4*Chess (four dimensional chess)

Here's a 4 dimensional chess game that can be played as if on a 2D plane. The game is played using sixteen 4x4 (2D) mini-boards that are arranged in four columns and four rows. A 16x16 board with appropriate spacing could be used even on a coffee table; it would be about the size of a Scrabble board (15x15 for that). Using a computer program for it, a player could check if he is making a legal move, and whether it is mate or stalemate, for example. It's based on a BASIC computer program I made for it in the 1980's, which took up less than 16K. I played a game of it with a friend, and it didn't take too long after a blunder, as we knew a 4*Chess King and 4*Chess Queen vs. 4*Chess King can win very quickly.


Note that some links are provided in the Notes section, for further reference. A Game Courier preset for play is available.

Setup

Pieces

In 4*Chess, some 3 and 4 dimensional moving pieces are introduced, and all the pieces may possibly move between the mini-boards when performing a move (note that 'coordinate' in these instructions refers to the rank or file of a square on a mini-board, or refers to the row or column number of a mini-board):

D=4*Chess Balloon (I'd nickname it Dirigible) - moves like a bishop except changes 4 coordinates as it moves & stays on same square colour (standard 4D fairy chess piece);

U=4*Chess Unicorn - moves like a bishop except changes 3 coordinates as it moves (standard 3D fairy chess piece);

B=4*Chess Bishop - changes 2 coordinates as it moves, like a bishop (does not change square colour even if moving to another mini-board);

R=4*Chess Rook - changes 1 coordinate as it moves, like a rook;

Q=4*Chess Queen - moves like a 4*Chess Bishop or 4*Chess Rook, or a 4*Chess Unicorn, or a 4*Chess Balloon;

K=4*Chess King - moves like a 4*Chess Queen, only one square/mini-board at a time (no castling);

N=4*Chess Knight (nicknamed Horse) - changes 1 coordinate by 1 square and 1 coordinate by 2 squares, like a knight;

P=4*Chess Pawn - moves like a 4*Chess Rook (unless capturing) except only moves forward one square at a time on a rank, or forward by one column or one row to another mini-board (but moves to same rank & file there). If it is making a capture it moves like a 4*Chess Bishop, except only moves by one square, or by one mini-board that's adjacent diagonally or by one row or by one column, and never retreats by rank or mini-board (i.e. by row or column). 4*Chess Pawn promotions occur on the last rank of the corner mini-board where the enemy 4*Chess King starts the game, and a 4*Chess Pawn may promote to any 4*Chess piece type (other than 4*Chess King). There is no double step or en passant, and it is possible for a 4*Chess Pawn to early on avoid being captured by an enemy 4*Chess Pawn simply by moving to the last rank of a mini-board (except for the appropriate promotion mini-board).

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Unicorn {(U)} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Bishop {[B]} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Knight {(N)} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Rook {(R)} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Balloon {(D)} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess Queen {(Q)} involved:

Here's a diagram illustrating all possible legal moves on an empty board by the White 4*Chess King {(K)} involved:

Here's a diagram illustrating all possible legal moves by the 2 White 4*Chess Pawns {(P)} involved:

The White 4*Chess P at ab43 in making non-capturing moves can advance like a 4*Chess Rook, but only forward, by a single step, towards the last rank of White's promotion mini-board in the upper right corner. That is, to either nearby marked square if it is unoccupied. In making capturing moves it moves like a 4*Chess Bishop, but only forward, by a single step, towards the last rank of the promotion mini-board. That is, to any square occupied by a Black 4*Chess Unicorn in the diagram {[U]}, taking it. Note that none of this White 4*Chess Pawn's possible moves result in it promoting since it will not have reached a square on the last rank of White's promotion mini-board (i.e. any of the squares da44, db44, dc44 or dd44). Similarly, the White 4*Chess P at db13 in making non-capturing moves can advance like a 4*Chess Rook, but only forward, by a single step, towards the last rank of White's promotion mini-board in the upper right corner. That is, to either nearby marked square if it is unoccupied. In making capturing moves it moves like a 4*Chess Bishop, but only forward, by a single step, towards the last rank of the promotion mini-board. That is, to any square occupied by a Black 4*Chess Rook in the diagram {[R]}, taking it. Once again note that none of this White 4*Chess Pawn's possible moves result in it promoting.

