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Hi Mats, 

your definition of 'Elk Chess' generates an asymmetry in the
possibilities for the initial position:  White can develop immediately its
elk on the king side with a knight move to the row in front of its pawn,
Black cannot do so.  At the queen side the situation is inverse.

You could reach easily symmetry by a little change in the rules:
 - for white swap the colors when the elk has to move as knight/rook;
 - for the black elk let the definition as it is.
(Also a converse definition would be possible, depending at which wing you
prefer a quick development of this piece.)

Shortly you could summarize this new rule as follow:
The Elk moves at squares of its own color like a rook, at squares of 
the opposite color as a knight (or the converse regulation). 

Alfred Pfeiffer