3 Dimensional Cubic Chess 8x8x8 Board

What about 8D chess? I saw 5D chess but its not enought. I want something Beyound mortal man, cosmic horror that will tear down my mind and sanity while only looking at the board or attempting ot move the pawn.

A single 3Dqueen can mate in the corner of a 3D board (supported by a 3-king). In multiple ways, in fact - there are 7 cubes (the analog of squares) where a 3Dqueen controls the corner cube and all adjacent cubes. Then there are multiple cubes for the attacking 3Dking to protect that 3Dqueen from.

Forcing it should be possible, a process of nudging the defending king nearer the corner by first getting it on a face, then an edge and finally the corner.

I conjecture that the generalisation of this procedure works to allow a NDQueen and NDKing to mate an NDKing on any NDrectangular board.

(I am not 100% sure. While a queen chops the board in 2 in 2D, it does not do so in 3, so proximity has to be enough to avoid the defender getting through the holes. I am assuming an NDqueen can move any vector where the elements of the vector are k, -k or 0 for some natural number k. Eg in 3D it can move in 3^3-1 directions, which is 26 directions. In n-D it is 3^n-1 directions. [Check - in 2D it is 3^2-1 = 8 directions, which is right].

Yeah unlike in 2d, the lines the queen controls don't cut the king off from many of the 26 possible directions, because the lines it controls are still one dimensional and to cut off a king in 3d would have to cut off an entire plane at a time to force the king to a face, let alone an edge. Obvious support checkmates work even just in the center of an edge-plane, with the queen directly on top of the king and the king defending in from any cube 1 away from it. I don't think a king and 1 queen could cut off all the possible escape cubes together though from a central position. Maybe 2 Queens, that was my next question, hard to visualize but could 2 Queens mate the king in the center of a 3d board, without the kings help?

I think one 3-queen (and 3-king) is enough to force the opposing king to the boundary. Probably even the corner, if necessary. Indeed I think this is true in n-D for all n. But I am not close to sure!

Below, you see 3 slices of a 3D board, all with a boundary on the left and bottom, and the 0 slice is also a boundary (so the bottom left of slice 0 is a corner).

I originally thought this example was fairly convincing, but black can do better and make it more difficult by going to slice 2. I believe white has a win, but to demonstrate this requires a lot more care than I have put in.

3 kings? I only have 1 king in my version. The question is how many Queens does it take with the help of the king?

Each row is a part of a position. There are two kings: K is white, k is black. The endgame is KQ v k.

The above is probably not critical line as black can move to the third slice on the first move, a knight's move away from the white queen (cube 220, where the digits are slice / horizontal from left / vertical from bottom). There could be a slice 3, so there is some corralling necessary.

Best would be to write a program to generate the K v KQ table base for the game. I presently believe KQ does mate K in the game discussed (say with 8x8x8 board).

Yeah that would be fascinating! Probably could only do 3 or 4 piece tablebases in 3d chess due ti the exponentially higher number of positions.

A slightly semi-relevant topic I'd like to bring up here as well, the 8 Queens problem. That is placing 8 Queens on a chessboard such that no 2 queens threaten each other; in 2d there are over 90 solutions. Is it even possible in 3d or an 8x8x8 board? Can 64 Queens be placed so that no 2 threaten each other in any direction in any plane, or on any 3d diagonals? I think avoiding 3d diagonals with that many is impossible, but what about just no two threatening each other on any of the 24 cross-section 8x8 planes within the cubic board?

I might made a new notation (If someone already made it then I’m sorry that I didn’t read all the comments) it works like this, the layer number (3d), file (2d), rank (1d)

EXAMPLE: 4e5 Pawn moved to the 4th layer, e file, and 5th rank

King and queen mate is definitely possible. King and rook mate becomes impossible and you need two rooks. King and three bishops is 100% possible. There is a position where 2 knights and a king mate the other king, but i am sure that 3 knights is possible.

Yes king and queen checkmate positions are obviously possible, as already mentioned several times in this thread. I also specified a specific 3d position where 2 knights and a king mate the other king in the corner. 2 bishops may be possible, but 3 definitely are. Remember bishops move along both 2d and 3d diagonals and only 8 cubes need to be guarded from a corner.

My notation is simply a 3 digit number for the little cube. The order makes it clear which.

Not only possible, the king can be mated by KQ on any boundary location. But I am sure (should really prove this) that all mates are "contact mates" unlike in 2D, even in a corner.

Actually, not only 2 bishops can checkmate, but they can do so on a face, not just an edge or a corner! A white king on 3e4 and white bishop on 2e4, stalemates a black king on 1e4. Any check by a 2nd bishop is mate, and it is the same thing with 2 pawns. And yes the queen has to be 1 cube away from the king to checkmate even if the king is in a corner.

I see. That 3D bishop is rather powerful. 20 adjacent cubes controlled, out of 26, compared to 4 out of 8 for 2D.

Yes, and I calculated a bishop in the center can control 65 total cubes, more than triple the rook's 21 possible moves. Not that a rook could get easily trapped by a 3d bishop. But here's how I got 65:

13 cubes to move to in each of the three perpendicular cross section planes = 39. Two sets of 13 moves each along the 3d diagonals = 26. 39 + 26 = 65. That's the thing with diagonals, they increase geometrically as dimensions are added! Whereas the rook is always the number of dimensions X 7 possible moves. That's what we were debating earlier, whether a bishop should have both 2d or 3d moves or have separate pieces for each.

Another cool factoid, a bishop that's restricted to only 3 dimensional diagonal moves and can't move 2d, can only access 1/4 of the board, not even 1/2. Because in every 2x2x2 section, a 3d bishop can only move between two out of the 8 cubes.

I made a table breaking down each piece into its 2d, 3d, and combined total number of moves: Note these are based on maximum possible number of moves from the most central location within the cubic board possible. I didn't even attempt to try and figure out the totals from less than optimal positioning..etc lol.

It is also staggering how many pieces can end up on the cubic board. With 64 starting pawns and the initial setup given, up to 67 Queens, 92 rooks, and 84 knights are possible via pawn promotions. Another major difference is that a knight can triangulate in 3d chess. It couldn't do it if we restrict the movement to any one plane at a time (only 2-1 moves), but a knight could back to its cube in an odd number of moves by combining two 2-1 moves and one 2-1-1 move.

Vast numbers of possibilities mean that all play in such a game will very likely be low quality, worse than bullet play in chess. It's just too much to consider - literally hundreds of legal moves at each step.

Games are likely to be unpleasantly long as well, unless people are willing to give up when the complexity has made them blunder.

A better game is a smaller board with the new pieces. One possibility is 3 x 3 x 8. This leaves room for 9 pieces, 9 pawns and an opponent at the same distance as on and 8 x 8 board. But perhaps the shape is unattractive. 5 x 5 x 5 might be ok, but 25 pieces and 25 pawns would be very cumbersome.