A while ago it was noted that this piece goes through a cycle of 6 groups of squares, so htat it can return to a square only in a multiple of 6 moves. This cycle is due to two overlapping cycles. As it has a subset of the Knight move, it is colourswiutching as the Knight is and so takes a multiple of 2 moves to return to a square of the same colour. It also always moves either 2 ranks forward or 1 back, so that it takes a multiple of 3 moves to return to the same rank. The same actual square is both the same colour and the same rank, therefore it can only return in a multiple of 3x2=6 moves.
[Note that a Crabrider can return in 4 moves e.g. White Crabrider g1-h3-g5-c3-g1, and in 5 e.g. White Crabrider g1-h3-d1-c3-e2-g1. How many other non-multiples of 6, odd or even, can you spot on an 8x8 board?]
Now consider the Lobster=Tusk+Crosscoward. Now that can return in an odd number of moves - a1-c3-b2-a1 - but still always moves either 2 ranks forward or 1 back and so takes a multiple of 3 moves - odd or even - to return to the same cell. So, there is after all, George Duke may be pleased to knoiw, a multiple-of-three equivalent of colourswitching. It is more evident on hex boards, where the Forewazir, Hindwazir, Foredabbaba, Hinddabbaba, and the Ringworld Chess Wazir-Dabbaba and Dabbaba-Wazir puppeteers have this characteristic. Perhaps there needs to be a new terminology. How about a 'Switching Cycle' which is 1 for non-Switching pieces; 2 for the likes of the square-board Wazir, Ferz, Knight, Camel, et cetera - as well as the hex-board Wazirranker, Wazirfiler, Dabbaranker, et cetera; 3 for the square-board Lobster and the above-listed Fore- and Hind- pieces; and 6 for the Crab here?
[Note that a Crabrider can return in 4 moves e.g. White Crabrider g1-h3-g5-c3-g1, and in 5 e.g. White Crabrider g1-h3-d1-c3-e2-g1. How many other non-multiples of 6, odd or even, can you spot on an 8x8 board?]
Now consider the Lobster=Tusk+Crosscoward. Now that can return in an odd number of moves - a1-c3-b2-a1 - but still always moves either 2 ranks forward or 1 back and so takes a multiple of 3 moves - odd or even - to return to the same cell. So, there is after all, George Duke may be pleased to knoiw, a multiple-of-three equivalent of colourswitching. It is more evident on hex boards, where the Forewazir, Hindwazir, Foredabbaba, Hinddabbaba, and the Ringworld Chess Wazir-Dabbaba and Dabbaba-Wazir puppeteers have this characteristic. Perhaps there needs to be a new terminology. How about a 'Switching Cycle' which is 1 for non-Switching pieces; 2 for the likes of the square-board Wazir, Ferz, Knight, Camel, et cetera - as well as the hex-board Wazirranker, Wazirfiler, Dabbaranker, et cetera; 3 for the square-board Lobster and the above-listed Fore- and Hind- pieces; and 6 for the Crab here?