Here's a diagram illustrating all possible legal moves by the 2 White 4*Chess Pawns {(P)} involved:

The White 4*Chess Pawn at square cb34 has earlier reached the last rank of a mini-board. In making non-capturing moves it can move to either nearby marked square if it is unoccupied. In making capturing moves it can take any Black 4*Chess Knight in the diagram {[N]}. It can also take the Black 4*Chess Queen in the diagram {[Q]}, in which case it is promoted to something since it has reached a square on the last rank of White's promotion mini-board. The White 4*Chess Pawn at square bb12 in making non-capturing moves can advance like a 4*Chess Rook, but only forward, by a single step, towards the last rank of White's promotion mini-board in the upper right corner. That is, to any of the 3 nearby marked squares if it is unoccupied. In making capturing moves it moves like a 4*Chess Bishop, but only forward, by a single step, towards the the last rank of the promotion mini-board. That is, to any square occupied by a Black 4*Chess Balloon (aka Dirigible) in the diagram {[D]}, taking it.

Rules

Stalemate is a draw, as in standard chess. 3-fold repetition and 50 move rule also are draws.

Notes

An implication of the above is that the following pieces have certain max. number of directions that they can move along in making a move:

4*Chess R: 8 directions max. (including the 4 if it stays on the same mini-board as it starts)

4*Chess D: 16 directions max.

4*Chess B: 24 directions max. (including the 4 if it stays on the same mini-board as it starts)

4*Chess U: 32 directions max.

4*Chess Q (or 4*Chess K): 80 directions max. (the sum of the above pieces' max. directions)

4*Chess N: 8 plus 8 plus 4x4 plus 4x4 = 48 directions max. in theory, but less than that since the mini-boards (and the number of them) are not large enough to ever allow it (actual max. = 24).]

An example legal first move in 4*Chess would be to move White's 4*Chess Pawn in front of his 4*Chess King one square forward (staying in the same mini-board). Then, Black could reply the same way. These first moves can be written in 4*Chess notation as 1. Paa12-aa13 Pdd43-dd42 if a game were to be recorded. Thus, all four coordinates (Column, then file, Row and rank) are given for where a 4*Chess piece or 4*Chess Pawn starts and finishes its move. If a 4*Chess Pawn promotes, this is recorded by tacking on the letter of the 4*Chess piece type selected after the promotion square's four coordinates. Similarly, a capture, check or mate can be indicated as in standard chess notation.

P.S.: With the given setup, White might develop an early initiative, at least by 1. Ncd11-dd13, at once attacking Black's queen. Perhaps some may wish to experiment with the setup, e.g. altering it by switching the location of each side's queen and rook that are on the same rank & file cell as their king's cell is. With this different version of the setup, note that without yet moving, each queen has potential deep scope on its column at least (besides seeming safer), which may not be an entirely positive thing. Plus, the relative positioning of each side's rooks in this version of a setup doesn't seem so aesthetic to me as in the original version.

Beyond easily checkmating a lone 4*Chess King with just a 4*Chess Queen, I've imagined checkmates of a lone 4*Chess King with other 4*Chess pieces (excluding 4*Chess Pawns), though these might not be even close to being generally forcible 'basic' mates if the starting point is not totally favourable (i.e. beyond mate in one move being available). Notwithstanding that, I conceived of possible mates in one move using any four such other pieces, but with at least two of them not being 4*Chess Balloons.

Exceptional cases requiring less than four such pieces where mate in one is possible that I've found include having a 4*Chess Rook plus two 4*Chess Bishops, or plus two 4*Chess Unicorns or plus two 4*Chess Knights (or plus a 4*Chess Bishop and a 4*Chess Unicorn), (or plus a 4*Chess Knight and a 4*Chess Unicorn), (or plus a 4*Chess Knight and a 4*Chess Bishop). Mate in one with three 4*Chess Rooks is also possible.

