5*4DChess (four dimensional chess)

Here's a 4D game I call 5*4DChess that uses twenty-five 5x5 (2D) mini-boards. Because it is not so easy to checkmate a K in many 3D or 4D chess variants, perhaps including my earliest 4D variant, 4*Chess, besides using standard 4D pieces in my second 4D variant, Super4*Chess, I added 6 powerful types of mann-like pieces. That was with the aim of not only making checkmating a king during a game easier, but also so that that game (and now 5*4DChess, too) still might be viable to play & enjoy. 

My original inspiration behind 5*4DChess was to give pieces (especially a 4*Chess N) chances for even greater scope than on 16 4x4 mini-boards, possible on many of the added & larger mini-boards. It also inspired me that the mann-like pieces (twice as many of which are included as in Super*4Chess) would have somewhat less strength, while still remaining useful for checkmating the opposing king. Note that 5*4DChess pawn movement and promotion rules are rather more like for standard chess than was the case for my earlier 4*Chess and Super4*Chess four dimensional variants, as I felt this would be more suitable for the larger 5x5x5x5 board used in this game, besides for the initial setup that I chose.


Note that some links are provided in the Notes section, for further reference.

Setup

Pieces

In 5*4DChess some 3D & 4D moving pieces are introduced, & 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 of a mini-board). Note a total of 14 piece types are used, 13 borrowed from my earlier 4*Chess & Super4*Chess 4D chess variants:

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 (stays on same square colour);

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

Q=4*Chess Queen - moves like a 4*Chess B or 4*Chess R, or a 4*Chess U, or a 4*Chess D;

K=4*Chess King - moves like a 4*Chess Q, only 1 square/mini-board at a time (no castling);

T=Super4*Chess Pilot - can move like a 4*Chess D or a 4*Chess K;

H=Super4*Chess Shaman - can move like a 4*Chess U or a 4*Chess K;

M=Super4*Chess Missionary (based on a piece from Shogi [promoted Bishop, or 'Horse', in that game]) - can move like a 4*Chess B or a 4*Chess K;

S=Super4*Chess Sailor (based on a piece from Shogi [promoted Rook, or 'Dragon', in that game]) - can move like a 4*Chess R or a 4*Chess K;

J=Super4*Chess Judge (based on a fairy chess piece [Centaur]) - can move like a 4*Chess N or a 4*Chess K;

X=Super4*Chess Mann (based on a fairy chess piece [Mann]) - moves like a 4*Chess K;

P=5*4DChess Pawn (a unit specific to this game that moves differently than a 4*Chess P) - moves like a 4*Chess R (unless capturing), but generally moves forward by only 1 square at a time on a rank, or forward by 1 row to another mini-board (but moves to same rank & file there). It may make an initial double step like a 4*Chess R moving forward two squares (i.e. not jumping over anything on a R move's path), that is either by rank, or by moving to another mini-board two rows forward on the same column (landing on same rank & file there). If it is making a capture it can do so like a 4*Chess B that moves 1 rank forward on the same mini-board, or it can do so like a 4*Chess B that moves to a mini-board that's adjacent to (but not behind) it, i.e. it must always advance by rank or row, & never retreat by rank or row. Capturing a 5*4DChess P that has taken a double step by en passant capture is allowed, with the side doing the capturing placing his pawn making such a capture on the square the enemy pawn would have gone to if it had taken only a single step in the same direction that it took the double step on the previous turn. 5*4DChess P promotions occur on the last rank of any mini-board that's on the same row where the enemy K starts the game, & a 5*4D*Chess Pawn may promote to any 4*Chess or Super4*Chess piece type above (other than 4*Chess K). It is possible for a 5*4DChess P to early on avoid being captured by an enemy 5*4DChess P simply by moving to the last rank of a mini-board.

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. (can be true in 5*4DChess)

If a game were to be recorded, all 4 coordinates (Column, then file, Row & rank) are given for where a 5*4DChess piece or 4*Chess P starts & finishes its move. If a 5*4DChess P promotes, this is recorded by tacking on the letter of the piece type selected after the promotion square's 4 coordinates. Similarly, a capture, check or mate can be indicated as in standard chess notation.

Beyond easily checkmating a lone 4*Chess K with just a 4*Chess Q, I've imagined checkmates of a lone 4*Chess K with other 4*Chess pieces (excluding 5*4DChess Ps or pieces specific just to Super4*Chess), 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 1 move being available). Notwithstanding that, I conceived of possible mates in 1 move using any 4 such other 4*Chess pieces, but with at least 2 of them not being 4*Chess Ds.

I'd guess the relative values of 5*4DChess pieces to be about as follows:

5*4DChess P = 1

4*Chess D (for within 5*4DChess, this piece for example could be called a 5*4DChess D instead, if one prefers) = 1.2

4*Chess R = 3

4*Chess B = 3.4

4*Chess U = 3.4

5*4DChess N = 3.6 (stronger than a 4*Chess N on its 4x4x4x4 board)

Note that on a 5x5x5x5 board, individual Bs and (especially) Ds can't always reach the same number of cells on an empty board as other individuals of the same piece type, so some Bs & Ds are worth, at the least, very slightly more than other Bs & Ds, respectively, on such a board.

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 5*4DChess K = 32 x (max. # cells 4*Chess K moves to [eighty]) divided by (# of cells in 4*Chess [six hundred and twenty-five]) = 4 approx.

I'd say a 5*4DChess X = 4 roughly (since it moves like a K as well).

Here are my estimates for the remaining pieces:

5*4DChess S = 6.6;

5*4DChess T = 6.8;

5*4DChess M = 7;

5*4DChess H = 7;

5*5DChess J = 8.6 (just as Q=R+B+P in value, J=N+K+P in value).

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 (the game is perhaps barely OK on this);

2. Low total number of pieces in the setup (barely adequate perhaps);

3. The pieces (other than pawns) are "natural" or "pure" to 4D Chess (no, not the Mann-like ones);

4. Good rules (& setup) for pawns (I'm not sure I'm satisfied having the initial double-step & en passant rules, but there's no ideal solution);

5. Some chance of early mate or relatively short game (scores adequately);

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

7. Variety of "major" and "minor" pieces (excellent);

8. K can legally attack opposing pieces/pawns (excellent, noting K not as powerful on 5x5x5x5 board);

9. Pieces (especially N-like) may obtain great scope (excellent due to 5x5x5x5 board);

Fwiw, in scoring 5*4DChess with these 9 criteria, giving a score of 0-4 for each, I found it had a total score of 22/36 (or a little over 61%), just slightly greater than for my earlier 4D variants, 4*Chess and Super4*Chess. I'd note though that IMHO either of these scores slightly better than 5*4DChess if a score of 0-2 for each criteria is used instead. Also IMHO, the game's best attraction may be that it nicely meets criteria #6-9.

Here are links to my Chess Federation of Canada website blog entries discussing 4*Chess & Super4*Chess:

Here's a similar link to 4D crazyhouse/bughouse variants based on 4*Chess (similar variants could be made arising from 5*4DChess instead):

Here's a link to 4*Chess as given on chessvariants.com:

Here's a link to Super4*Chess as given on chessvariants.com:

A similar link to another 5x5x5x5 variant of mine (one which uses 74 pieces):

Last revised by Kevin Pacey.