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# 3D Offset Chess

Alberto Monteiro and Marek Ctrnak have emailed me some valuable suggestions about improving my Funny Notation.

In the course of this discussion, and perhaps as a result of it, some new chess variants have been inspired.

The following is an idea that somewhat spoils the attempt to extend the notation so that it covers 3D Chess.

## 3D Offset Chess

One of the problems of 3D Chess is that it is physically difficult to reach into the middle of a board on a lower level because the upper boards get in the way.

Another problem of 3D Chess is that the number of pieces on an 8x8x8 board is simply too huge.

One way to solve both difficulties at once is to play on a smaller board with diminishing levels. For example, if the lowest level is a 3x8 chessboard and the next level is a 2x8 chessboard and the top level is a 1x8 chessboard, you have a three-dimensional playing area of a mere 48 squares, and if each level is centered above the previous level it should be fairly easy to reach into the middle of the bottom level without knocking anything over.

However, in this case it is difficult to define the 3D motions of the pieces in normal terms: if we go straight up from the square 1b1, we see half of the square 2a1 and half of the square 2b1, so which one is "straight up"?

Of course, the answer must be that "straight up" includes all of the squares above the starting square, and therefore the normal 3-dimensional movements of the pieces will be too powerful, and therefore we must rule that pieces on this sort of board can only go "straight up or down".

This is not bad; we are playing on a small board with a simplified 3D rule, so it makes an excellent introduction to 3D Chess.

My personal preference is to say that Pawns may not make 3D moves in this game, and everything else can move one square "straight up or down".

For example, you might have a small game with a 3x6 board above a 3x7 board. This would have exactly 39 squares, so it could be called Constitutional Sesquicentennial Chess, in honor of the 150th anniversary of the Constitution of the USA (an event which occurred in 1939).

It is interesting that a double pyramid, a 3x8 board with a pair of 2x7 boards directly above and below, and a pair of 1x6 boards directly above those, has exactly 64 squares. Unfortunately, this board would sacrifice the physical convenience gained by having the smaller boards atop the larger.

If you don't mind reaching under a board of width 3, then a board of 4x8 + 3x7 + 2x6 + 1x5 would give 66 squares, a pretty nice size. One obvious setup would be

```    level 1     level 2     level 3     level 4
a b c d      x y z        b c          y

1  N X Q N      R Z R
2  P P P P      P P P        B B          K
3  . . . .      . . .        P P          P
4  . . . .      . . .        . .          .
```
The pieces X and Y could be Chancellor and Archbishop, for example.

You'll notice that an interesting variation of algebraic notation is suggested. You'll also notice that the Kings in the attic are very far forward and close to each other. In fact, the White King at 4y2 is directly above the Pawn at 2y2, and is therefore out in front of the Pawns on level 1.

For this reason, it seems better to have a setup like this:

```    level 1     level 2     level 3     level 4
a b c d      x y z        b c          y

1  R K Q R      X Y Z
2  N B B N      P P P        P P          .
3  P P P P      . . .        . .          .
4  . . . .      . . .        . .          .
```
In this case, the attic is empty. The Bishops can not get out without Pawn moves, and the Pawns form a more solid front.

You will notice that a normal 3D Knight move from 1a2 to 2b2 would still have the same geometrical shape on this board, and thanks to the strange numbering I've used for the squares it would have its normal notation. The problem is that a normal 3D Knight move from 1a2 to 2c2 would geometrically land on the corner shared by the squares 2y2 and 2z3, and allowing it to choose any of the 4 would make the 3D Knight much too strong.

## Circular 3D Offset Chess

In order to avoid being forced to think about squares of different sizes, I assume that a Circular 3D Chess board will always have the same number of ranks (making a circuit of the board will always cover the same number of squares, no matter what level or what file you do it on).

It is impossible to avoid mentioning here my unwritten game "Towers of Hanoi Chess". In Towers of Hanoi, the hole in the middle of each level of the board is always the same size, and therefore "Towers of Hanoi Chess" uses a Circular 3D Chess board but NOT a Circular 3D Offset Chess board.

In "Towers of Hanoi Chess", the a-file is the one nearest the center, and so the top level of the board has only an a-file, which is directly above the a-file of the bottom level.

In Circular 3D Offset Chess using a 3 level board, the top level would have only the b-file which would be directly above the b-file of the bottom level.

In any form of Circular 3D Chess, it is pleasant to have a rule that allows you to rotate one or more levels of the board; making this physically possible imposes certain constraints on the manufacture of the board, and so I suppose we are unlikely to see a physical 3D Circular Chess board that allows level rotation unless or until somebody builds a "Towers of Hanoi Chess" board!

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