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  "body": "\n## Definition\n\nA *cell* is a unit square $[a, a+1] \\times [b, b+1]$ with $a, b \\in \\Z$.\nA *domino* is the union of two cells that share an edge, so it is a $1 \\times 2$ or $2 \\times 1$ rectangle with integer corners.\nLet $R \\subseteq \\R^2$ be a region that is a union of finitely many cells.\nA *domino tiling* of $R$ is a finite set $D$ of dominoes such that:\n\n1. every domino of $D$ is contained in $R$;\n2. the union of the dominoes of $D$ is $R$;\n3. two distinct dominoes of $D$ have disjoint interiors.\n\n## Remarks\n\nExample: the $2 \\times 3$ rectangle $[0, 3] \\times [0, 2]$ has exactly three domino tilings.\nThey are: three vertical dominoes; one vertical domino on the left and two horizontal ones on the right; and the mirror image of the second.\nMore generally, the $2 \\times n$ rectangle has $F_{n+1}$ domino tilings, where $F_1 = F_2 = 1$ and $F_{k+1} = F_k + F_{k-1}$ are the [[Fibonacci numbers|0GA9K57A39]].\n\nNon-example: remove two opposite corner cells from the $8 \\times 8$ board.\nColour the cells like a chessboard; each domino covers one black and one white cell, but the two removed cells have the same colour.\nSo the remaining $62$ cells have $32$ of one colour and $30$ of the other, and no set of dominoes satisfies clauses 2 and 3.\n\nEquivalently, a domino tiling of $R$ is a [[perfect matching|0NHK9MFFW9]] of the graph whose vertices are the cells of $R$, two cells being adjacent when they share an edge.\n",
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