Comments/Ratings for a Single Item

What do you do about this? In another guise, this problem of chaos is
addressed as the slippery king problem. How can you interdict the king in
higher dimensions? In 2D, the king has 8 moves, in 3D, it's 26, and in 4D,
81, assuming the king moves 1 square 'in any direction'. And, because
checkmate is a specific rule for a specific piece, it poses a different
problem than the general problem of chaos, which is being unable to
forecast the board, and thus, unable to choose the best moves. So 2 related
but different problems must be solved for a humanly-playable game.

A number of approaches have been taken. The most obvious is to restrict the
number of dimensions in the game. I do not recall seeing any chess variant
described of higher than 6 dimensions. Most games are 3D, but there are a
growing number of 4D games, including Alice Raumschach. 'Alice'-ing, or
Alicing, is itself a nice cheat, a mutator that adds a dimension to any
boardgame, without all the associated problems of the extra diagonals. They
are all subsumed in seeing if a square is available to land on, in that
higher dimension [along with a little associated confusion about en
passant, as well]. The game is played as the lower-dimensional game,
effectively. But this approach is too limiting. Players are looking for
that extra dimension of freedom of movement.

Alicing represents an extreme form of limiting the size of a dimension,
which is a very common method of controlling higher dimensional games. But
I think it can easily go too far. 

Take boards of sizes 8x8x2 or 6x6x3, for example. These are a pair of 3D
boards that have a common flaw, in my opinion. They are too narrow in one
dimension. A dimension of size 2 or 3 is too small to give the standard
chesspieces enough scope to move as themselves. These narrow dimensions are
in some senses more gimmick than full higher-dimensional boards. 

My rough rule of thumb is that a dimension of length L [Length of side in
squares] should be large enough that a knight placed on a 2D LxL sized
board can move from any square on that board to any other square on that
board. This gives a minimum size of 4 for any dimension used. Admittedly,
that's personal prejudice, and I've used smaller boards myself. Learned a
good bit from using different boards, including the bit about liking the
knight to be able to move freely. In the games with less than all sides of
length 4, the knight does not have full x-dimensional freedom. And of all
the standard chesspieces, the knight is the most beautiful in higher
dimensions.

Raumschach uses a board of 5x5x5, 125 positions, double the standard 64 of
FIDE. This is more than big enough for free knight moves, and it also seems
like a lot of places to move to, but it's not, really. Look at it this way
- from the center, a king, using 3D movement, can move to any square on the
board in 2 turns. On a standard FIDE 8x8 2D board, it's 4 turns, so in
that sense, the FIDE board is twice the size of the Raumschach. And that
size difference means a lot, because the smaller the board, the worse the
edge effects.

As someone who actually found out about these pages while searching for 3d
nomenclature I have always found it odd that anyone should want to go to four or more dimensions. We experience the world in just three spatial dimensions, so that's the most in which an accurate game board can be built. It is no accident that most 3d variants, including most of my own cubic ones, are designed for relatively few board sizes - 5x5x5, 8x4x4, 8x8x3, 6x6x6, 8x8x4, 8x8x8. They gravitate toward boards that might already have been built for some other game. This is one of the reasons why I have never considered 4 dimensions, either for my own variants or for piece names. Has anyone ever had a successful game of a 4d variant?
	I hope that Redistribution 3d Chess is a bit more than a 'name game' - a 3d contribution to the Short-range project, at least. Or are you referring to my recent process of moving 3d pieces round the alphabet in Man and Beast 18?

Ah, I've been slower than I wanted to be with these posts. Charles, I
liked the idea of your Redistribution 3d Chess very much, and started a
comment, but wound up with this, instead, some days later. My apologies for
the rather offhand reference to your style of game design. I was almost up
to R3d [need a better shorthand for this design, it sounds like an
unfinished robot] in this commentary, and had hoped to engage you in a
discussion of shortrange 3D pieces, since we have designed different kinds.
Let me post the little [heh, as if my comments are 'little'] I had
completed before I saw your comment, then take this up again.

