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## The Diagonal that Isn't There

I've been using a method for mapping normal 2D chess moves into the third dimension that does not allow the use of the diagonal diagonal: a piece on a1 of one level cannot see a piece on b2 of the next level.

Similarly, the move from a1 to be-of-the-next-level looks natural but is simply impossible.

Rather than invent new powers for the pieces, I've thought of a different mapping that uses the normal set of pieces, takes advantage of these new squares, and seems generally more natural and satisfying.

## Changing The Colors

My old mapping assumed that colors must alternate if you move vertically up or down, that is, I assumed that the square above a black square must be a white square.

Under this assumption, the Bishop should move from a1 of one level to a2 of the next level, and so on. This seems unnaturally like aRook move, even though it is mathematically correct: simply rotate the cube 90 degrees and you will see.

However, if a1 is always a black square on every level, the Bishop move must be different. From a1, it can go to b2 of the same level, or to b2 of the next level up, or to b2 of the next level down.

In other words, for every step it takes, it ascends one level, descends one level, or stays on the same level. The Bishop now may move in twelve directions.

The Rook, in addition to the same rising, descending moves, may also move straight up or down, and so it moves 14 different directions.

The King or Queen move in 26 different directions; the King is at the center of a cube of places to which it may move. This is an excellently logical way to extend its powers into 3D, since the King in 2D Chess is at the center of a square of places to which it may move. There are 8 exterior squares in the 3x3 square, and 26 exterior cubes in the 3x3x3 cube.

The Knight in 2D Chess moves to all 8 of the squares in a 5x5 square to which a Queen could not go. In 3D terms, there are 72 exterior cubes in a 5x5 cube to which a Queen could not go; this seems excessive, and some of the destinations do not feel all that Knighty.

If we follow a strict construction of the Knight's move as "one, two, and over", it can go to 12 places, which seems too few (although its number of directions is multiplied by 3, same as the Bishop); but if we allow the "ascend or descend" addition too naively, the Knight can go to some places that the Rook or Bishop could reach.

Let's look at the vectors: The Bishop goes (1,1,0) [or (1,-1,0) or (-1,1,0) or (-1,-1,0)] when it makes a traditional move that stays on the same level. In these 3D rules, we have also allowed it to go (1,1,1) [or, of course, (1,1,-1) and so forth]. The Rook goes (1,0,0) or (0,1,0) in two dimensions, but we now allow it to go (1,0,1) or (0,1,1), or even (0,0,1) in three dimensions.

The 2D Knight is (1,2,0) or (2,1,0). The strict-construction 3D Knight adds (0,1,2), (0,2,1), (1,0,2), or (2,0,1). So if we add the ascend/descend rule (so that the Knight uses the same meta-rule as all the other pieces), the Knight can vary its move by one in whichever of the three dimensions its vector coordinate was zero.

With this rule, the Knight can go to 48 different squares, and there are still 24 squares in the 5x5x5 cube that neither a Knight nor a Bishop can reach. These unreached squares are really a new direction, not a very Knightish move.

From c3 of level 3, the Knight can go to the 8 traditional squares on level 3; or on levels 2 or 4 it can reach c1,a3,c5,e3 (the strict-construction squares), and also a2,a4,b1,b5,d1,d5,e2,e4 (varying its height by one from the squares on level 3, OR varying its rank or file by one from the squares on level 2 or 4); or on levels 1 or 5 it can reach the strict squares c2,b3,c4,d3 plus the added squares b2,b4,d2,d4 (varying rank or file by one).

In the 5x5 cube around c3, neither a Queen nor a Knight can reach from 2c3 (square c3 of level 2) to 4a1, 5b2, and so on: the coordinates of this new direction are (1,2,2) [and (2,1,2) and (2,2,1)].

Pawns, of course, are obvious.

## A Brief Digression to Look at the Old Mapping

In the old mapping, the Bishop could go (1,1,0) or (0,1,1) or (1,0,1): always two ones and a zero. The Rook in the old mapping could go (1,0,0) or any set of two zeroes and a one.

Knights were able to go to the "strict-construction" squares mentioned in the preceding discussion.

## Back to the New Mapping

In the new mapping, the 3D pieces are more powerful, and yet I think that it will be easier for players to see the possible moves. This mapping is just as correct as the older mapping.

The only disadvantage is that the new mapping is polarized. If you look at the cubical board from the side, you may see Bishops moving Rookwise and Rooks moving diagonally; and of course the colors of the squares mean that one of the 3 axes is permanently defined to be the vertical one. This seems minor, and I prefer the new mapping.

## Is This a Game?

Can you play 8x8x8 3D chess with this new mapping, using the 3D board I designed? Yes, but maybe not.

The only technical problem might be that the King is too hard to checkmate. At the end of the game, you will need more than a King and Rook to checkmate a bare enemy King.

However, in 2D Chess, a Rook is about one-eight of your entire starting force. With these rules of 3D Chess, you need considerably less than one-eighth of your starting force to checkmate the bare enemy king. I think it should be okay.

The practical problem is that because you start with 8 times as many pieces, there are probably 8 times as many moves in an average game; and with all these pieces on the board, you may need to think longer for each move than you would in a normal game.

Even so, it is clear that the only real 3D Chess is on an 8x8x8 board, each player having 64 Pawns, one King, and 63 other pieces which may or not be "16 each of Rooks, Knights, and Bishops, 8 Queens, and seven pieces that move like the King but do not need to worry about check"; that would be the simplest setup, but of course all of us know about lots of nice pieces that could be used instead of having so many of the same thing....

Because the 8x8x8 game is so large and unwieldy, many people over the years have attempted to devise more practical compromises that would give some of flavor of 3D chess without being so huge. Many of these games were enormously clever, but of course, they are not Chess. The only real 3D Chess is on an 8x8x8 board.

It is obviously true that I love chess variants, but it is equally true that I know the difference between Chess and chess. For example, my games of "Chess with Different Armies" are Chess with a capital C, but my games such as "Wand Chess" are merely chess.

Well, IMHO, YMMV, HAND.....

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