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I understand why a king's movement is aperiodic, because it can return to the start point in 2 or 3 movements. If the king starts at 2,2 the moves for the 2 movements scenario would be 2,2 -> 2,3 -> 2,2 and for the 3 movement scenario would be 2,2 -> 2,3 -> 3,3 -> 2,2. Since 2 and 3 have a highest common factor of 1 the chain is aperidic.

However for a bishop I can't find a scenario where it would take an odd number of movements for it to return to the start position. Since it can only move diagonally wouldn't it return to every cell it moved to so it can get back to the start position. This is the only scenario I can think about for a bishop returning to the start point. 2,2 -> 3,3 -> 2,2 or 2,2 -> 3,3 -> 4,4 -> 3,3 -> 2,2.

Thank you very much, any inout would be extremely helpful.

MrFuzzy
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  • I can't say for sure but I feel like this question is missing a large portion of its context. It seems like you are working in the context of a specific problem which you have not included. – Sriotchilism O'Zaic Dec 06 '17 at 23:51

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The bishop in chess can move $1,1 \to 3,3 \to 2,2 \to 1,1,$ or $a1 \to c3 \to b2 \to a1$ in chess notation, and also $a1 \to b2 \to a1,$ so one can construct an aperiodic movement.

Reiner Martin
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  • I see, thank you very much! For some reason I thought that the bishop was limited to one square per move. The trick here is that each state is a move not a square. Thanks again! – MrFuzzy Dec 07 '17 at 03:59