The value of similiar pieces to knight(constant distance)

[Uri Blass]

I put this subject in this forum because I know that H.G.Muller programmed a lot of non standard pieces and I wonder if he investigated part of this problem by testing with a program that play with different pieces.

The knight can move exactly distance of sqrt(5)

I wonder what is the value of the knight if you change the distance to a different constant number.

For example if you have constant distance of 5 squares then white has the following legal moves in the opening position:

Nb1-b6(of course bad move)
Nb1-e5
Nb1-f4
Ng1-g6
Ng1-c4
Ng1-d5

possible game with the five-knight may start Nb1-e5 Ng8-d4 Ne5xa8
when black can choose between Nd4xa8 and Nd4xh1.

If you have constant of sqrt(68) or sqrt(73) then the original position is illegal because both king are under check but I suggest distances that does not prevent knight in the centre to move so only distances of
sqrt(a^2+b^2) when 0<=a<=4 and 0<=b<=4 are relevant and basically we have the following distances:

1)0(the knight only help the player to play null moves)
2)1(clearly weaker than rook because rook has more moves)
sqrt(2)(clearly weaker than bishop because bishop has more moves)
3)2(we cannot say that rook has more moves because rook cannot jump but inspite of it I think that it is a weak knight)
4)sqrt(5)
5)sqrt(8)
6)3
7)sqrt(10)
8)sqrt(13)
9)4
10)sqrt(17)
11)sqrt(18)
12)sqrt(20)
13)5(lucky to have more directions so in a big board it could have 12 moves but 8*8 board is not big enough)
14)sqrt(32)