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George Duke wrote on Sat, Jul 12, 2008 08:52 PM UTC:

We can circumscribe our understanding of all leapers by looking at Tim
Conway's Angel from 1970's. Here 27.July.2007 explains more fully that
Angel jumps to any square that can be reached in n King moves. That means
always every other square on board having both dimensions < (n+1). What is
value of Angel on 8x8 n=7? Any convenient value > 1.0 but 60 is promising
for comparisons. How could Angel possibly be strengthened? By not being
able to be captured. (A complete Rules-set would have provisions for King
escape.) Rook has been implemented as compound leaper to (0,1), (0,2),
(0,3)...all the ''Rook squares.'' (Chatham or Paulowich may remember
where.) Call that piece Rook-all. Then our standard Rook is lame Rookall
or Rookall darter same difference, a complete definition because it is
understood we are interested in only one pathway. The most useful
divisions for piece-types are actually Multi-path, Rider and Leaper, but
there are even some unclassifiable there.

George Duke wrote on Sat, Dec 27, 2008 06:39 PM UTC:

''Piece Values'' thread revived should be reminded there are dozens of
other areas of discussion of piece values. They range from Ralph Betza's
articles to my ''Game Design Analyses'' under Large CVs 3xx or 400.
Any values estimation has to have a reference point. And only convention
says Pawn is the reference, and that it is 1.0. Tim Conway's Angel is just as
good a place to start. We could key off instead a piece such as Angel that is
as powerful as we could ever possibly want. Actually it takes two values to
get a standard started, but they could as well be Angel and Rook, not
involving Pawn at all. Now Angel jumps to any square that can be reached
in n King moves. (See comments this Shatranj thread for more detail.) With n=7 on 8x8, Angel goes immediately to any square.
Provision needs to be made in rules of any specific embodiment as to
Angel-King allowability of interaction, removing anomalies. Angel at 60
points may be convenient, but we have to be careful. That is because Angel
is not all-powerful, and there are stronger pieces: (a) stipulate Angel
cannot be captured for a greater Angel (b) stipulate cases of multiple
occupancy, so that King or other piece has more difficulty finding a
square to move (c) on board 10x10, n=7 Angel is weaker than n=8 Angel; and
so on. For most purposes regular Angel with n equal to the longer dimension
would be the most valuable piece, all others will be lesser value, keying
A=60 for good relative valuations. Then we are at liberty to find or
design a piece, say, 1/6 the worth of Conway's Angel (invented 1970s).

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