The gryphon is a neat fairy chess piece functioning kind of like a fat rook. Suppose a gryphon sits at (0,0). The unmodified original piece can move to (1,n) where n>0, plus reflections. Here are some possible variations of it:
Remove the constraint that (1,1) must be empty, giving it jumping ability like a knight.
Allow it to also move to (1,0). This makes its attack area shaped like a hash symbol.
We can also imagine an analogous fat bishop. It is not colorbound; in fact it alternates colors each move like a knight.
The crooked bishop is also somewhat like a fat rook, attacking not just the rank or file it is on, but also the adjacent files. We could imagine a hybrid: a colorbound gryphon. Several possibilities of, what squares must be empty to permit the colorbound gryphon to progress? One possibility is it has crazy jumping ability, kind of like a nightrider, actually dabbaba-rider.
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