How many points can you place on an n x n grid without having any three of them lie in a straight line? It turns out, we don't know for sure! Find out how tantalizingly close we've come to solving this century-old "No-three-in-line problem", and see how mathematicians approach unsolved problems in general. We'll go through general math principles, a couple of side quests into algebra, and even...just taking a guess? Let's explore!

This video is an entry in 3Blue1Brown's Summer of Mathematical Exposition 3.

Chapters:

0:00 Intro
0:28 Starting off
1:28 An upper bound
3:02 A lower bound
7:16 A better lower bound
9:19 Taking a guess?
12:07 Conclusion

Corrections:

0:44 The check marked point should not be in line with points 6 and 8
5:50 The bottom equation should say "mod 3" instead of "mod 5"

Citations:

Wolfram Mathworld
52 x 52 solution: Achim Flammenkamp
Paul Erdős: Kmhkmh, CC BY 3.0
Richard Guy: Thane Plambeck, CC BY 2.0
End Music: HOEDOWN by Jason Shaw, CC BY 3.0
Manim community edition: by The Manim Community Developers. https://www.manim.community/
Hall et. al. paper: https://doi.org/10.1016/0097-3165(75)...
Guy & Kelly paper: https://doi.org/10.4153/CMB-1968-062-3
With gratitude to Gabor Ellmann for his email.