Comments/Ratings for a Single Item

Oh, and note the spelling of my surname!

Dear Mr. Gillman.

A N+Q would be a Sherazade?

After all... Alfil + Dabbabah= Alibaba...

What is required to succesfully drive a King into the corner is very difficult to answer. For one, as the 12x12 checkmates thread shows, it is dependent on the size of the board. When you solve the end-games by retrograde analysis, there are several possibilities:

Usually you start with very many checkmates, along the edge, none of them enforcible. (Unless you have something like a Queen). Pieces like Bison that cover only 2 orthogonally contiguous squares do not have those, however, and solely rely on a handful of corner checkmates. Usually all longer mates are ancestors of those corner mates.

Sometimes the longer mates die out immediately, like in the case of Gnu. There exist mate-in-1 positions, but no mate-in-2 positions, so afterthat, you are done. This is just an unlucky coincidence of the piece not being able to make the critical step between a position needed to force the bare King to step into a corner, and the square where it needs to be for a corner checkmate.

More often the number of longer mates increases very slowly with their duration, and then hovers for a long time around a very moderate number. These are the positions where the bare King is already trapped on the edge, and has to be driven into the corner with very precise play. On large boards, the number of longer mates even tends to decrease again, because it becomes easier and easier for the bare King to actually flee towards the corner voluntarily, the attacking pieces not being able to all follow it quickly enough to keep it trapped there, so that it can then escape along the other edge. So you have to confine the bare King more precicely as you are further away from the corner, not only cutting off its way back to the middle of the edges, but also preventing it gets too much of a headstart towards the corner. 

This leads to a decrease of the number of positions. On boards that are too big the number of mates actually decreases to zero before you retrogradely reached the middle of the edge, and the game is generally drawn: there is no way to drive the bare King towards the edge without it reaching the edge in the middle between corners, and there is no way to drive it over such a large distance to the corner without it escaping in one direction or the other.

If you survive (in retrograde time) until the middle of the edges, though, the number of longer mates suddenly starts to explode. It is usually very easy to drive a bare King to the edge, if you don't care where it will hit the edge. Unless the board is really big. But on 8x8, if your own King is in the center, the opponend is already driven onto the second rank. So in the early phase, from very unfavorable position, almost any sequence of moves that step your King plust its lieutenant towards the center (using opposition to drive the opponent out of it) is a direct route towards the checkmate, and there are very many possibilities for this. 

On all boards I have tried, (upto 16x16) once you reach the point where the number of longer checkmates starts increasing again, it usually fills up the entire space of positions. Either that, or it slowly peters out before it ever got big.

Yes, orthogonal contiguity of capturing moves is necessary. The Bison (Camel + Zebra) is just strong enough to force the lone King into a corner and checkmate it. See H. G. Muller's [2008-07-15] comment in the 12x12_checkmate thread for details.

The DN piece is sometimes called a Carpenter. The path to checkmating the lone King is complicated and nonintuitive. On this comments page, back on [2005-06-23], I wrote:

'I believe that the Knight-Dabbaba piece is sufficient mating material on the standard 8x8 board. Not sure about 12x12 and larger boards.

Here is a computer-verified endgame position from 1999. White to move and mate in nine. WHITE: King (c6) and Knight-Dabbaba (h8). BLACK: King (c8).'

Later [2008-07-06] H. G. Muller wrote: 'King + Carpenter can almost always perform checkmate on 10x10, but hardly ever on 12x12.'

I do understand that a Bishop does not win, but that was due to the first condition (orthogonal contiguity of capture moves). What I did not understand was the board-opposition stuff.

And what the heck is a Donut??? 

(And why does it lose rather than draw? I can't say that your last posting clarified matters much...)

The problem is that I don't understand what you mean by 'opposition to the nearest sides of the board of their moves'.

I am not sure about declaring stalemate a loss would have muh impact. It is true that with almost any reasonable piece stalemate positions are possible, but that does not mean that it can be forced. I used to have a version of my tablebas program that would equate stalemate to a loss, (can't find it anymore... :-( ), and from what I remember in most end-games hardly made eny difference. For instance, I don't believe KBK would be generally won under this rule, despite the fact that a Bishop is quite strong for a piece without mating potential.

The Shatranj baring rule has a much bigger impact in this respect.

Indeed, on 8x8, K+Q4 vs K+R is a general draw. K+Q5 vs K+R is a general win.

K+A vs K+R is also a general draw. K+(BNN) vs K+R and K+(BNW) vs K+R are general wins.

A fortress draw originally meant a position from which you can prove that the weak side can hold out forever. This in contrast to drawn positions where the weak side draws by gaining a piece. The latter occur a lot in end-games like KFFWK, when the bare King can chase an F or W cut off from their allies into an edge or corner, after which the remaining KFFK or KFWK is a draw.

An example of a fortress in KQKBN is this:

. . . . . K Q .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . n . . . .
. . . . . . . .
. b . . . . . .
k . . . . . . .

All black pieces are defended, and the white King cannot approach the bishop to attack it a second time. This fortress holds out even against an Amazon.

