consider a fairy chess piece that can make a subset of knight moves: starting from (0,0), it can jump to (1,2) (-1,2) (2,-1) (-2,-1). it goes forward more quickly than backward. it is called a "crab".
it requires a surprisingly large number of moves to reach nearby squares. here is its 7x7 neighborhood, annotated with number of moves.
6 3 6 3 6 3 6
7 4 1 4 1 4 7
2 5 2 5 2 5 2
3 6 3 0 3 6 3
4 1 4 7 4 1 4
5 8 5 2 5 8 5
6 3 6 9 6 3 6
starting at (0,0) in the center, it takes 3 moves to move to reach (0,1), same as regular knight. 4 moves to (1,-1). 5 moves to reach (0,1). 5 moves also to reach (2,1) or (1,-2), the knight's moves it cannot do. shortest closed circuit is 6 moves. 7 moves to (0,-1). 9 moves to (0,-3).
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