The outside of a 5x5x5 is 98 cubes, and the inside is 27 - just under 80%
edge cells on this board. Going up a dimension makes it much worse. A
4x4x4x4 has 256 cells, with 16 in the center, and 240 on at least 1 edge.
Edge effects cripple diagonal pieces, reducing the number of cells they can
reach compared to starting in the center, but don't affect orthogonal
pieces at all in that way. This does counterbalance the increasing mobility
of diagonal pieces in higher dimensions, but is an order of magnitude too
small to be truly effective, unless you make the edges so close together
diagonal pieces diverge off the board within 1 or 2 cells in any direction.
This is somewhat like using the alfil in shatranj. It has a certain freedom
of motion, but it doesn't really go anywhere much. It can either kill you
or miss you completely, and is rather easy to avoid.

Okay, let's see if I can weave the 2 parts of my last comment together
here. Was discussing higher D diagonal pieces, and comparing them to alfils
because they may have rather limited movement. Another comparison is to the
lance, the 1D rook, a piece that can strike deep into the heart of the
enemy position, but has limited lines of action. 

But neither comparison is really valid, because the fully higher D diagonal
piece, once it gets into a position, can devastate it. Since such a piece
has far more degrees of freedom to move, a fully 3D diagonal slider can
easily fork half a dozen other pieces located all over the board. You have
to tame the pieces.

The traditional way to control the sliders is to clog the board with pawns.
Chess starts with the pieces hemmed in by two pawn walls. 8x8x8 3D uses a
pawn plane for the same purpose, 64 pawns occupying an entire 2D 'board'
in front of the 64 pieces in the game - for each player. 128 pawns can clog
up a 3D board pretty well, but this game must be a nightmare to play. 

This idea seemed so much more compact in my imagination; seems to be
sprawling out here, though, much like that 3D game in the last paragraph.

If 8x8x8 is too big, and 4x4x4 is the smallest 'real' chessboard in 3D,
then Raumschach 'goes small', controlling diagonals by having close edges
which limit movement and thus attack potential. It also uses only 10 pieces
and 10 pawns per side, far more handle-able than 5 or 10 dozen. Still,
there's a degradation of the pawn wall. Bishops can capture opposing pawns
on the first move, showing gaping holes in that wall. To close that up,
you'd need 2 more rows of pawns 'above' the white pieces and also
'below' the black pieces, giving 4 rows of pawns - twice as many as
pieces - per player. That's kind of ugly, in my opinion. 

Let's take a slightly geometric look at pawns and pieces, and their
structures. In the beginning of the standard 2D game, a line drawn from
anywhere in your camp [back rank] to anywhere in the enemy camp
[opponent's back rank] must pass through 2 pawns, minimum, at the
beginning of the game, one friendly and one enemy pawn. For higher D games,
when such a line is drawn, how many does it pass through - 3, 2, 1  or no
pawns? The line: is a chain of cells that touch each other, whether at
corner, edge, face [etc]; goes through the minimum number of pawns
possible; and is drawn as straight as possible [no extra wiggles]. With a
symmetric setup, the answer, for 3D, is either 2 or 0, and the 0 line would
have at most 2 bends in it, in reasonable, and reasonably simple, setups.
What 0 means is that pieces have a way around the pawns, making them less
relevant.

So what can you put up with? Unless one is willing to have what could be
considered an unwieldy number of pawns [at least], then the double pawn
wall on which so much of modern chess depends no longer exists. A piece in
a 3 or 4D game can move off its starting cell to somewhere near enough to
the middle of the board on the first turn, without any pawns moving. From
that spot, the piece will be able to either attack an enemy piece, not
pawn, or move to another spot from where an enemy piece can be attacked -
if we use infinite sliders. And here, Charles, is where we come back around
to your nicely-conceived idea and the use of short-range pieces as a
partial but fair solution to the problems posed by 'infinite sliders'. I
don't suppose it's surprising that I recommend short range pieces, but
that doesn't make me wrong in this case.

The board will still likely be in a probably unguessable state in just a
few turns, but since the pieces are short range, the chaos of the situation
is limited in scope. Players can see an attack gathering/coming, even if
they can't see just where it might land, giving some chance to react. Face
it, attacks will be far more flexible in higher D, so giving the defense a
little more flexibility gets a bit of the imbalance back.

