ 89, and proved an algebraic lower bound of a(p) ≥ p² for every prime p.
Identified an anomalous layer structure at n = 5 where requiring outer layers to achieve planar maxima yields 39 points compared to the global optimum of 40.

Cite This Study

Aleksei Kudriashov (2026) studied this question.

synapsesocial.com/papers/6a895f74ca7ade938187e16f https://doi.org/10.5281/zenodo.22019280

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Implication

Computational study determines exact values and bounds for 3D grid point configurations without collinear triples, establishing foundational limits in discrete geometry.

Certified exact values for the no-three-in-line problem in the three-dimensional grid

Discussion

Member takes

Implication

Key Points

Cite This Study

Certified exact values for the no-three-in-line problem in the three-dimensional grid

Discussion

Member takes

Implication

Key Points

Cite This Study

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