Five 4*Chess Balloons plus one of any other type of piece may make a mate in one possible, too. I've also conceived of possible checkmate positions with exactly eight 4*Chess Balloons (the number one starts the game with).

In all these cases of mate in one (i.e. excluding a 4*Chess Queen or a 4*Chess Pawn), the lone 4*Chess King was in an extreme corner square, with the opposing 4*Chess King very close.

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by a White 4*Chess Queen {(Q)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 2 White 4*Chess Unicorns {(U)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 2 White 4*Chess Bishops {(B)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 2 White 4*Chess Knights {(N)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by a White 4*Chess Bishop {(B)} plus a White 4*Chess Unicorn, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by a White 4*Chess Knight {(N)} plus a White 4*Chess Unicorn {(U)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by a White 4*Chess Knight {(N)} plus a White 4*Chess Bishop {(B)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 3 White 4*Chess Rooks {(R)} and a White 4*Chess King {(K)}:

Here's another diagram illustrating a Black 4*Chess King {[K]} checkmated by 3 White 4*Chess Rooks {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a possible leadup to the previous diagram; it's White to play and mate in 2 moves:

After 1. Qbd34xQcd44+ (if 1. Rcb44xQcd44+ or 1. Rcd34xQcd44+ then 1... Kdd44-dc44 is possible) now if 1... Qdc34xQcd44 White plays 2. Rcb44xQcd44 mate, reaching the previous diagrammed position.

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 8 White 4*Chess Balloons, aka Dirigibles {(D)}, and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 5 White 4*Chess Balloons, aka Dirigibles {(D)}, a White 4*Chess Knight {(N)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 5 White 4*Chess Balloons, aka Dirigibles {(D)}, a White 4*Chess Rook {(R)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 5 White 4*Chess Balloons, aka Dirigibles {(D)}, a White 4*Chess Unicorn {(U)} and a White 4*Chess King {(K)}:

Here's a diagram illustrating a Black 4*Chess King {[K]} checkmated by 5 White 4*Chess Balloons, aka Dirigibles {(D)}, a White 4*Chess Bishop {(B)} and a White 4*Chess King {(K)}:

I'd guess the relative values of the 4*Chess pieces to be about as follows:

4*Chess P = 1

4*Chess D = 1.2

4*Chess R = 3

4*Chess B = 3.4

4*Chess U = 3.4

4*Chess N = 3.4

Note that I didn't wish to put the seemingly humble 4*Chess R below 3, as it seems it could deal with at least 3 passed pawns in endgames at times, and it is not colour-bound like a 4*Chess B (which, however, reaches a much greater number of cells on average, and has much more influence on other mini-boards' cells than a 4*Chess R would). Also note that the rather humble 4*Chess D is not only colour-bound, but has two additional forms of binding (plus lacking any influence on its own mini-board, and reaching a relatively low number of cells on average, though an intangible is that it might develop deep scope on a critical 4D diagonal). In some perhaps rather uncommon circumstances, a 4*Chess B can restrain 3 passed pawns in an endgame, so that's some justification for valuing a 4*Chess B as worth more than 2 pawns (plus, it seems to be more of value than, say, a B in Circular Chess, or worth more than a 4*Chess R here).

As for the 4*Chess U and 4*Chess N, they are judged to be similar in value to a 4*Chess B, in terms of their attributes. A 4*Chess U is not colour-bound, but lacks a 4*Chess B's influence on its own mini-board (otherwise they reach about the same number of cells on average). A 4*Chess N is not colour-bound and has its leaping ability, plus it influences cells on its own mini-board. However, it reaches less cells on average than a 4*Chess U (or 4*Chess B), which is also more of a long range piece. The latter is also true for a 4*Chess B compared to a 4*Chess N; plus, as noted above, a 4*Chess B does influence cells on its own mini-board, but it is colour-bound to offset this).

Just as a chess Q = R+B+P in value,

4*Chess Q tentatively = ((4*Chess R + 4*Chess B + 4*Chess P) + 4*Chess D + 4*Chess P) + 4*Chess U + 4*Chess P = 14, but actually I penalized a 4*Chess D by two pawns worth for its additional forms of binding, so I think the value of a 4*Chess Q = 14 + 2 = 16.