As someone who actually found out about these pages while searching for 3d
nomenclature I have always found it odd that anyone should want to go to
four or more dimensions. We experience the world in just three spatial
dimensions, so that's the most in which an accurate game board can be
built. It is no accident that most 3d variants, including most of my own
cubic ones, are designed for relatively few board sizes - 5x5x5, 8x4x4,
8x8x3, 6x6x6, 8x8x4, 8x8x8. They gravitate toward boards that might already
have been built for some other game. This is one of the reasons why I have
never considered 4 or more dimensions, either for my own variants or for
piece names. Has anyone ever had a successful game of a 4d variant?
	I hope that Redistribution 3d Chess is a bit more than a 'name game' - a
3d contribution to the Short-range project, at least. Or are you referring
to my recent process of moving 3d pieces round the alphabet in Man and
Beast 18?

Again, I am slow in getting my thoughts into the aether. Following is what
I wrote yesterday but did not post. I'll then continue to answer your post
in my next, which, with luck, will appear shortly.

Charles, you asked 'why 4D chess?' Because it's beautiful.

I mean that sincerely. Some configurations pieces take during play in [my
version of] Hyperchess are fascinating and can be actually beautiful. Well,
to a CV player, anyhow. 

But you are right that these games don't get much play. There's only one
complete GC game here and it ended when one player let college distract him
from the game, blundered badly, and then resigned. If I recall, every
completed game I know of has ended in resignation, not mate. 

I cannot honestly say that mate is all that achievable, although with the
'Held King' rule it is certainly possible. And that is the key to the
reason most people who try these games don't play many - not only are they
relatively hard to play, but they are almost impossible to end, because a
higher D king is so insanely mobile. What do you do about the 'Slippery
King' problem? How one solves that problem determines a lot about the
game.

What characterizes a satisfactory solution to the problem of checkmate?
I'd like to suggest that it do as little violence to the standard game's
concept of checkmate as possible. By restricting the game to only 3
dimensions, you make it easier to track down the king, somewhat. [And I
have been assuming they are all spatial dimensions. If you are looking for
a time travel game, Gary Gifford has an excellent one.]

Charles, I found these pages while searching for something to do with
Hyperchess, my first CV and a rather decent 4D spatial variant. While I was
waiting for Tony Q to post it, I designed Walkers & Jumpers, my 2nd
variant, one that is a mixed 2D-4D variant. I guess what fascinates me is
the connectivity of higher dimensions. [I read Abbott's Flatland at an
early age; subversive book, that.] Anyway, few see the beauty of even a
simple 4D dance of pieces, and far fewer actually try to play. 

I admit that I'm afraid to play Jim Aikin's Chesseract, and I find
Raumschach intimidating. They have pieces that use all the available
orthogonals and/or diagonals. I like simple pieces, to the extent that I
wrote a short article on that in the CVwiki called Diagonals, Dimensions,
and Draws, that looks at some of the problems of complexity, chaos, and
checkmate in higher D games. 

I was glad to see your venture into shortrange higher D games, hoping that
you would influence people, and get more action going in higher D games.
Lord knows my efforts have not met with overwhelming success. And your
views are more mainstream than mine; that might help. Throw in the use of
independent halves of shortrange power pieces that can combine and separate
at will, and you've got a very attractive game idea. I recently posted a
simple preset for a 6x6x6 shortrange game [lol, disappeared instantly].
While my time is limited these days, if you are willing, we can put
together a preset of your game and push pieces around in both games to
playtest them, or just play [slow] email games of both. I think running the
2 games side-by-side would be both interesting and advantageous.

Finally, I managed to get in a rather unintentional zinger from my offhand
reference to your design style with the 'Name Game' comment. Just as I
see one designer as an openings/development designer, another as piece
designer, a craftsman as artist, the power designer... you I see as someone
who designs suites of pieces both named and themed, and builds the games on
them. Sadly, I have a terrible memory, and I always get lost in your names.
So I tend to think of your variants in general as 'Name Games'. For what
it's worth, upon consideration, I figure at a CV picnic, I'd be the one
nicknamed Shorty.

Hello, David Cannon. I took the liberty of grabbing your general comments
on 3D chess from a rating you gave someone else's game. Higher-dimensional
chess is something I've been fascinated by for a long time, and I've made
a couple tentative forays into the field. Let me preface the replay of your
comment by saying that I'm glad to have found someone to argue the other
side of the design questions for at least 3D chess, and maybe higher
dimensions, if I'm lucky. Your critique could be taken, with an exception,
and applied to my recent posting, 3D Great Shatranj. Following is the text
of your comment, slightly edited to remove a name:

''Poor	I don't regularly comment on others' games and when I do, I hate
to give a negative grade. However, as there are already quite a number of
3D variants, I can only compare this game to some of them.

