My 3d Chess Game

[anderscolingustafson]

3d Chess 1.png

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3d Chess 2.png

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3d Chess 3.png

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3d Chess 4.png

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3d Chess 5.png

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I was thinking about all the moves pieces can make in 2d and what would be all the equivalent moves in 3d and all the different pieces that could be produced with all the different combinations of moves. I was also thinking about what types of pieces I would need in order to have 64 none pons and found that these two questions produce the same results in terms of the number of types of pieces. I have a series of pictures to represent a 3d Chess cube with each picture represtenting a cross section of the chess cube.

As with 2d Chess a piece can capture a piece from the other side by moving to the cube that that piece is in. With the exception of the Soldiers, Pilots, and Troopers as well as castling pieces may not move through other pieces. A piece is captured if a piece from the other side moves to occupy the cube it is on. Immediately after a drone has moved two spaces forward in the first move it may be captured.

Captains can move any distance through the chess cube, and if a Captain moves a none zero distance in one dimension it does not move in the other two dimensions. Stepers can move any distance through the chess cube, and if a Steper moves a none zero distance in one dimension it must move the same distance in another dimension and not move in one of the three dimension in which a Steper does not move. Heros can move any distance through the chess cube, and if a Hero moves a none zero distance in one dimension it moves the same distance in the other two dimensions. Engineers can move either in the way that a Captain can move, or in the way that a Steper can move. Divers can move either in the way that a Captain can move, or in the way that a Hero can move. Gliders can move either in the way that Stepers can move, or in the way that Heros can move. Angels can move either in the way that Captains can move, in the way that Stepers can move, or in the way that Heros can move.

Soldiers cannot move more than two cubes in any direction and if a soldier moves two spaces in one dimension it moves one space in one dimension and does not move in one of the three dimension. Pilots cannot move more than two spaces in any direction and if a pilot moves two spaces in one dimension it moves one space in the other two dimensions. Troopers can move either in the way that a Soldier can move, or in the way that a Pilot can move.

A God can move in any direction like an Angel but it can only move one space with the exception of castling. Also a God cannot move to a space that would make it vulnerable to getting captured.

A drone can only move forward unless it is capturing a piece from the other side. If a drone is capturing a piece from the other side it must move one space forward and one space in at least one of the remaining dimensions. On its first move a drone may move forward two spaces but on any other move it can only move one space forward.

If a drone makes it to the other side of the chess cube it must be promoted to a none drone piece that is not a God.

In order to castle a God moves two spaces in the direction of an Engineer that is two spaces to it's side. This also causes the engineer to move two spaces in the opposite direction.

If someone puts an opponent in danger of having the God captured then the player has put the other player in check. If a player cannot get out of check then it is checkmate and the one putting the other in checkmate wins the game. If the God is at risk of getting captured the one playing it must do whatever he/she can to prevent it from getting captured if the God is not in checkmate.

[gonegahgah]

Hi Anders.

I found this an interesting new subject that you have introduced.
Thinking about it I have done up the first layer of a different type of 4D chess board and wonder what you think of it?

First off, it stems out of my preference for our 4D beings to have 3 legs, 3 arms, 3 eyes, etc.
This would tend to be the minimum for a 4D creature and I'm a fan of evolutionary minimalism.

When I started I was doing similar to what we would do to go from 2D to 3D like you have with yours going from 3D to 4D.
That is taking the 2D board (a series of while/black lines behind each other) and standing it up and extending this foward to make the checkered grid we are familiar with.
Standing the 3D board up and extending it forwards achieves something similar to what you have done; which is certainly cool looking.
Well done for coming up with a range of characters too.

Trying to convert it to 3 sideways axes instead of the 4 that you have gave me some interesting results that also lay question to the sideways orientation of 'cuboidal' like positions.
Here is my attempt so far (though I am yet to work out how it progresses forward to the opponent):


As you can see, each of the square faces, that would extend forward to form a piece position, has a slightly different orientation to each other - which is allowed in 4D.
This gives us an interesting back 'row' for our 4D chess pieces that conforms with my preference that our 4D creatures think - like their arms - in 3 axes of travel for their board.

