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  "body": "\n## Problem\n\nFor an integer $n \\ge 1$, consider the $n \\times n$ grid $\\set{1, \\dots, n}^2$ of points in the plane.\nFind the largest number of points of the grid that can be chosen so that no three chosen points are [[collinear|0AFNNTRTJE]].\nIn particular, decide for which $n$ this number equals $2n$.\n\n## Remarks\n\nHenry Dudeney posed the problem in 1900.\nEach of the $n$ rows can contain at most $2$ chosen points, so by the [[pigeonhole principle|0HGVCWQGY3]] at most $2n$ points can be chosen.\nThe bound $2n$ is attained for all $n$ up to at least $46$, by solutions found with computer searches.\nFor a prime $p$, the $p$ points $(x, x^2 \\bmod p)$ with $0 \\le x \u003c p$ have no three in a line, an observation of Erdős.\nIndeed, three of them on a line would give three roots in the field $\\F_p$ of a nonzero polynomial of degree at most $2$.\nHall, Jackson, Sudbery and Wild refined such constructions in 1975.\nThey showed that for every $\\varepsilon \u003e 0$ and all large $n$, at least $(3/2 - \\varepsilon)n$ points can be chosen.\nIt is conjectured that for large $n$ the maximum is strictly less than $2n$.\n\n## Explanations\n\n### L1 human\n\nOn an $n$ by $n$ board of dots, choose as many dots as you can so that no three of your dots lie on a straight line, in any direction.\nTwo dots per row is the most you can hope for, which gives $2n$.\nFor small boards one can reach $2n$, but whether this works for all boards is unknown.\n",
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      "first_stated": "1900",
      "posed_by": "Henry Dudeney"
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    "kind": "problem",
    "msc": [
      "52C10",
      "51M04"
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    "name": "No-three-in-line problem",
    "same_as": {
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      "wikipedia": "No-three-in-line problem"
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        "url": "https://en.wikipedia.org/wiki/No-three-in-line_problem"
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    "tag": "TS442VKWAG",
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  "link": "https://tamnd.github.io/math2/node/TS442VKWAG/",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/TS4/TS442VKWAG.md",
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  "tag": "TS442VKWAG",
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