For 
,
it is possible to select 
lattice points with ![x,y in [1,n]](/images/equations/No-Three-in-a-LineProblem/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
 |  |  |
(1)
|
 |  |  |
(2)
|
(OEIS A093602; Guy, pers. comm., Oct. 22, 2004), correcting Guy and Kelly (1968) and Guy (1994, p. 242) who found 
.
Selected configurations are illustrated above. Some large known no-three-in-a-line configurations are summarized in the following table. The entries through 
are listed by Flammenkamp.
 | rotation class | discoverer |
| 52 | rot4 | Flammenkamp |
| 65 | rct4 | Heule (Jun. 18, 2026) |
| 67 | rct4 | Heule (Jun. 21, 2026) |
| 68 | rot4 | Prellberg (Mar. 23, 2026) |
| 69 | rct4 | Heule (Jun. 21, 2026) |
| 70 | rot4 | Heule (Jun. 17, 2026) |
| 72 | rot4 | Heule (Jun. 25, 2026) |
Here, rot4 denotes quarter-turn rotational symmetry and rct4 denotes quarter-turn symmetry except on the long diagonals. (The full symmetry is either half-turn rotational
symmetry or both diagonal reflections.) Heule's solutions were found using a Boolean
satisfiability (SAT) solver. The 
, 67, and 69 discoveries fill the remaining gaps up to 
,
so configurations with 
points are known for every 
grid with 
. Heule's 
configuration is the largest currently known. Flammenkamp's
page illustrates the 
configuration and many additional configurations and counts.
Pegg (2026) gives a Wolfram Language notebook for checking configurations by searching for extraordinary lines, i.e.,
lines containing at least three selected points, and reports Prellberg's enumeration
of 118057 solutions for 
.
A minimum variant asks for the smallest no-three-in-a-line subset of an 
grid that is maximal, i.e., such that adding one more
counter on any vacant grid point produces three in a line. The corresponding numbers
for 
,
2, ... begin 1, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 10, ... (OEIS A277433;
Aichholzer et al. 2023). Aichholzer et al. (2023) study this problem
as a general-position version of the geometric dominating
set problem. Pegg's June 2026 post gives additional examples and computations
for this minimum variant.