A chess K has a fighting value of 4 (even though it cannot be exchanged); this value in my view might be rather oddly expressed (for lack of a known formula) as chess K = 32 x (max. # cells chess K moves to [eight]) divided by (# of cells on a chess board [sixty-four]) = 4, and similarly,

the fighting value of a 4*Chess K = 32 x (max. # cells 4*Chess K moves to [eighty]) divided by (# of cells in 4*Chess [two hundred and fifty-six]) = 10, which seems in the right ballpark, given a 4*Chess K's great influence in mid-board.

It could be an arduous task to discover all forcible 'basic' 4*Chess mates vs. a lone Black 4*Chess K; I'd be speculating to say that the total value of the pieces involved (excluding 4*Chess Pawns or the White 4*Chess K) should be worth at least 16 points (the value of a 4*Chess Q).

How would I try to assess the strengths & weaknesses of this 4D variant? In attempting to invent a number of 4D variants, I came up with 9 equally weighted (sometimes slightly conflicting) criteria, to try to compare these variant ideas with each other:

1. Low total number of piece types (IMHO, the game scores well);

2. Low total number of pieces in the setup (good score again);

3. The pieces (other than pawns) are "natural" or "pure" to 4D Chess (scores perfectly, like Raumschach similarly does for 3D Chess);

4. Good rules (& setup) for pawns (I like them; there's no ideal solution);

5. Some chance of early mate or relatively short game (just adequate);

6. Variety of viable exchanges of differing piece combinations (adequate);

7. Variety of "major" and "minor" pieces (adequate, but Q is the only major piece);

8. K can legally attack opposing pieces/pawns (good score, but powerful K);

9. Pieces (especially N-like) may obtain great scope (a 5x5x5x5 board would be better);

Fwiw, in scoring 4*Chess with these 9 criteria, giving a score of 0-4 for each, I found it had a total score of 21/36 (or a little over 58%). IMHO the game's most pleasing feature may be that it fully meets criteria #3.

For an older link to a discussion of 4*Chess:

Note 4*Crazyhouse would be the same game as 4*Chess except it would be a crazyhouse version. Regarding drops, in 4*Crazyhouse a 4*Chess Pawn may not be dropped on a player's own 1st rank on any mini-board, nor on a promotion square.

A link about plain (8x8) Crazyhouse:

4*Bughouse would be the same game as 4*Crazyhouse, except it would be a bughouse version (two teams of 2 players, using two 4*Chess setups in bughouse fashion).

A link about plain (8x8) Bughouse:

Another link, which discusses 4*Crazyhouse & 4*Bughouse:

I'd note that (even without using a crazyhouse or bughouse version) 4*Chess may well be rather compter-resistant as far as giving fairly skilled humans a chance vs. an engine, at least for a while (due to, e.g., the large board, like for standard 19x19 Go, and also due to the large number of legal moves on average per turn).

Here's a chessvariants.com link to Super4*Chess (4x4x4x4 variant that uses some K-like moving pieces, which was inspired by 4*Chess):

Here's a similar link to 5*4DChess (5x5x5x5 variant inspired in turn by Super4*Chess):

A similar link to Open King 4*4DChess (4x4x4x4 variant that uses 48 pieces, inspired partly by Super4*Chess):

A link to Open King 5*4DChess (5x5x5x5 74 piece variant inspired partly by Open King4*4DChess):

A link to 4D Quasi-Alice Chess, an 8x8x2x2 variant partly inspired by Alice Chess:

Finally, here's a setup position for a 4x4x4x4 4D variant that uses just 80 pieces, with the same rules as 4*Chess, which I'd dub Slim 4*Chess, in case one prefers to have just 2 knights & just 2 rooks for each side at the start (though personally I prefer my setup for 4*Chess, if only since each side has an even 6 mini-boards, plus all 4 columns, at least partially occupied by some of their pieces):

Last revised by Kevin Pacey.