It also creates some more direct opportunities for movement - which I was struggling to find in your original design (but perhaps I'm wrong)?
In this one you can have:
- one direction of travel that goes colour 1, colour 2, colour 1, colour 2, etc. (similar to our forward/sideways movement
- one direction of travel that goes colour 1, colour 1, colour 1, etc. (similar to our diagonal movement)
- one direction of travel that goes colour 1, colour 3, colour 2, colour 3, colour 1, colour 3, etc. (new movement unique to 4D)

Also, it makes me like to think that our 4D creatures might have 3 genders: the male, the bearer, and the feeder.
So instead of a king beside a queen we might have a triplet royalty which I have shown with the letters: I, S and O.

I haven't explored, as much as you have, the various pieces this could give rise to.
The design I have made has 57 back positions as opposed to yours which has 64 back positions.
I would welcome any of your thoughts on what the pieces should be and how they might move?

[joan]

@Anders: have you looked into existing 3D chess variants? There are several existing already.

My favorite is Raumschach. It's played on a 5x5x5 board, the motion of the pieces is very intuitive in regards to regular chess. There is only one additional piece which is called a Unicorn, it can only move along space diagonals. All other pieces basically do what you would expect. It's very playable.

You can play online against a computer here (click on the green play button): https://www.jocly.com/#/game/raumschach

[quickfur]

It's relatively easy to make a 4D chess game: just lay it out as 2D slices in an n*n grid, each cell of which is an n*n subgrid. So the same (x,y) position in one subgrid would be connected to the (x,y) positions in the subgrids around it, and so on. A lot of directions possible.

The only challenge is that individual pieces will have to be tiny, or the width of the board (n) will have to be relatively small, as otherwise you wouldn't be able to fit it all on the screen at once. For a "standard" n=8 chessboard size, you'd need to fit a 64x64 grid on the screen, so on a screen of, say, 1600x1200 resolution, the chess pieces can only be ≤18 pieces tall and about ≤25 pixels wide, probably less because you need space for drawing dividing lines and stuff so that the grids are understandable.

OTOH, a 64^4 chessboard has 4096 cells, which is a LOT of cells. If you're willing to sacrifice some width, more realistic display options are possible. Say if we reduce it to n=6, we'd have an available height of about 28 pixels, which is much more practical in terms of having recognizable icons that don't require a microscope to see. The board will still have 1296 cells, so it's still nothing to sneeze at. You could devise the movement rules of the pieces in such a way that the resulting game will still be as challenging, if not more challenging, than the standard 8x8 chessboard.

[gonegahgah]

I realised, after trying to generate a 4D chess board, that my approach would have to use 6 colours also.
I'm thinking I might change these to light and dark variations rather than the primary and secondary colours... [done and edited in]
Here's a render of a possible 4D chess board.

It is made up of lots of octahedrons with 8 sides equal but not equal to 4 equal sides.
This provides 8 directions of opposite face movement, 8 directions of same end face movement, 8 directions of edge movements and 6 directions of point movements.
That's a total of 30 directions of movement I think! Classical chess has 8 directions of movement.

I'm going to try to make a lesser positions one as well. I'll see how it looks?

[joan]

You could devise the movement rules of the pieces in such a way that the resulting game will still be as challenging, if not more challenging, than the standard 8x8 chessboard.

The real challenge in my opinion is to create a game that is still playable instead of being too challenging. Even in 3D there are a lots of variants but most of them are unplayable because they are too hard to wrap your mind around. At the primary level of reasoning when you move a piece you have to consider all adverse pieces and their specific motion, like knights leaping one + two, to avoid hanging pieces. Thinking two steps ahead would already be very hard and coming up with any kind of mid-term strategy even harder. Being able to play only step by step + general intuition is probably not going to provide the joy of 2D chess.

I would say the maximum to even consider for being actually playable would be a board of 4⁴ = 256 hypercubes. Knowing that Raumschach is 5³ = 125 cubes, it seems a good step up.

