{
  "id": "2405.19589",
  "version": "v1",
  "published": "2024-05-30T00:50:00.000Z",
  "updated": "2024-05-30T00:50:00.000Z",
  "title": "Knights are 24/13 times faster than the king",
  "authors": [
    "Christian Táfula"
  ],
  "comment": "7 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime $b>a \\in \\mathbb{Z}_{\\geq 1}$ such that $a+b$ is odd, define the $(a,b)$-knight and the king as \\[ \\mathrm{N}_{a,b}= \\{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\\}, \\] \\[ \\mathrm{K}=\\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\\} \\subseteq \\mathbb{Z}^2, \\] respectively. One way to formulate this question is by asking for the average ratio, for $\\mathbf{p}\\in \\mathbb{Z}^2$ in a box, between $\\min\\{h\\in \\mathbb{Z}_{\\geq 1} ~|~ \\mathbf{p}\\in h\\mathrm{N}\\}$ and $\\min\\{h\\in \\mathbb{Z}_{\\geq 1} ~|~ \\mathbf{p}\\in h\\mathrm{K}\\}$, where $hA = \\{\\mathbf{a}_1+\\cdots+\\mathbf{a}_h ~|~ \\mathbf{a}_1,\\ldots, \\mathbf{a}_h \\in A\\}$ is the $h$-fold sumset of $A$. We show that this ratio equals $2(a+b)b^2/(a^2+3b^2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-30T00:50:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B75"
    ],
    "keywords": [
      "times faster",
      "infinite chess board",
      "fold sumset",
      "ratio equals",
      "knight reach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}