1. 3D chess creates extra paths on which pieces may move. The most obvious
of these is the so-called 'trigonal' path (like a diagonal path, but not
colour-bound); various variants have introduced a UNICORN or a MACE as a
line-piece to move on this path. A variant that misses this is really
lacking something important, I feel.

2. One weakness with the starting position is that a couple of simple moves
by the Rook or the Queen can check the opposing King. As white moves first,
this gives quite a head-start to white, which is not fair.

3. The piece density is too low. FIDE chess has 32 pieces for 64 cells
(50%); Shogi has 40 pieces for 81 cells (almost the same, although Shogi's
pieces are somewhat weaker). Changgi (Korean Chess) has a lower density of
32 pieces for 81 points (40%). Now, there's nothing sacred about these
percentages, but they have stood the test of time. Having designed quite a
few games and playtested them on Zillions of Games, I've found that
Changgi's 40% density is close to the lower limit at which one may play a
satisfactory game. With too low a density, the players just chase each
other around the board forever. (My own Diamond Chess 306, whose Zillions
file you can download from this site, has a 38% density, but in two of the
variants each piece is really a three-piece compound that can be
unpackaged). The popular 5x5x5 variants have a 16 percent density, which I
find too low. Your own 32 pieces per 320 cells is only a 10 percent
density. You've got to be joking. 

Now, some suggestions ... 1. Either borrow a Unicorn/Mace-like piece from
other variants to ride the Trigonal path, or modify one of the existing
pieces to utilize it. 2. Give us something most of the existing 3D variants
haven't got. Leapers, for example. Most existing variants simply
extrapolate the Knight's move (one orthogonal step plus one diagonal step)
to the 3D board. How about bringing in some new leapers that cover the
orthogonal plus trigonal, and diagonal plus trigonal, steps? 3. Increase
your piece density! Either reduce the size of the board, or increase the
number of pieces. (I know the solution is problematic - I'm working on the
idea myself, and it's tying my brain in knots - but we've got to
try).''
(continued in next comment)

(cont.)
The exception is that the shortrange pieces in 3D GtS don't quickly offer
check, and white doesn't have an excessive first-move advantage. But I do
think I should only get a half-pass in this area, because about half the
pieces I use in higher-D games are 'infinite sliders', even if they are
effectively short range pieces, too. It's close enough to a legitimate
description of some of my stuff that I certainly can't argue too long and
hard:'not me!' 

Now, to start somewhere, David, I have a problem with high piece densities.
Take a simple 3D or 4D board - 6x6x6 or 4x4x4x4. Both are 'common', and
have over 200 squares. Why should they follow a semi-accidental feature of
a trio of very small chess games, and try to have somewhere between 40% and
50% starting piece density? I've found, with my own experiments in excess,
that very roughly 50 pieces is about all you really want per side. I've
tried 100, and it kinda sucks. Of course, I'm a minimalist in a lot of
ways, and I can see chess variants with 100 pieces per side and very
complex rules mimicking wargames, but there I'm discussing a rather
different thing, and not entirely a chess variant. 

Sticking to chess variants, an 8x8x8 or a 4x4x6x6, which I'm just looking
at, takes the number of 'squares' up to the 500-600 range. Now you're
talking maybe 150 pieces/side. How many different kinds of pieces does this
represent? How in heck are the players going to keep track of how all those
pieces move, much less what your opponent might do in 2-3 turns? To me, it
seems the game would bog down to long series of exchanges which would
reduce the pieces by 40-80%, then the players might start being able to
attack each other's royalty with a decent chance of success. Assuming they
could actually play the game reasonably well, and to grant that, we'd need
to explore the particulars a bit.

Charles Gilman, I'm waiting for one more shoe to drop before starting our
2-game playtest, assuming it's not a sabot. I'll let you know soon when
we can start.

It does appear there are at least 2 other people interested enough at this
time to post on 3D chess. I would be willing to organize a playtest for
both your games, Oskar and David, and anyone else who might want to
participate. This includes David's apparently only partly-conceptualized
game. Or, we could just discuss them. However you all want to play it. I'm
already doing a dual playtest/discussion on 4D games with Ben Reiniger,
and we're beta-testing Hyperchess while trying to make TessChess stable in
the opening - the problem you complained about earlier, David. Whoever's
interested, please comment.

