No-three-in-line $D(21)$

The largest number of points of the $21 \times 21$ grid with no three on a line.

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: $42$ Checked construction
  no-three-in-line-21-lower-42, 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: $42$
  Trivial or easy according to the secondary source.
  [via No-three-in-line problem - Wikipedia]
• 1990 Lower bound: $42$
  Gerken, personal communication to H. Harborth (1990), cited on the no-three-in-line records page.
  [via The No-Three-in-Line Problem, Achim Flammenkamp, 2026-10-28]