Comments/Ratings for a Single Item

Sam, for this game I investigated many initial positions, even non-mirrored.
This was the only one I found that was good, I think. Why I didn't choose
the 'natural' positions of bishops and knights had to do with the fact
(if I recollect correctly) that the enemy mastodon would lose its natural
development square on b3, b6, i3, i6. Also, the flank pawn would also be
initially threathened by the bishop, which would be a hindrance to
castling. It's possible that my 10x10 version of MastodonChess is better, at 
least more strategical. Thank you for the information on TamerSpiel. 

--Mats

As an aside, another of Greenwood's variant with this piece is Tamer Spiel (a 2002 variant) where it is called the 'Lion'.

- Sam

I made a test where I put an Archbishop + King versus Mammoth + King on 
an otherwise empty board in Zillions. On an 8x8 board Zillions evaluates
Archbishop and Mammoth as equal. But on an 8x10 board Zillions evaluates
the Archbishop as significantly stronger than the Mammoth (so that the
smiley looks unhappy when making the Mammoth move). On a 10x10 board 
the difference increases yet more in the Archbishop's favour, but not that
much. So Zillions thinks that the Mammoth has about the value of an
Archbishop on an *empty* 8x8 board, but this evaluation changes when the
board is bigger. This corroborates what has been said recently, although
I'm unable to interpret Zillions' numbers.

Hi Mats, your arguments remind me at those of Ed Trice. This is also true
for basing piece values on 'save' king threats. But I have to insist,
that tactical considerations have nothing to contribute to average piece
value calculations. There are of course board situations, where a Pawn
could capture a Queen, but that is completely irrelevant for those
figures. In a similar understanding end-game considerations are highly
influenced from tactics, thus conclusions basing on table bases are merely
of partial value for fixing average piece values as needed during the whole
game before. 

P.S.: using SMIRF's 10x8 values a Mammoth or an Archbishop both are about
equal to a Rook's value + one Pawn unit or equal to two Bishops.

For an index to Ralph Betza's work, see About the Values of Chess Pieces -- Contents. His early attempts at STANDARD values (including Ferz, Alfil, and King/Commoner) can be found here.

I have investigated the theoretical endgame properties of the Mammoth and
it proved to be significantly stronger than the Rook in this area.

(1) M + K  vs. Q + K  = draw
(2) M + K  vs. R + K  = draw
(3) M + K  vs. B + K  = win
(4) M + K  vs. N + K  = win

Comparatively, a Rook + King cannot win against neither Bishop nor Knight,
in the general case. This implies that the Mammoth is stronger in
theoretical endgames. Moreover, Rook + King generally loses against 
Queen + King, but the Mammoth draws against Queen.

Also, in a theoretical endgame, a Mammoth is well suited for escorting a
passed pawn to the promotion square. Bishop, Knight, or Rook, cannot
achieve this.

So it is stronger than a Rook. I contend that it compares to Rook + Pawn,
i.e. 6. But I refuse to believe that it's as strong as an Archbishop (6.8).

Hi Mats, yes, I handled the Mammoth as a fully jumping piece. Your
arguments whether the piece is not save towards distant threats is a
TACTICAL argument, which has nothing to do in average piece value
considerations. Be aware, that a Mammoth also has the chance to escape
such threats by his numerous move possibilities. And - as already earlier
stated - real exchanges would have to consider the actual positional
implications of all pieces within the actual board situation.

P.S.: I should mention again, that my scheme is calculating AVERAGE piece
values, which have to be completed by positional detail evaluations when
evaluating real board positions.

P.P.S.: See also at http://www.chessbox.de/Compu/schachveri1_e.html

Are you sure that you got this right? The Mammoth (Mastodon) *jumps*
 one or two steps diagonally or orthogonally. It cannot be as valuable
 as 6.8. Those evaluation systems don't work because they don't take into 
account how pieces *relate* to each other. Nor does the simple counting of 
squares work, i.e., its factual power, because it doesn't take into account 
that the Mammoth has to flee to every threat and cannot strike back. For 
instance, if a rook is threatened by a queen it can strike back on the 
orthogonals. The Archbishop can strike back against a threatening bishop, 
queen, or knight, but a Mammoth must generally back off before any threat. 
I argue that the Archbishop must be clearly more valuable than the 
Mammoth.

The Mammoth seems ideal for testing the reliability of evaluation systems.

