# 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 \(k \le 2n.\)

Questions:

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 \(n = 1,\,2,\,3,\,4,\,5.\)