No-three-in-line $D(9)$

18 points of the 9 by 9 grid with no three on a line

The largest number of points of the $9 \times 9$ grid with no three on a line.

Value: $18$ Verified lower bound

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

Lower bound: $18$ Checked construction
no-three-in-line-9-lower-18, submitted by daniel-munro, 2026-11-05
[construction from The No-Three-in-Line Problem, Achim Flammenkamp, 2026-10-28]
Upper bound: None yet

Updates from the literature

Upper bound: $18$
Trivial or easy according to the secondary source.
[via No-three-in-line problem - Wikipedia]
1975-05 Lower bound: $18$
R.R. Hall, T.H. Jackson, A. Sudbery and K. Wild, Some advances in the no-three-in-line problem, Journal of Combinatorial Theory, Series A 18 (1975) 336-341.
[via The No-Three-in-Line Problem, Achim Flammenkamp, 2026-10-28]