recently I've been investigating many things in higher dimensions (chess being one)

This is how moves a king would have (one space in any direction) in different parts of the board:

1D:

2 moves away from the sides

1 moves at the corner

2D:

8 moves away from the sides

5 moves at an edge

3 moves at a corner

3D:

26 moves away from the sides

17 moves at a face

11 moves at an edge

7 moves at a corner

4D:

80 moves away from the sides

53 moves at a cell

35 moves at a face

23 moves at an edge

15 moves at a corner

5D:

242 moves away from the sides

161 moves at a teron

107 moves at a cell

71 moves at a face

47 moves at an edge

31 moves at a corner

6D:

728 moves away from the sides

486 moves at a peton

323 moves at a teron

215 moves at a cell

143 moves at a face

95 moves at an edge

63 moves at a corner

(Hope I've caculated all that right and got the right terminolgy!) btw I could go beyond that!

I've set rules for pieces so that:

An extra piece for each dimension (for each new type of diagonal)

a knight moves (1,2,3,...etc) one square more each direction (or one square orthogonally and diagonly which means the same thing)

Both of these mean that the number of "spaces" each way increases by two:

8X8, 10X10X10, 12X12X12X12 etc.

I having trouble with one aspect right now:

How many of each piece shold each person get! Any ideas?