# Professor Halfbrain and the fantasy knight

Professor Halfbrain owns a $99\times99$ board for fantasy chess, whose rows are numbered consecutively from $1$ to $99$ and whose columns are also numbered consecutively from $1$ to $99$. A fantasy knight can jump from a square in the $k$-th column to any square in the $k$-th row (and can jump to no other square); note that if the knight can jump from square $x$ to square $y$, then this does not mean that it can also jump from square $y$ to square $x$.

The professor claims that there exists a closed fantasy knight tour on the chessboard that makes the knight visit every square exactly once, and in the end takes it back to its starting square.

Question: Is Halfbrain's claim indeed true, or has the professor once again made one of his mathematical blunders?

Let $(x,y)$ be the square in row $x$, column $y$, so that a fantasy knight can move from $(x,y)$ to $(y,z)$. A closed tour is described by a cyclic sequence $$x_0,x_1,x_2,\ldots,x_{99^2-1},x_{99^2}=x_0,$$ where the knight moves from $(x_0,x_1)$ to $(x_1,x_2)$, then to $(x_2,x_3)$, and so on up to $(x_{99^2-1},x_0)$, then finally back to $(x_0,x_1)$. Each square is visited exactly once, so this is an example of a de Bruijn sequence (specifically a $99$-ary de Bruijn sequence of order $2$). De Bruijn sequences are known to exist (the Wikipedia article describes a construction), so Halfbrain's claim is true.