'like a diagonal path, but not colour-bound'
Depends how you colour the board! The cubic Unicorn certainly is bound, to
a quarter of the board, but that quarter comprises a quarter of each
Bishop
binding rather than half of just one. The intersection of the two kinds of
binding (one eighth of the board) is the Dabbaba binding on a cubic board.

There are a number of ways to increase the number of pieces without getting too complicated. One way is simply to have more of the familiar range of pieces - e.g., four Rooks instead of two, etc. Another is to make new pieces with new moves, or to borrow pieces from other well-known variants (such as the Elephant and the Cannon from Korean Chess). A third way is to have compound pieces (a number of my own games explore this possibility).

I appreciate your offer to 'host' both Hafsteinn's version and mine (which is still on the drawing board). I'll have to crystallize my ideas a bit more and get back to you about that as soon as possible.

Joe Joyce started this thread by commenting: 'What happens is that the game becomes chaotic; players have no way to forecast the state of the board even a few turns in advance because there are so many possible moves.' Another complex shortrange piece is the Hippogriff, which can leap (1,1,2) to 24 different cells in the 6x6x6 variant Monster 3D Chess.

SOME NOTES ON LARGE VARIANTS: my ideas for a 6x6x6 variant started with comments in the old 3-D Chess thread. Placing 16 White Pawns on the second level and 16 to 20 other White Pieces on the first level results in 64 to 72 pieces on 216 cells, which seems like a reasonable 3D density. Dai Shogi appears to have 130 pieces on 225 squares. Joe Joyce once put 64 pieces on the 256 cells of a sort of 4-D game.

'Edge effects cripple diagonal pieces, reducing the number of cells they can reach compared to starting in the center, but don't affect orthogonal pieces at all in that way.' writes Joe on April 5. Even on a 6x6x6 board, with 152 'edge cells' and 64 'interior cells', the Unicorn remains a weak piece, worth perhaps 40 percent of a Knight. I now prefer giving each army four Unicorns, serving as expendable units, similar to the Alfils in Shatranj.

Royal Exclusion Zone

Here is a new approach to the Slippery King problem on 6x6x6 boards, somewhat inspired by the rules of Chinese Chess. Both checkmate and stalemate count as wins. Let the King move in six directions like a Wazir, but forbid all other Kings from entering any of the 26 adjacent cells. This choice of words will also exclude a 'friendly' King in a four player game.

   level 1    -    level 2    -    level 3    -    level 4    -    level 5    -    level 6
  
   *   R   *  |  *   *   *    |    *   *   *  6
 *   *   *    |    *   *   *  |  *   *   *    5 
   *   *   *  |  *   *   *    |    *   *   *  4
 *   k   *    |    *   *   *  |  *   *   *    3
   *   *   *  |  *   *   *    |    K   *   *  2
 *   *   *    |    *   *   *  |  *   *   *    1
 
 a b c d e f  -  a b c d e f  -  a b c d e f
 
WIN IN FIVE: 
1.Rook 1d6-1d4  King 1c3-1b3
2.Rook 1d4-1c4  King 1b3-1b2
3.Rook 1c4-1c3  King 1b2-1b1
4.Rook 1c3-1c2  King 1b1-1a1
5.Rook 1c2-1b2 STALEMATE

Hi, David Cannon. Heh, I've actually cheated a bit - most of my original
higher-D thinking was done about 4D. Later, I applied those ideas to 3D. 

I have a distinct philosophy about higher-D games, and that is to let the
board star. It came out of my decades-long struggle to design a good,
playable 4D game. I learned [the hard way, which is apparently the only way
to learn - for me, anyhow] to not clutter up the board with pieces, and
also that even regular pieces do tricks in higher-D. You don't really need
complex pieces to enjoy the thrills of higher dimensions. David Paulowich
quotes something from the beginning of this thread about chaos. I believe
too many pieces, with too many different moves, lead to chaos in a game.
But that outcome is certainly not inevitable, especially in a 3D game.

I think we are misled by the starting piece densities in the 3 main forms
of chess. Very few people realize just how small the boards are in these
three games. Instead of thinking 50% is the perfect number, turn it around
and look at what else FIDE chess could be. In simplest terms, could you add
another row of pieces to each side? Could you take a row away from each
side? The 8x8 board has certain limitations inherent in its small size. 
Now look at the Eastern games. With bigger boards, they drop piece density.
They could ramp up densities to the 60% or more level, but don't. Instead,
with larger play area, both games thin out the pieces. Admittedly, people
don't tend to look at it that way, but people don't intuit statistics
well. 