Probably a good chess variant in 4D would provide analogies and extensions to the usual tactics, pin, fork, skewer and discovered attack. There are probably new concepts only doable in 4D. A single regular knight would be able to attack 48 cells at once I think, imagine the royal fork you could do with that. (But on a 4⁴ grid it would always be close to a side).

The introduction of new pieces should probably be kept minimal and straightforward IMHO.

[joan]

Care should also be taken to make the game "finishable".

I just spent more than half an hour in the end game against the computer at Raumschach, even though I had two Queens and a Rook against a lone King. The problem is that unlike in 2D where you can easily push the King to a border and then mate it, in 3D even a Queen is not enough to force it against the side. There are always holes through which it can sneak in above or below or in diagonal. In the usual case it has 26 possible escapes. I feel that in 4D it would be extremely hard to force a mate.

[quickfur]

It's true that the higher the dimension, the more ways there are to escape. Because of this (and other such effects) a straightforward lifting of 2D chess pieces and rules to 3D/4D is unlikely to produce a well-balanced game; probably a lot of play-testing and refinements are needed to produce something that's not only playable, but has good game balance.

One interesting thing about going to higher dimensions is that the number of axial directions (i.e., parallel to the coordinate axes, a generalization of vertically/horizontally-constrained piece movements) increases linearly, but the number of diagonal directions increase exponentially. In a high-enough dimension, pieces that are constrained to axial movements will be of very limited use, whereas pieces that can move in any diagonal direction will have great usefulness. Things like queens that can do both will be immensely powerful. Even kings, that can move diagonally even if the distance is limited, become very powerful because now it has available so many more ways of escaping. Limit the king to only axial movement, and it should become a lot easier to checkmate.

(Perhaps a bit too easy, though, so something in between may work better, such as allowing only "pure" diagonal movements, i.e., either you move axially, or if you move diagonally then it has to be diagonal w.r.t. all axes. I.e., in 3D, if you consider a 3x3x3 cube centered on the king, then the only allowed movements correspond with the 6 face centers and the 8 corners, but not the 12 edges. This roughly reduces the number of escape routes by a little less than half. In 4D, you'd have a 3x3x3x3 tesseract and only the 8 cell centers and 16 corners are allowed, not the 24 face centers nor the 32 edges. This doesn't stop the exponential increase of the number of pure diagonal movements, of course, but at least it makes it a little more manageable, and perhaps easier to balance against say 2-3 opponent pieces trying to corner the king.)

[quickfur]

Also, an interesting thing about 4D is the relationship between the tesseract and the 24-cell.

Consider the 3x3x3x3 tesseract: if we allow all movement directions, that's 8+16+24+32 = 80 directions. Interestingly enough, if we take only the "pure diagonal" directions, i.e., just the 8 cell centers and the 16 corners, we have 24 directions, and these actually correspond with a 24-cell. If we take only the 24 face-centers as allowed directions, that actually forms the dual 24-cell. This seems to suggest that you can have "bishop-like" pieces, where one is allowed the first set of 24 directions, and the second is allowed the 2nd set of 24 directions, and they would mutually complement each other as duals. Then the remaining 32 edge directions can be covered by other piece types.

Furthermore, there's this interesting decomposition of a 24-cell into 3 16-cells. If we allow only the 8 cell-center directions, then it corresponds to a 16-cell. However, there are two other 16-cell direction sets that we can obtain, by alternating the 16 corners of the tesseract. These also form 16-cells! Except in two different orientations from the original 16-cell. Because of this correspondence, they can be thought of as a kind of "skew axis-aligned movement", sort of like the movement of a rook except skewed in two different directions. All together, these 3 different "skewed rook" directions complement each other in perfect symmetry, together forming the directions of a 24-cell. The dual 24-cell, corresponding to the 24 face centers of the tesseract, may also be decomposed into 3 16-cells in the same way. Together, then, you have 6 sets of complementary "skewed rook" direction sets, in two groups of 3 mutually-complementary sets.

4D is fun, isn't it?

[gonegahgah]

You make a very interesting point quickfur about how to put the "king" into check mate.
That is an interesting idea to explore.
I hope I can explore the paths available sometime soon...