Sliders and Leapers

Power of Sliders

In standard chess, we're happy with the notion that the bishop is colorbound, and that the rook has the very convenient power of blocking a king's movement. Things become more "interesting" in multiple dimensions. Here I'll work with all possible diagonals for sliders.

Rook

We quickly see the rook lose power. In N-dimensional chess, it takes a rook N moves to reach from one square to another. More devastating, the rook no longer provides the blocking power it once had. On the other hand, it does of course continue to be able to reach every square on the board. The rook has $2N$ directions of movement when he is unobstructed by edges or pieces.

Bishop

Perhaps somewhat surprisingly, a properly colored board will still keep the bishop on one of two colors, regardless of N. The bishop has $2N(N-1)$ directions of movement when unobstructed. Notice that this is considerably larger than that for the rook when N becomes large.

Unicorn

In 3D chess, the board can be colored with four colors so that a unicorn is limited to squares of one color. In higher dimensional chess, this is no longer the case, and the unicorn can reach any square. The unicorn has $(4/3)(N)(N-1)(N-2)$ directions of movement.

Balloon

In 4D chess, the board may be colored with 8 (!) colors so that a balloon is restricted to squares of one color. In higher dimensional spaces, the balloon reaches half the board. The balloon has $(2/3)(N)(N-1)(N-2)(N-3)$ directions of movement.

Everything

Outside of the more common sliders listed above, there are of course infinitely many others, although they will see increasingly less actual gameplay. I believe the following characterize an n-dimensional slider in N-dimensional space:
• If n=N, then the board can be colored by $2^{n-1}$ colors;
• if n<N and n is even, the board can be colored by $2$ colors; and
• if n<N and n is odd, the slider can reach any square on the board.

• The slider has $\displaystyle \frac{ N! \cdot 2^n}{n! (N-n)!}$ directions of movement.

So it seems that even-dimensional sliders and sliders of the highest/lowest dimensions have a disadvantage.

Power of Leapers

As far as I can tell, most multidimensional games have used only one leaper, usually some sort of extension of the knight. An asymmetric leaper like the knight will benefit more from extra dimensions than a symmetric leaper. For instance, if each piece is unrestricted by the edges of the board or friendly pieces, the number of moves available are:

 N wazir knight ferz (1,1,1) (1,2,3) 2 4 8 4 - - 3 6 24 12 8 48 4 8 48 24 32 192 5 10 80 40 80 480 6 12 120 60 160 960