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Unsolved Math: The No-Three-In-Line Problem

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.