For ,
it is possible to select lattice points with such that no three are in a straight line
(where "straight line" means any line in the plane--not just a horizontal
or vertical line). The number of distinct solutions (not counting reflections and
rotations) for , 2, ..., are 1, 1, 4, 5, 11, 22, 57, 51, 156 ... (OEIS A000769). For large , it is conjectured that it is only possible to select at most
lattice points with no three collinear, where
(1)
(2)
(OEIS A093602; Guy, pers. comm., Oct. 22, 2004), correcting Guy and Kelly (1968) and Guy (1994, p. 242) who found .
Solutions for and with quarter-turn () rotational symmetry are illustrated above. The solution was found by Flammenkamp. On Mar. 23, 2026,
Thomas Prellberg found two solutions for in Flammenkamp's rot4 class (quarter-turn rotational symmetry),
which are the largest currently known. Configurations with smaller true symmetry
are also known for intervening sizes, including , 59, 61, and 63 in Flammenkamp's rct4 class (quarter-turn
symmetry except on the long diagonals, with true symmetry either half-turn rotational
symmetry or both diagonal reflections), and further rot4 configurations are known
for ,
60, 62, 64, and 66. Flammenkamp gives many additional configurations and counts.
,
it is possible to select 
lattice points with ![x,y in [1,n]](/images/equations/No-Three-in-a-Line-Problem/Inline3.svg)
such that no three are in a straight line
(where "straight line" means any line in the plane--not just a horizontal
or vertical line). The number of distinct solutions (not counting reflections and
rotations) for 
, 2, ..., are 1, 1, 4, 5, 11, 22, 57, 51, 156 ... (OEIS A000769). For large 
, it is conjectured that it is only possible to select at most
lattice points with no three collinear, where
.
and 
with quarter-turn (
) rotational symmetry are illustrated above. The 
solution was found by Flammenkamp. On Mar. 23, 2026,
Thomas Prellberg found two solutions for 
in Flammenkamp's rot4 class (quarter-turn rotational symmetry),
which are the largest currently known. Configurations with smaller true symmetry
are also known for intervening sizes, including 
, 59, 61, and 63 in Flammenkamp's rct4 class (quarter-turn
symmetry except on the long diagonals, with true symmetry either half-turn rotational
symmetry or both diagonal reflections), and further rot4 configurations are known
for 
,
60, 62, 64, and 66. Flammenkamp gives many additional configurations and counts.
