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Beeing on the discussion on color binding I noticed that I don't understand the concept well enough. How do I figure out if a piece is color binded. For square tilling it is easier but for hex is more difficult. For 3D is also a bit more difficult that for squares. I'm mostly interested about (m,n) leapers on a hexagonal board. Are there color binded  twice leapers that are (m1,n1)&(m2,n2)? For example (4,1)&(3,0) is a third board bounded. What about 3D? Can anyone help me with a general math answer?

Color binding is a hard problem. For one, it does not only depend on piece moves, but also on the over-all board topology. On a cylinder board of odd width a Bishop is not color bound, and on a toroidal board with the proper helical pitch a Shogi Lance can access the entire board.

Even on regular Euclidean boards there can be surprises. E.g. the Chiral Knight (frbllfrbN) has a pretty severe meta-color binding, and can access only 1/5 of all squares. While in terms of normal colors one would have guessed it is a color alternator. In general, pieces that leap only in two bipolar directions cannot access all squares on a non-warped board, with the Wazir as only exception, because it makes the smallest possible step in both directions, and thus skips nothing.

BTW, color alternation is also a kind of binding, even more subtle than color binding, which imposes a conservation law on the evenness of the turn and square shade combined. This then prevents the piece to triangulate.

So, are the hex camel and hex zebra color binded?

Bishops VS Zebras has finished!

Bishops wins:100

Zebras wins:74

draws:26

Bishops points:113

Zebras points:87

The last result is pretty bad as zebras did better against bishops than the stronger camel!

Sorry, my elaborate analysis got lost because CVP suddenly decided I was not logged in. Anyway, I have no idea what Camelor Zebra or (N,M) leaper means on a hexagonal board.

Could you pull together and redo your elaborate analisys. I'm sure you have cool things to say!

On a hex board the knight is an (2,1) leaper, an (m,n) leaper would be an leaper that jumps to all squarez (actually 12-if  n<>m) that can be reached through a hexrook move of m and then a move of n, or first n and then m, as in regular board chess. This is what I meant!

You mean two hex-rook moves with an angle of 120degrees between the two? So something like this:

. . R C B C R . . 
 Z C R N N R C Z
Z B N R F R N B Z
 C N F W W F N C
R R R W O W R R R
 C N F W W F N C
Z B N R F R N B Z
 . C R N N R C Z
. . R C B C R . . 

The C definitely is not color bound then: from O it can reach two cells that are 2 hex-rook steps removed from each other. Do that twice, and you move four hex-rook steps. OTOH there also are C cells that are 5 hex-rook steps from each other. So you can move a single hex-rook step in 6 moves. And if you can do a single hex-rook steps you can go anywhere. Hex-zebras alread kan reach an adjacent cell in two moves.

The 12 moves that oblique leapers have can be split into two chiral sets of 6: the right-bending and left-bending. Pieces that only have the 6 moves from one such group always suffer a form of meta-color binding: they can make a triangle by making 3 leaps at angles of 60 degrees to define a triangle, and the plane can then be tiled with such triangles (sharing the corners). These corners are then the only cells that can be reached. Pieces with non-oblique moves, i.e. (N,0)- and (N,N)-leapers always have only 6 moves, and except for the hex-wazir (which makes the smallest possible step,and thus sever skips anything) they will all have high-order color binding.

Wow, thanks! This is what I meant!

Apothecary 1 Knights (NmZ) vs Champions (WAD) has finished

Knights wins: 86

Champions wins:96

draws:18

Knights Points :95

Champions Points:105

Apothecary 2 Elephants(FAmH) vs Camels(LmW)

Elephant wins :97

Camels wins: 81

draws:22

Elephant points:108

Camels Points:92

Apothecary 1 Champions(WAD) vs Bishops (B) has finished:

Champions wins:112

Bishops wins:67

draws:21

Champions Points:122.5

Bishops Points:77.5

Apothecary 1 Knights(NmZ) VS Wizards(LF) has finished:

Knights wins:97

Wizard wins:83

draws:20

Knights Points:107

Wizard Points :93

Now the minor pieces vs pawns tests have started. In these tests 1 (and just 1) minor piece is deleted and that side receives an extra pawn in compensation. The opposite side has 2 pawns deleted for a total of 3 pawns for the piece. The first apothecary 2 such test has finished with the following results:

Zebra VS 3 Pawns

Zebra Wins: 101

Pawns Wins:72

Draws:27

Zebra Points :114.5

Pawns Points:85.5

I was personally expecting balance here, so I'm a little surprized.

Apothecary 2 Camel (LmW) vs 3Pawns has finished

Camels wins:96

3Pawns wins:73

draws:31

Camel Points:111.5

3Pawns Points :88.5

Apothecary 1 Wizards (LF) vs 3Pawns has finished

Wizards wins:117

3Pawns wins:68

draws:15

Wizards Points:124.5

3Pawns Points :75.5

Apothecary 1 Champions(WAD) vs 3 pawns has finished

Champions wins:116

3 pawns wins:68

draws:16

Champions points:124

3 pawns points:76

Apothecary 2 Elephant (FAmH) vs 3 Pawns has ended.

Elaphants wins:114

3Pawns wins :64

draws:22

Elephants Points:125

3Pawns points:75

It's been a long time since I last posted here mostly because of beeing busy with playing Civilizations (I'm over it now).

Today, 30th of November 2016 on my 32 birthday I vow I'll add to Greg Strong's ChessV my apothecary 1&2 variants. I have chosen ChessV instead of Fairy-Max because it offers more flexibility in implementing the rules of the game, provided you know c++ (which I do). For know I've managed to implement the griffin but I'm searching for a way to test the code.

to Greg Strong: Thank you for providing this oportunity

mostly to H.G. Muller

Hello again H.G., long time no see, my fault.

Thanks for teaching me the statistical methods with which to evaluate different pieces. Experiments done with fairy max have proven useful espeacially in setting up the strengh order of minor pieces. Thanks very much for that H.G. . Although I'll redo most experiments with  ChessV once everything is setup.  I have chosen chessV because with it I can implement the proper promotion rules and I hope to be able to engineer a fool (that imitates the last move of the opponent).

I've notcied you have done a lot of wonderfull work with the interactive diagrams that represent variants. I need for the 2 apothecary games the following facilities:

1. Generation of random 1/12 intial positions. *

2. Very difficult implementation of the fool. You've said before that there is no concept of a turn in the diagrams and I accepted that, but can't the fool be set to have the move (with a default of nothing) of the last used piece friend or foe, and let the user be aware of move turn. I also noticed that the concept of starting in hand is not hard to implement is actually possible in the default implementation.

* the twelve starting positions come from having the two knights in or bishops in options that I have mentioned before. Also the three major pieces begining sqares get permuted for 6 other posiibilities. 6*2=12

Great news, Aurelian.  If you want to email me the code, I'm happy to take a look at what you've done.  I won't get to it until the weekend, though, so no need to send before then.

Greg,

Send me an email at [email protected], with the adress where to send you the code!

H.G.,

Have you seen my previous message on this post?

As the whole apothecary series (more on that later) is more about me tinckering with chess variants ideas maybe people carring about them should listen to this video: https://www.youtube.com/watch?v=m4p7T9O_tqg