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  "body": "\n## Description\n\nLet $k \\ge 1$ and $n \\ge 1$ be integers, and let $A$ be an alphabet with $k$ letters.\nA *de Bruijn sequence* of order $n$ over $A$ is a cyclic word $w = a_0 a_1 \\cdots a_{k^n - 1}$ of length $k^n$ with the following property.\nEvery word of length $n$ over $A$ occurs exactly once as a cyclic factor $a_i a_{i+1} \\cdots a_{i+n-1}$, with indices taken modulo $k^n$.\nSuch sequences exist for all $k$ and $n$.\n\n## Remarks\n\nExample: for $k = 2$ and $n = 3$, the word $00010111$ is a de Bruijn sequence.\nIts eight cyclic factors of length $3$, starting at positions $0$ to $7$, are $000, 001, 010, 101, 011, 111, 110, 100$.\nThese are all the $2^3 = 8$ binary words of length $3$, each once.\n\nExistence follows from graph theory.\nThe de Bruijn graph is the [[directed graph|0KFYGSHHF5]] whose vertices are the words of length $n - 1$, and in which each word $a_1 \\cdots a_n$ is an edge from $a_1 \\cdots a_{n-1}$ to $a_2 \\cdots a_n$.\nEach vertex has in-degree and out-degree $k$, and the graph is strongly connected.\nBy the directed form of [[Euler's theorem|020422MDQF]], it has an [[Eulerian circuit|05NWGBHKSZ]].\nReading the last letters of the edges along this circuit gives a de Bruijn sequence.\n\nThe number of distinct de Bruijn sequences of order $n$ over $k$ letters, counted as cyclic words, is $(k!)^{k^{n-1}} / k^n$.\nFor $k = 2$ and $n = 3$ this is $2^4 / 8 = 2$: the words $00010111$ and $00011101$.\n\n## Explanations\n\n### L1 human\n\nA de Bruijn sequence is a shortest circular string that contains every possible block of a given length.\nThink of a lock with a keypad that opens as soon as the last $n$ digits pressed are the code.\nTyping a de Bruijn sequence tries every code while pressing far fewer keys than typing each code separately.\nFor binary codes of length $3$, the circular string $00010111$ contains all eight codes.\n",
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