Friday, April 12, 2019

[lejnsnaj] Parity of fairy chess pieces in higher dimensions

We previously defined movement of higher dimensional analogues of orthodox chess pieces.  We now consider some fairy chess pieces, paying special attention to parity (colorboundness).

The ferz moves one hypercube in any direction a bishop can move, the colorbound subset of king moves.

The wazir moves the complement of ferz, the king moves that the ferz cannot do.  Two notes: the wazir has some possibly surprising space diagonal moves, like the higher dimensional rook previously defined.  The wazir necessarily alternates colors each step, consistent with 2D.

I think the knight, previously defined, similarly alternates colors each move, like in 2D.

Consider a piece which jumps to any hypercube reachable by two consecutive wazir moves, not necessarily in the same direction.  Obviously avoid moves constructed from steps that cancel each other out.  By the above mentioned parity property of the wazir, this piece is colorbound.  In 2D, it is a ferz + dabbaba compound.  We can subtract the ferz moves to define dabbaba.  This might be equivalent to two wazir moves in the same direction.

Jumping to the destination of two ferz moves in the same direction gives the higher dimensional alfil.  Is there anything new in two ferz moves not in the same direction?  In 2D, this was a dabbaba.

Consider a piece which jumps to any hypercube reachable by four consecutive wazir moves, not necessarily in the same direction.  Subtract moves reachable by a (jumping) queen (e.g., moves 2 or 4 king steps in the same direction).  Hopefully this gives the higher dimensional analogue of the camel, a modification of the knight to be colorbound.  In 2D, the camel is the (1,3) leaper.  In higher dimensions, jumps like (1,1,2) are also possible.

Do the higher dimensional alfil and camel keep their 2D abilities to jump over higher dimensional pawn chains (sheets)?

Also consider 3 consecutive wazir moves.  This will be a superset of knight moves.

As the number of dimensions increases, the Small wazir becomes becomes relatively much weaker than the other pieces.

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