Nice numbers, David. The Unicorn [BNN] alternates between 4 and 12 squares at any given range, equalling the queen's # of squares attacked at every even-numbered distance [2 squares, 4 squares...], and falling 4 squares short for odd-numbered distances. It doesn't interdict, though. As boards get larger, the balance of power shifts more and more to the queen and the unicorn, but on an 8x8, how does the BNW do? [By the way, what the heck is a Fortress Draw?]

Finally, how does the shape [the geometric footprint] of the attacked squares affect the piece value? I'm beginning to doubt that the stronger of 2 pieces is always the one that attacks more close squares. Comparisons like knight vs camel fail, because the camel is colorbound, thus having an additional source of weakness.

Piece - @1 square - @2 squares - @3 squares - @4 squares 
 U          4          12            4           12
 Q          8           8            8            8
 S          8          12            4            4
 C          4          12            4            4
 A          4          12            4            4

The Unicorn (Bishop + Nightrider) and the Queen are the most powerful pieces at a distance. I rate them equal on a 10x10 board. The Super Archbishop or Super Cardinal ('S' here and 'X' in Joe Joyce's note) has an impressive shortrange punch, but I still rate it halfway between the Queen and the Chancellor on a 10x10 board.

Years ago I considered the simple endgame K and X vs K and R as a guide to piece values. X=Queen is almost always a forced win, given sufficient skill and patience. X=Q4 (moves up to four squares like a Queen) is probably not a forced win, according to a small sample of (FIDE rules) endgames I have examined, where the winning Queen move involves: [1] moving at least 5 squares or [2] giving check from at least 5 squares away or [3] attacking the Rook from at least 5 squares away. [EDIT] Dave McCooey's Endgame statistics with fantasy pieces on the 8x8 board states that K and Q vs K and R has no Fortress Draws, while K and A vs K and R has Fortress Draws around fifty percent of the time. The Archbishop (A) is called a Pegasus (G) by McCooey.

Recently, HG Muller has demonstrated the values of the queen (BR), Chancellor (NR), and Archbishop (BN) are all actually similar, with the BN within about a pawn value of the queen. Adding the wazir moves to the BN should, I think, kick the value of the piece above that of a queen. John Smith has talked about the coverage within 2 squares of this piece. Let's look at the numbers.
Piece - @1 square - @2 squares - totals/24 - @3 or more
 Q          8            8          16/24        8
 C          4           12          16/24        4
 A          4           12          16/24        4
 X          8           12          20/24        4

The queen is the most powerful at a distance.
The queen, within 2 squares, by virtue of attacking more close squares, is more powerful than the chancellor or archbishop.
X is significantly more powerful than the other 3 pieces up close. 

In my opinion [developed in the CVwiki under 'Attack Fraction'], X is the most powerful of the 4 pieces shown. It is only the queen which ever attacks more squares, and that requires the queen be able to move 4 or more squares unobstructed in all 8 of its possible directions. As a minor note, the Q can't jump, X does. 

Consider counterattacks. A can't attack X unsupported, while X can attack A freely from 4 squares. Q and C attack X freely from 4 squares, X attacks both freely from 8. 

Finally, consider interdiction and checkmate power. If I'm doing this right, Q and C can interdict, but not checkmate by themselves. A can checkmate but not interdict by itself. X can do both.

Conclusion, X wins, hands down, in my opinion.

is anybody interested in a piece that is as strong as a Queen but still
different from a Queen?  How about a Royal Cardinal? (combines the powers
of a commoner, bishop and knight).

I suspect that this piece is just as strong as the Queen -- but without
the long range Rook moves .......... comments??

I very much want Pawnless Ecumenical Chess to work, but I'm not sure there is a setup that does. Maybe this one though, a very slight alteration of the second one, simply transposing cardinal with gnu.

A fair point, in fact it could even prepare to capture the Camel, that is,
1 h2-g5 f8-g6 2 g5xh8 g6xh8. So what about my suggestion for a fix?
	For the record, the array on which this comment was made has 2nd rank
CNBDGBNC. For some reason not of my doing the page seems to have reverted
(at the time of writing) to its original form.

'In Pawnless E.C., what's to prevent White from opening with h2g5,
winning the exchange of the black rook on h8?' To start with, the better
prize of the Marshal on f8! It certainly demonstrates that the Camel has
its strengths on a FIDE board that it lacks on Really Big Boards.
	Would moving the Caliphs forward to the middle of the second rank with
RVDQKMGR on the back rank, fix it? It certainly leaves Black guarding b5
and g5, and as far as I can see the most that White could get through
forced exchange for a Caliph would be the weaker Knight (1 d2-e3 ~ 2 e3xb7
c8xb7). Still, best to check if I've overlooked anything again before
posting as modified.
	A variant combining these pieces with Glenn Overby's is an interesting
idea, although the names Pegasus and Roc suffer from mutiple uses. His
Pegasus is the same as my Gamewarden, as when I submitted FUO I was
planning another use for Pegasus. His Roc I am planning to term a Caribou
as part of a quartet with Kangaroo (from Outback Chess), Carpenter (N+D)
and Casbah (C+D), for extrapolation much like the pieces from this
variant.
	In Dai Shogi, the Ferz+Elephant compound is called a Flying Dragon, so
the name Caliph for Bishop+Camel has the advantage of not being ambiguous.