Clearly I have a minority opinion. But I like opinions much much better
when they have solid facts as their base. The only real way to get any
facts is to run tests. And pushing pieces often helps clarify design
issues. So I'd like a show of hands on keyboards: who is interested in
participating in a 3D [and maybe 4D, if wanted] group playtest session?
Even if only to watch and maybe kibbitz.

Wich chess game have highest number of dimensions ever (of corse, it
probably may have more dimensions if played according to Alice chess rules
and other mutators, wich makes more dimensions)?
I invented one 6-dimensional game, but it's not playtested and it's very
likely that it's not playable.

Daniil: this EXTERNAL LINK leads to a list of games created by W. D. Troyka, including 6D Chess, with a 2x2x2x2x2x2 board.

L. L. Smith invented an interesting 4x4x4 variant called Chess Cubed, with all the pieces moving one step only. His 'Knights' are single step Unicorns. I am thinking about creating a preset for this game, but changing the initial setup to: White pieces on the lowest level (same as Raumschach).

Thanks. My 6-dimensional chess is also played on 2x2x2x2x2x2 board.

I would play some 3+ dimensional variants.  As Joe might be able to confirm
by now, my chess playing skills might be a bit lacking, but I think I can
hold my own at design discussions.

Dan Troyka includes an impressive history on high dimensional variants in
his 6D chess download.

I have proposed an (countably)infinite-dimensional chess game before, but
the complete lack of visual ability dampens hope for ever actually playing.
 :P

Unless anyone minds, I might collect and edit some ideas from here into the
wikidot.  It would be nice to have a relatively central source for these
things.

The comment about Eastern games having a lower piece density applies, at
least as national standard games goes, only to the Chinese and Korean ones.
The Burmese, Mongolian, and Thai ones all have the same board and size of
armies that European forms of Chess inherited from Shatranj. Shogi, despite
its larger board, has very nearly a 50% piece density, and more or less
retains it as the pieces keep coming back - what one reference book
describes as preventing the kind of endgame to which Europeans are used. So
if we are to learn anything from the East to apply to 3d Chess, it should
perhaps be to allow captured pieces to return, to maintain enough strength
to bring Checkmate about.

Now to weave a few threads together and hope it doesn't tangle too much. 

Hi, Ben, Daniil, glad to see you here. Welcome to the intellectual
free-for-all. I am most happy to have 4D and 6D represented. Off the top of
my head, I only know of VR Parton, LL Smith, and Dan Troyka doing 6D
variants, and there is a headline article in a recent Variant Chess
magazine where I believe the editor, John Beasley, analyzes a 2x2x2x2x2x2
game. Oh, and Larry will deny it. ;-) He doesn't believe in more than 3D.
He sees higher than 3D boards as nothing more than complicated 3D boards.
Didn't stop him from designing the darnedest 6D board - it has a hole
right through the middle of the 4th, 5th, and 6th dimensions. The games
I'm bringing to this are my version of Hyperchess, and the David
Paulowich-mentioned Chess on Two Boards. The comparison of these last 2
games illustrates where I got some of my ideas from. The comparison with
Ben's TessChess, which we are both working on now... well, we're pushing
pieces and I'm making some strange suggestions. 

David P and I have discussed some higher-D ideas, and his 3D games propose
some ideas that promise interesting times in the chesscube - 3D Grand
Acedrex, for example:
http://play.chessvariants.org/pbm/play.php?game%3DGrande+Acedrex+2007%26settings%3DGrande+Acedrex+2007

Check out his rules, in the 3D chess thread. Those games are worth looking
at. Charles Gilman has proposed a radical shortrange 3D game himself, that
is ready to playtest. And those games that aren't quite done? David C,
Hafsteinn, set up a board with some pieces and push the pieces and kick
some ideas around in company. Or we could kick the pieces around and push
ideas at one another. As long as there's no blood; it makes the monitor
all sticky and hard to see. I've got a few irons in the 3D fire myself. As
well as 3D Great Shatranj, Directed Alice III is an entry here. 

There's a lot of ideas around, many contradictory. The nice thing about
variants is that all these ideas can find games to demonstrate their
usefulness, we just have to look. Imagine what kind of a tournament we
could hold with the games that come out of here.