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All this discussion on Crabriders, and all this time, and nobody thinks to even mention (much less post a diagram for) a more conventional Crabrider that just goes in a straight line like the Nightrider?
Crabflyer
Crabsidler
Crabdueller
Before dissecting the short-range Lobster, recall Jaguaribe suggested the right Knight and the left Knight split, http://www.chessvariants.org/index/displaycomment.php?commentid=17921. Good compounds to fulfill these two could be Water Rook and left Knight, Land Rook and Right Knight, Water Rook and right Knight, Land Rook and Left Knight, making four paired piece-types for 10x10 Hatch Fantasy Grand different armies. Each piece-type should be valued over 4.0 for 45-point army getting equivalence by adding one Queen-type centrally to the other Fantasy Granders.
Still (re)familiarizing with some of these geometric ways of making/breaking piece-types, I think the Crabrider's paths to return have to be 4, or 5, or 6, or multiples of them or additions of them, the latter like (4+5), (5+5), (5+6). Then player so inclined, or skilled to plan ahead, naturally knows there is no '7'. The detail here is interesting where there are more gaps in possibilities like Crab. Does the board have to be bigger than 8x8 for some or at some point? Apparently not these two. Now inclusive would be 4, 5, 6, 8, 9, 10, 11, 12..., Crabrider having no more omissions, and the '8' case is next up from what Gilman already mentions. '8's can vary hundreds of ways on 64 squares, probably not tens of thousands (they are very countable). Crabrider is more problemists', but regular Crab is worth recognizing as much better intriquing piece-type than C.D.A. Fibnif despite 2:3 values. At the section ''Crab as a Piece,'' Betza in fact recommends Crab, two of them, for C.D.A. improvement of Amazon Army.
[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?
Another interesting idea is the crab eqivalent of a camel (e4 to d7, f7, b3, and h3), which is even weaker than a crab (colorbound). This piece doesn't have a name, so I'll call it a mirage. The 'mirage' can be combined with a ferz, alfil, or dabbah to make an interesting colorbound piece, or with a wazir to make an interesting non-colorbound piece. 'Mirage' + wazir is probably slightly less powerful than a knight on an 8x8 board, but slightly more powerful than a knight on a 10x10 board.
- Sam
If there were ratings on comments (not my first use of this hypothesis) Robert Shimmin's would deserve an Excellent. It should perhaps be included as a notes section in the otherwise short text. Each subset comprises cells a sideways Dabbaba or f/b Camel move away -- contrast the subset which is the whole of a Heavenly Horse's binding, cells a sideways Dabbaba or f/b Knight move away. I suspect that the big future use of the Crab may be in Fusion-style variants, where it can combine with other pieces to take short cuts. This would also be true of equivalent pieces on other types of oblique -- a Cram, with the corresponding Camel moves (Cram to Camel as Crab to Caballo!) would have a cycle of 10 moves without such a feature.
Interesting side note: the crab, although able to visit every square on the board, must change 'color' in a much more complicated way than the fully-powered knight. You can divide the squares of the board into six sets in such a way that the crab must cycle through these sets as it moves. Just as the knight must change color with each move, the crab's moves must go from squares of type 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 1... This makes using the crab a rather tactical experience, since once moved, it takes a total of six moves to get back to the square it started on. In practical terms, this means that if it ever relinquishes an attack on a particular square, it is unlikely to ever be able to return to it.
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Many of the early postings are about normal Crabriders; only one mentions crooked ones. I don't think anyone would need an explanation or diagram to understand how a normal Crabrider moves.
This discussion on 'high-order color alternation' is interesting, btw. I wonder how far this can be pushed. In other words, which fixed move pattern would need the most moves to return to the same square? I suppose there is no limit: by combining moves with incommensurate strides in one dimension you force the return duration in that dimension to be their sum. E.g. fNbH (or fNbZ) are color alternating, but since their vertical stride is +2 or -3 it would need 5 moves to return to the same rank, and 10 moves to return to the same square. Returning to the same file could take longer if there was left-right asymmetry. With only a +2 and -1 stride it would take 3 moves. So a lfNrfAlbCrnZ would take 15 moves to return to the same square.
[Edit] One can also increase the cycle by creating the left-right asymmetry by removing one if the moves. E.g. a 'Fiddler Crab', lfbsN needs 12 moves to return to its starting square.