Why is the movement of a Chess King aperiodic?

Question

Imagine we have a king by itself on a chess board, making random moves around the board. Although it is apparently aperiodic, wouldn't the corresponding Markov chain to the King's movements be periodic since the King could only return to a square i on the board at moves 2,4,6 ... etc. (great common denominator of 2) after initially being at the square?

By definition, aperiodic chains have return times to i with a g.c.d. of 1.

Kings can move diagonally can't they? So you can get a cycle of length 3 by doing left, up, diagonally down to the right? — David, Oct 06 '15 at 00:53
A king can move *up, right, and then down-left*, returning to the same place in 3 moves. — Zev Chonoles, Oct 06 '15 at 00:53
A king can return to its original square in any number of moves greater than $1$. — Brian M. Scott, Oct 06 '15 at 00:53

score 1 · Accepted Answer · answered Oct 06 '15 at 01:02

Your mathematics is correct but your understanding of the rules of chess is not. A king on any square that is not on the edge of the board can move to any of eight squares: the four that share a common edge with the one he's on and the four that share only a common vertex.

Why is the movement of a Chess King aperiodic?

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