Comments by JoostBrugh

For two Camels, two Knights or a Wildebeest we can prove that the longest
forced mate is one move, because the geometry allows only a few mating
patterns. This two-move example with two cannons is not trivial. Is it
possible to prove that two moves is the maximum. Known is that the mating
patterns is always with the Black King on the side (X1), White's King on
X3 or on b3 against when X = a. And White's cannons are at Y1 and Z1 with
Y between X and Z and Y not adjacent to Z. The last move is a vertical move
by a cannon (C YA-Y1 or C ZA-Z1). Blacks last move is a horizontal King
move, which can only be forced if the end file is involved (Second rank
squares can only be covered by the White King), so this must be Ka1-b1,
which means that X = b. One retromove by a cannon later, c1 must be
covered. This is impossible. With two (Cannon + passive Bishop)-pieces
(passive Bishop is a Bishop that does not capture), it should work (from:
White CmB on c4 and c5, White King King b3, Black King b1, Black to move)
1. ...,Kb1-a1 2. CmB c4-f1, Ka1-b1 3. CmB c5-g1#. Probably (not certainly,
it should be possible to force this with two CmB's and a King against a
lone King.

It would be interesting to prove this (and of course the
King+Cannon+Knight against King)

A very important point in Pawns. In FIDE chess the Pawn skeleton is a key
strategic element. Pawns on adjacent files protect each other. The idea
'Pawns are the soul of chess' certainly applies more for FIDE chess than
for Xiang Qi. Piece strategy in the middle game and in the endgame are much
related to Pawn structure. The Pawn structure defines your playing space in
the middle game. If you want to penetrate through the opponent's Pawn
fortification (with brute force), you have to sacrifice at least a piece
with thrice the value of the Pawn (Knight or Bishop). In Xiang Qi, a Pawn
isn't worth much less than an Elephant (at least when the Pawn moved
twice, getting it across the river). In the endgame, Pawn promotion is a
much bigger issue in FIDE chess. With little material the mobile FIDE King
isn't easily checkmated. The idea of the endgame is to use the King as an
attacker and the goal is to get a Pawn across the board. In Xiang Qi, the
goal of the endgame is still to attack the King, not to eliminate Pawns
with the King.

A problem for the real mathematician about DVONN: How high can a stack
maximally grow in the game when:
a. The stack contains one or more DVONN-pieces (so it will survive by
definition)
b. The stack contains no DVONN-pieces
bi. The stack dies later in the game by becomming disconnected
bii. The stack gets a DVONN-piece later in the game (and thus survives)
biii. The stack survives without getting a DVONN-piece

Some one-dimensional examples. Assume these lines as isolated islands: A
number is a stack without DVONN-piece, X is a high, immobile stack without
DVONN-piece, D is a single DVONN-piece.
bi: X - 1 - D. The singleton has to move and stack X dies. It can only
become X+1 upon dying. The X+1-stack never really lived.
bii: X - 1 - D - 1. The stack can only be saved by using the rightmost
singleton to put the DVONN-piece on the stack and it can grow to X+3 with
a DVONN-piece, but had highest size X without DVONN-piece.
biii: X - 2 - D. The stack stays connected and survives.

Note that White and Black play together to get the high stack. But the
rules must be obeyed. The problem can be simplified by disregarding one or
movre rules.

Could there be a systematic way to solve this problem.

This is not for making a ZRF. I already made an ugly ZRF in which I used
25 as maximum, becuase when higher stacks are brought back to 25, the same
moves are possible and the outcome (win/loss/draw) will always be the same.
Only the point difference can be different.

The Harpy is not just a piece, but a whole idea. Like in Shamanic Chess,
the piece can go in move-mode or in fight-mode. In move-mode, they are
more mobile and in fight mode, they can capture. You can make a whole
chess variant (or a 'Chess with Different Armies'-army. For example,
strong fighting pieces that can't move to any square in move-mode, but
for example just like a queen. Or pieces that have an effect when
deployed, but can't move then (fight-mode), for example the
Ultima/Rococo/Maxima-Immobilier (that does not immobilize in move-mode and
does not move in fight (immobilize)-mode. There are many possibilities.

I don't know. But I got my ass kicked by Zillions several times. But I
don't know any DVONN tactics, so probably it is just a sign that I'm
still not good at it.

My ZRF is built with the idea that the only things that matter for a stack
are its size, its owner and whether or not a DVONN-piece is in there. I
didn't bother about stacks larger than 25 (because they and larger stacks
are equally immobile and equally winning when surviving). Rules like the
'No move with enclosed pieces' are trivial to implement. After each
move, an administrator (?-player) must remove all disconnected pieces.

I used a pass-detector that detects when players pass. Then I create dummy
pieces to make high stacks count for that many pieces and then carefully
trigger the count-condition.

But I think that the maximum height of a stack is more than 25. Take the
leftmost positions as building position. Then try to get stacks with
heights 1, 2, 3, 4, etc. on positions on the center row. From four other
positions, stacks can directly be moved to the target. Still, I don't
think the answer is 49 (or 46 for a DVONN-less stack), but probably they
are close.

I played it a few times. I think I figured out the algorithm.

After placing the tiles, there are 12 ranks with total number of squares
64. Define: Area(n) = Number of squares on ranks 1..n. As there are 12
files, Area(1) = 0..12, Area(2) = 0..24 etc. Area(12) = 64. White may
place pieces on rank r if Area(r) is 32 or less. For Black, it works the
same, except that rank 12 is now rank 1 etc.

For example, if White and Black construct a Chessboard on ranks 2..9 (with
eight squares on each rank. Then, for White: Area(1) = 0, Area(2) = 8,
Area(3) = 16, Area(4) = 24, Area(5) = 32, Area(6) = 40 Area(n) > 40 for
n>6б, so White can place pieces on ranks up to 5. For Black the same
results in ranks from 12 down to 6.

If Area(n)=33, you just can't place pieces on the n'th rank. The maximum
number of squares on the n'th rank is 12, so Area(n-1) must be at least
21. This is enough space to drop the 16 pieces. The game, however gets
stuck if you have to drop your King into check.