No three in a line

Below are some of the questions that I have, regarding these no 3 in a line puzzles. Feel free to contribute any thoughts or additional questions of your own.

Known result:

No 3 points can occur in a row. Thus, a crude upper-bound for the maximum number of points \(k,\) in an \(n \times n\) grid, is \(k \le 2n.\)


In the video, solutions are demonstrated for \(n \times n\) grids. What about \(m \times n\) grids?

With the \(4 \times 4\) grid, we can place at most 8 points without having 3 in a line. What if our goal was instead to avoid placing 4 in a line? How many points could we place in that case? 1 and 2 are uninteresting, as 1 point always sits on a line, and 2 points always define a line.

Given a triangular grid, \(n\) points wide, what's the maximum number points we can place without having 3 in a line?

What is the maximum number of points that we can place in an \(n \times n \times n\) grid, such that no 3 points form a line. Is this question interesting? I ask this because 2 random lines in the plane likely intersect, but 2 random lines in 3D likely don't.

Besides triangular grids, and 3D grids, are there any other arrangements of dots that are interesting?

This is a very old problem. What interesting results, related problems, generalizations, etc. can you find by Googling the problem? I see Wikipedia has a page titled "No-three-in-a-line problem," which is probably a good place to start.

Call this a width 5 triangular grid, because there are 5 empty circles along each side. How many circles can you fill without making 3 in a line?

I'm pretty sure that 6 is the max you can place for a width 4 triangular lattice:

The max you can place for a width 3 triangular lattice is 4:

I'm pretty sure the solution for \(n = 5\) is 7. It's very difficult to find! The solution is pictured below. As you can see, there are lots of lines to consider! Even having this solution, I wonder if there are others. I'm also wondering about \(n = 6\) now. The sequence so far is 1, 3, 4, 6, 7, for \(n = 1,\,2,\,3,\,4,\,5.\)