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A Discussion of the Mapping

When you think about 3D Chess, you start by mapping the moves of the well-known 2D pieces into 3 dimensions.

The first mapping I came up with turns out to be the "traditional" mapping. I didn't like it, so I looked further.

The new mapping I came up with is chosen especially because it is easy to use, easy to visualize and learn, and is reasonably clean. In this mapping, the square on top of a white square is still a White square; a Bishop makes a normal Bishop move, but as it moves it may fly level, descend, or ascend; a Rook does the same, but may also move straight up or down; and Kings or Queens combine both directions.

Vertical and horizontal views of these moves do not look the same.

The Pawn and Knight, however, do have symmetrical moves. The Pawn makes normal Pawn moves on its own level, and may capture in any of its other 8 "forward" directions; the Knight moves to all the (0,1,2) squares as well as to all the (1,1,2) squares.

Look at the pattern my 3D Knight makes; how pretty it is! But is it too strong? My 3D Knight moves in 6 times as many directions as the 2D Knight, but my 3D Bishop moves in only 3 times as many. Has the Knight improved too much? Too much for what?

In the game as it stands, the 3D Knight seems to be a good piece, and it seems like a good thing that it is as strong as it is, because of the difficulty of checkmating a 3D King.

I am convinced that the "strict construction" 3D Knight would be too weak; although it moves in 3 times as many directions as the 2D Knight, too little of its gain is in the forwards directions, and its pattern of square coverage is very sparse.

Postulating instead a Knight that moves to any of its 2D squares, or to any of the squares one above or below, we get a 3D Knight with 3 times the mobility of the 2D Knight, a balanced gain forwards, and an easy pattern to learn; but it isn't 3D enough, and I don't like it. This Knight can't move from 1a1 to 2a3, which is a "strict-construction" move.

Therefore I must vote to keep my 48-square Knight, but perhaps that causes a problem in the next section.

3D Chess With Different Armies

It would be nice if piece values were so well preserved by their 3D mappings that 8x8x8 Chess With Different Armies could be played using the standard different armies.

If Knights gain more from their 3D mapping than other pieces do, then the Nutty Knights will be too strong, and will always beat the Colorbound Clobberers, won't they?

I'm not at all sure that this is a problem, and I believe that the overall values of the armies will still be close enough to make the game worth trying.

Of course, you can also ask if it's boring for each player to have 8 copies of the same army. What if each player had a set of Clobberers on the bottom level, a set of Rookies on the second level, and so on? (Ah, but this is not Chess anymore because it's too hard to coordinate so many different kinds of pieces; let's reserve this for Great Chess.)

By the way, I think I do not need to explain the way these pieces would move in 3D because their moves are well-defined by the mapping rules.

8x8x8 3D Chess Variants

The fact that the moves of the 3D pieces are defined by a simple mapping from the 2D rules means that most Chess Variants already have a well-defined 3D version.

Alice's Chess in 3D? Not for me, anyway; but it's well-defined.

Simple games such as Cylindrical Chess, Billiards Chess, Chessgi, Knight Relay, Progressive, Avalanche, Dynamo, Grid, and so on, are all possible and enjoyable, and are well-defined as they stand. Others fall into this category; the ones listed are those on which I have spent a few moments of thought.

Momentum Chess in 3D seems well-defined but too confusing.

Games of the co-chess family would need some adjustment of their rules to be playable.

Trapdoor Chess in 3 dimensions is an interesting idea; likewise Conveyer Belt Chess, Rotating Grid, Pinwheel Chess, and other variants with moving boards. The rules would need some work, to redefine the way that the board moves.

Liar's Chess was an attempt to create a form of Kriegspiel without a moderator; limited lying in the 3D game would be interesting: you can only have fibbed about which level you went to.

Newton's Chess is my name for the game that I have discussed as a form of Momentum Chess, and which has come up in some email discussions of Billiards Chess: the pieces have different weights and speeds, and their continuing motions are defined by Newtonian mechanics. In 3D Chess, you could include gravity.

(Nobody has been able to make a proper game out of Newton's Chess because of the problem of fractional speeds or positions -- if a Pawn bumps a Rook, the Rook gets a speed of 0.2 squares per turn. Nobody wants a game with fractional speeds.)

Berolina Pawns are simple and interesting.

Why All Those Kings?

In my setup of the starting posiiton, I gave each side one King and seven Commoners (pieces that move as King but can't be checked). (Maybe this piece should be named the Pretender?)

This was done "by default", as the simplest sort of setup to start with; you can use 8 normal chess sets and simply put a crown on the one royal King. However, it is not a bad setup because the Commoner is an especially good piece to have when you are in the late endgame and are trying to mate the lonely enemy King with your few remaining pieces.

However, nobody with chess variants in their blood can look at that setup without wanting to put in some Chancellors and Archbishops and Knightriders and so on....

Alternative Opening Setup

Instead of stacking 8 normal chess sets vertically, you could put 28 Rooks on the outermost places, 20 Knights within, 12 Bishops, 3 Queens and a King. I prefer the stack of 8 standard sets.

Other Kinds of Pieces

The Crooked Bishop is crooked in only one dimension? If you want it to be; however, it's actually well-defined how it could swerve in all dimensions.

The Snark is a new piece that moves to all the squares at (1,2,2) or (2,1,2) or (2,2,1). There is no 2D mapping for the move of this piece. As a piece in its own right, it is rather weak; as a power added to other pieces, it is probably as strong as the Wazir, Ferz, Dabaaba, or Alfil.

In 3D, the NS can be considered a new kind of Augmented Knight.

A piece that combines the powers of W, F, A, D, N, and S can jump to any square of the 5x5x5 cube surrounding its starting position.

Interesting thought: in one dimension, the W and D are all you need to cover this range; in two dimensions, the WFADN do the trick; and in 3 dimensions you only need one new piece. The fact that you only need one new piece might be taken to mean that my mapping was too generous? Perhaps there should have been more kinds of new pieces?

How many pieces are needed in 4 dimensions. Oops. No, I will not start in on 4D Chess.

The "Rose" is a circular Knightrider; the two-dimensional Rose might start from e1, move to g2, if g2 is empty continue in the same move to h4, then g6, e7, c6, b4, c2, and return to e1.

Special rule: it is considered unfair for a 3D expert to use the 3D Rose against a 3D novice. Entertaining, yes, but definitely unfair.

The same special rule applies to the Corkscrew, as described later.

It would also be possible to have limited 3D pieces, for example, a variation of the Rook that moves only either on its starting level or straight up and down; or a variation of the Rook that makes all the Rook moves *except* on its starting level or straight up+down. The same for Bishops, of course.

My funny notation wouldn't handle this at all.

Just as the Pawn can only go forward, the Nemesis can only move towards the enemy King, and the Horace can only go West, there could be a 3D piece that only descends.

A Corkscrew move would be a 3D spiral. For example, a spiral 3D Queen starting on 2b1 would have, as one of its many possible moves, the ability to go to 3b2, and if that is empty to continue as part of the same move to 3c3, and on to 2c4, 1c5, 1b6, 1a7, and 2a8.

You could also have a corkscrew Rook or a corkscrew Knightrider.

It is impossible to represent the corkscrew move on the 2D board.

You could have a Rook that rises/descends at half the usual rate, that is, one-half level per square traversed. From 1a1, it would go to 1a2, and if empty to 2a3, 2a4, 3a5, 3a6, 4a7, 4a8. This is the same as a piece that makes a W move followed by an F move, then W then F and so on; and I think it's described in the crooked-Bishop page.

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