Mats, of course a jumping Mammoth is stronger. My 10x8 scheme gives a
value of 6.7778 to it then, which is nearly identical to an Archbishop's
value of 6.8580. Regards, Reinhard.

then you prosume wrong, he he, it is a leaper +
i made the comment 2 down, forgot to enter name, i was just saying that i
thought the mammoth has to be stronger than the rook, it is alfil +
dabbaba + wazir + fers, that is a heavy piece, i know just the alfil +
king moves is a strong piece, the mammoth is a killer at short range, all
over the pieces like oprah on pork chop nite :))

Reinhard, the Mammoth *jumps* two squares. This significantly increases its
strength and makes it such an interesting and valuable piece on the big
board.

The problem of knight and bishop also has to do with the fact that their
values are so flexible. A centrally positioned knight in the middlegame,
on a square which cannot be threathened by a pawn, acquires the value of a
rook. In the endgame, when pawns are located only at one wing, the knight
is often more valuable than the bishop, sometimes winning the game,
despite equal material. However, the cooperation of two bishops can
increase the value of a bishop considerably. 

The derivation of piece value seems to be a complicated science. If we use
Taylor's notion that the value of a chess piece is proportional to its
ability to safely check an enemy king on another otherwise vacant board,
then the Mammoth is less valuable than a rook, since the rook can give
check from many more squares. It seems that this piece has great strengths
and great weaknesses.

Mats

Hi Mats, I have calculated the average exchange value for the 10x8 board
Mammoth using my approach. Then a Mammoth with 5.4198 compared to a Rook
with 5.8148 seems to be about 1/2 Pawn unit weaker. Additionally I want to
mention, that when exchanging pieces not only their average piece value
will vanish from the board, but also their positional influence. Thus it
is quite often when exchanging Knights with Bishops, that a Knight will
have the bigger positional influence at the board thus making an exchange
nearly equal. But this does not hold for the whole game. Regards, Reinhard
Scharnagl.
P.S.: In my calculations I prosumed, that a mammoth has to slide to its two
distant targets, thus not jumping over pieces like a Knight.

it is easily stronger than a rook, no doubt, it is alfil + dabbaba + wazir
+ fers, and that is a very powerful piece.

Michael, please give a concise account of 'Betza's Atomic Theory'. I've
read Scharnagl's papers on piece value, and one can immediately see in
this attempt, and others, that the result is not wholly correct. Had the
bishop's value exceeded the knight's value to this extent, then
exchanging a bishop against a knight would lead to minor advantage. But
practice has shown that bishops and knights have equal value. However, as
the player can sometimes steer the game into positions where the bishop is
stronger, it is often a good strategy to defer exchange.

If the Mammoth (Mastodon) is very strong due to its manœuvrability then a
paradox ensues. It cannot manœuvre because it's so valuable and must fall
back before the lighter pieces. This means that it's not so strong after
all. Unlike a Rook it cannot threathen at a distance. It must go forward
to make threaths, and this means that it exposes itself to threaths.
Moreover, the Mammoth is not only vulnerable at a short distance, it's
also easily exposed to threaths from long distance, by all other pieces
except the king. That is, other pieces can easily threathen the Mammoth
without being threathened themselves.

It seems like the 'vulnerability factor' is high with the Mammoth (is
'vulnerability' included when determining piece value mathematically?).
It is also a slow piece. It takes four moves to move it across a big
board. I don't believe it compares to a Cardinal (Archbishop, B+N) in
strength. The latter is faster and much less vulnerable. Intuitively, I
would say that the Mammoth compares to the value of a Rook.

Mats

The Mammoth is a strong piece. Betza's Atomic Theory suggests a 4-atom
value, equal to a Cardinal. Its lack of range is compensated by
unblockability and excellent coverage of nearby sqaures.

Hi Mats, to answer your question (By the way, I wonder how people evaluate
the strenght of pieces. I see that they value them as '8.9', etc.) have
a look at: http://www.chessbox.de/Compu/schachansatz1_e.html and the
following pages like: http://www.chessbox.de/Compu/schachveri1_e.html .
Regards, Reinhard.

I filed for a copyright for my Chess for Three game, and it was returned
'Denied.'  The explanation was that since the board is a geometric
shape, it can't be copyrighted (as opposed to other board-games whose
boards contain artistic graphics,like, say, Risk or Monopoly) the precise
text of any particular version of a rules page can be copyrighted,
although its intellectual content can't, and the sculptural patterns of
the pieces physically produced in 3d can, although their moves cannot.
I don't see anything wrong with just giving something to the world once
in a while; share the joy!

See also:

http://www.chessvariants.org/large.dir/contest/mammoth.html