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.