Comments by JorgKnappen

It's a pity. Michael's creations were interesting and of good game design
quality. I like his names for some pieces (e.g., Jerboa instead of
Tripper), too.

Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.

This gives the following formula for the value of a full rook:

R = R1 * (1 + p + p²+ p³+ p³+ p⁴+ p⁵+ p⁶)

Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.

The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is ≈0.715.

The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.

The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.

Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.

I'd suggest changing the "punchline" to something more descriptive than "http://www.spartanchessonline.com". Suggestion: "The spartan army with 2 Kings and novel pieces fights against the persians (standard chess army)"

The punchline occurs in several listings on this site, including the favourite games listing.

I learned that there was a german edition of this game published in 1972 by Parker under the title "Schach dem Schlaukopf". The pieces are Dummkopf (Ninny), Schlitzohr (Numskull), and Schlaukopf (Brain).

Source: http://de.wikipedia.org/wiki/Schach_dem_Schlaukopf

I like the idea of circular riders moving on exact circles, and the generic
name Orbiter is a good fit. In particular, I find those orbiters
interesting that have more squares on their circle than just the minimal
number (4 for straight or diagonal distance, or 8 for skew distance).
Unfortunately most of them are much too large to play well on usual
chessbords (eben 16x16 is small for them). And it needs some training to
visualise their possible pathes. They have so many directions to go!

P.S. A less symmetric version are orbiters orbiting around the center of an
edge, the simplest variant has four squares marking a rectangle.

P.P.S. One of the orbiters (the circular King) is alreay found in Betza's
article here:

http://www.chessvariants.org/d.betza/chessvar/16x16.html

I think a got a proof for the hex geometry.

We orient the hexes such that there is a horizontal line of rook movement,
and denote that direction by 1. The other directions of rook movement are
denoted by \omega and (\omega-1) [the use of the letter \omega is
inspired by Eisenstein numbers]. The centre of a hex is given by a+b\omega
with a,b integer numbers.

First step is a drawing: When we go horizontally firs and vertically as a
hex bishop second, we can reach only one half of the hexes (a+2b\omega).
We repeat this for the other rook directions and mark the hexes
accordingly. They fall in two classes: (i) hexes which can be reached in
one way only (ii) hexes that can be reached in all three way.

The second class forms a grid described by 2a+2b\omega (both coordinates
must be even.

Finally we map these to rook and bishop moves. The path to a three-way
reachable hex (2a+2b\omega) using horizontal and vertical moves
(elementary vertical bishop step: (2\omega -1)) consists of b bishop steps
and b+2a rook steps. Therefore the number of rook and bishop steps are both
odd or both even, giving an even SOLL.

The other direction: Take r rook steps and s bishop steps and demand that
r+s is even. Then we go to r+s*(2\omega -1) = (r-s) +2s\omega. This is a
three-way reachable square again, because (r+s) even implies (r-s) even.

Good to have your chess variants back online!

I browsed though them again and found Matron Chess very interesting. Just a
little rule change to make Queen exchange more difficult, but very
different game dynamics. The rule change is in some sense the opposite of
the rule on Chu Shogi lion exchange: With the Matron it is more difficult
to initiate a Queen exchange while Chu Shogi makes it difficult to complete
the Lion exchange by capturing the Lion back. The Matron variant leads to a
more offensive play which seems to be a good thing.

The Jelly is actually a nice suggestion for the Queen. I estimate it a little (about 0.5 to 1 pawns) weaker than a Queen. This weakness is overcompensated by the overall strength of the rest of the Bakery Bombers.
Since the Jelly is an extended Bison (LJ or Camel-Zebra compound) it has the can-mate property.

The Jelly is a tactically very dangerous piece because it has many immanent threats against the pieces on the opposite baseline. Against the FIDE army, it can enforce "Queen exchange" with the manoeuvre 1. Jelly b3 e6 2. Jelly e5 -- Black gets two moves for a nominally bad exchange, maybe not that bad. Black can save the castling rights at the expense of one move by answering 2. ... Nc6.

I checked that there are no immediate other dangers, 1 ... e6 is an almost universal weapon against early Jelly attacks. 

I have not tried the other canonical armies of Chess with Different Armies yet, they may have weaknesses against a Jelly on d1. I also have not yet checked whether other piece may orchestrate an early Jelly attack.

Upgrade Chess has extremely weak armies. Counting the levels, Upgrade Chess has 15 levels per side (compare to Shatranj with 29 levels or FIDE Chess with 56 levels).

