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  "body": "\n## Statement\n\nA theorem called \"the ultimate Gilbreath principle\" states that, for a permutation $\\pi$ of $\\{1, 2, 3,\\dots, n\\}$, the following four properties are equivalent:\n\n- $\\pi$ is a Gilbreath permutation.\n- For each $j$, the top $j$ cards $\\pi(1),\\dots \\pi(j)$ are distinct modulo $j$.\n- For each $j$ and $k$ with $kj\\le n$, the $j$ cards $\\pi\\bigl((k-1)j+1\\bigr),\\pi\\bigl((k-1)j+2\\bigr),\\dots,\\pi(kj)$ are distinct modulo $j$.\n- For each $j$, the top $j$ cards are consecutive in $1,2,\\dots, n$.\n",
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