{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nCall a region $R$ in the plane a *worm cover* if it has the following property.\nFor every [[plane curve|EWS4WFTA7W]] $\\gamma$ of [[length|0TH78S7GBZ]] $1$, some [[isometry|02HPP5VCZH]] $g$ of the plane satisfies $g(\\gamma) \\subseteq R$.\nFind the smallest possible area of a worm cover.\n\n## Remarks\n\nThe problem is open; most work concerns convex worm covers.\nA disc of radius $1/2$ is such a region, so the answer is at most $\\pi / 4$.\nIndeed, let $m$ be the point of $\\gamma$ at arc length $1/2$ from an end.\nEvery point of $\\gamma$ is joined to $m$ by an arc of length at most $1/2$, so it lies within distance $1/2$ of $m$.\nMuch smaller convex worm covers are known.\n\n## Explanations\n\n### L1 human\n\nPicture a worm of length one that can curl up in any way.\nWhat is the smallest blanket that can always be placed over the worm, whatever its shape?\nA round blanket of diameter one works, but smaller shapes also work, and the best one is not known.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "EWS4WFTA7W"
    ],
    "history": {
      "first_stated": "1966",
      "posed_by": "Leo Moser"
    },
    "kind": "problem",
    "msc": [
      "52C15",
      "52A40"
    ],
    "name": "Moser's worm problem",
    "same_as": {
      "wikidata": "Q15904307",
      "wikipedia": "Moser's worm problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Moser%27s_worm_problem"
      }
    ],
    "tag": "DTVFW3QMNW",
    "uses": [
      "0TH78S7GBZ",
      "02HPP5VCZH",
      "EWS4WFTA7W"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/DTVFW3QMNW/",
  "links": [
    {
      "field": "about",
      "target": "EWS4WFTA7W"
    },
    {
      "field": "uses",
      "target": "0TH78S7GBZ"
    },
    {
      "field": "uses",
      "target": "02HPP5VCZH"
    },
    {
      "field": "uses",
      "target": "EWS4WFTA7W"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DTVFW3QMNW.md",
  "msc": [
    "52C15",
    "52A40"
  ],
  "name": "Moser's worm problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DTV/DTVFW3QMNW.md",
  "status": null,
  "tag": "DTVFW3QMNW",
  "uri": "m2://DTVFW3QMNW",
  "url": "https://tamnd.github.io/math2/wiki/mosers-worm-problem/",
  "writing": {
    "tag": "DTVFW3QMNW",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}