Probably the winner of the first battle will win the whole game, making opening theory very important and almost a mathematical puzzle. Here is one opening (not a very good one, but illustrating some features of the game): 1. e4 d5 2. exd5?? c6 When the pawn goes on capturing it will be taken by the Crab on b8 which promotes to NN2 -- winning advantage for black. 3. c4 Kd7 Black brings the King to the front. There is no danger because of the weakness of the armies, and by capturing with the King he can distribute the additional levels to a piece of his choice.

BTW, Stackable piece would make a nice physical implementation of the game.

I don't see how the Cylindrical Cinders can capture a Rook on move 1. For cylindrical chess a board glued between the files h1--8 and a1--8 is usually assumed. A cylindrical bishop can move, e.g.,  from a1 to h2, but not from a1 to, say, b8. Even though the rooks are initially unprotected, they are not directly reachable for the Cinders behind their wall of pawns.

Gluing the board the other way round, along the ranks (a-h)1 and (a-h)8 is very unusual and you would start up with the opposite Kings in direct contact.

Here are a few more names for the same piece: Thoat (from Jetan, Edgar Rice Burroughs), Emperor (problemist's usage), and Marquis (from Scirocco, Typhoon, and Jupiter by Adrian King, also used in Töws' generic chess piece creation system, in Derzhanski's list of chess pieces, in the Sweeping Switchers by myself, and in Thronschach by Glenn Overby II)

BBC thought about the flag question almost a year ago ... here are some
designs

http://www.bbc.com/news/magazine-25205017

(BTW, I find the "German Jack" in black, red and gold quite funny)

and here are 25 more designs:

http://www.bbc.com/news/magazine-25222891

(BTW, I like Dave Parker's and Michael Elliot's designs)

An excellent to Carlos Cetina for the really nice diagram.

All Knight moves in the first step are "equal" (in the sense of symmetry), but the continuations fall in two classes that Jelliss terms "3D" (crossing the diagonal, the pure trajectories in Carlos' diagram) and "3L" (crossing the lateral, the "impure" trajectories in Carlos' diagram). Here's a reference on the terminology:

http://www.mayhematics.com/t/2b.htm#%282%29

Splitting the Quintessence into a diagonal and lateral piece is surely feasible and the pieces should both be very playable.

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The German and French term (Spiralspringer and Cavalier spirale) are generic (like crooked Nightrider), for further precision they are qualified (German: enger Diagonalspiralspringer = wide [sic!] diagonal crooked Nightrider etc.)

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Yes, the Nachtmahr army put a lot more of strength on the board than the FIDEs: Just exchange whatever Nightrider against Queen, the Rooks, and one Bishop and you are left with a stronger rest of the army. And because of the huge forking power of the Nahctmahrs, I don't see a chance for the FIDEs to avoid this.

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On relative piece strength: The here termed "wide" pieces are clearly the strongest: They have an enormous "capturing density" and the can-mate property. The classical Nightrider is the weakest, the others are in-between.

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I'd like to see your design of Nachtmahr II (allthough I cannot promise to have time for discussion)

Sounds trivial, but I mention it:

*) You cannot drop a piece onto an occupied square (all pieces in all drop
variants I am aware of)

Restrictions on Check/Checkmate:

*) You cannot drop a piece giving checkmate (Shogi P)

I like the idea of the many interesting new endgames. I just hope that the endgames 3 vs. 2 are decisive (at least when one of the 3 is an adjutant and the left over piece from the 2 is a minor one); otherwise the game will be very drawish.

No, RoAR didn't come out yet. It turned out to be more difficult than expected to create an initial array that is playable (and it may be impossible in an 8x8 setup).

BTW: Thanks for the PBM setup :-)

Great game!

There is a minor glitch in the first diagram: It has two back Leopards (artefacts from an earlier version that was discarded?) in e/i 12.

With the new results on the relative strengths of the different armies, how can they be fine-tuned to the FIDE standard?

For the Nutty Knights several proposals exist (replacing the charging knight with a drunken night or with a charging moo, e.g.); but what about the other armies?

The Rookies can be weakened in two obvious ways (a) Replacing the Short rook R4 with R3 or (b) making the Woody Rook WD a non-jumping R2. I think both adjustments will have the right size of effect.

The Colorbound Clobberers are more difficult because the adjustment needed is smaller. Maybe replacing the Bede (BD) with a BzF2 (Bishop + Crooked Bishop aka Boyscout restricted to 2 moves) has the right size of effect. What would be a good name for the BzF2?

EDIT: Changing the notation from BzF2 to BzB2 suggests the nice name "Busy Beaver" for this piece.