4D Quasi-Alice Chess

Here's an experimental 8x8x2x2 4D board variant idea of mine that was inspired by Alice Chess, Directed Alice Chess III and by my desire for there to be a completely viable 4D variant that is somewhat similar to standard (2D) chess.


A Game Courier preset for play is available. Note that links concerning Alice Chess and Directed Alice Chess III are given in the Notes section further below.

Setup

Pieces

All 6 piece types that are used for standard chess are also used exclusively in playing 4D Quasi-Alice Chess.

Rules

4D Quasi-Alice Chess is played using four 2D boards, three of which are empty at the start of the game. All pieces move on these four 2D boards as in standard chess, stalemate, 3-fold repetition and 50 move rule being draws. However, in turn each player makes a single move on any of the four 2D boards according to the five rules that follow, which are somewhat similar to those of Alice Chess:

1) A move must be legal on the 2D board where it is played. Note that a piece may block a check by transfering from another 2D board (although transfering from a 2D board the king is not on would temporarily leave the king in check on its 2D board, that does not matter, because that check is not on the 2D board the piece is transfering from). Also, a king may not move into check (on the 2D board that it begins its move on) to escape check by then transferring to another 2D board. Moving into check on the king's 2D board would normally be illegal by the rules of chess, and so counts as illegal.

2) A piece that is not a king can only move or capture if any choice of destination 2D board for transference has at least one of the two kings on it (unless the 2D board that a piece is moving from has both of the kings on it already, in which case any of the other three 2D boards are choices) and the corresponding destination square on at least one such choice of destination 2D board is vacant. The same applies if the piece that is to move is a king, except that it may possibly move to a corresponding destination square that is occupied by an enemy piece (see next rule), if the choice of destination 2D board is otherwise in accordance with the restrictions of this rule.

3) After moving, if the piece is not a king, it is transferred to a vacant corresponding square of that player's choice (if any available), on one of the other 2D boards, according to the restrictions stated by the previous rule. However, if the piece is a king, it may (in addition to transferring to a vacant corresponding square, according to the restrictions of the previous rule) possibly capture an enemy piece on the chosen corresponding square in the process of completing its move (though see next rule, concerning castling). Thus a double capture by a king in a single move is legally possible.

4) When a king and rook castle, the spaces they move to should both correspond to empty squares on a destination 2D board. Also, once the castling is complete, both pieces transfer to the corresponding squares on the destination 2D board.

5) A pawn may capture by en passant any enemy pawn that lands beside it (on the same 2D board) after making an initial double step move - so long as the corresponding square on another 2D board that the capturing pawn can be chosen to transfer to is a square that is also empty.

Notes

Regarding the relative values of 4D Quasi-Alice Chess pieces, I'd tentatively estimate them as being worth what they would be in Alice Chess, except for the fighting value of a 4D Quasi-Alice Chess K (which I deal with further below), and for having the value of a Quasi-Alice N=([{3+3/2}/2+1]+[3+3/2+1])/2=4.375, as it's not necessarily colour-bound. My own values for Alice chess pieces would be N=(3+3/2)/2+1=3.25; B=3+3/2+1=5.5; R=5+5/2+1=8.5; Q=9+9/2+1=14.5, with an Alice Chess K having a fighting value of 4+4/2+1=7 (though it cannot be exchanged).

Note that I value a piece's moves to a board transferred to in Alice Chess by it as being worth 1/2 its chess value (due to these always being non-capturing moves), on top of the chess value of the piece on the board that it moves on before transferring, with a pawn added to these two component values to get the particular piece's Alice Chess value, similar to how Q=R+B+P in chess, for that compound piece. In case of the knight in Alice Chess, it's moves/(piece value components) are judged by me to be worth half as much in value, since it's colour-bound on each board in Alice Chess.

Zillions of Games, in comparison, values Alice Chess pieces as having the Knight at 3.2 Pawns, the Bishop at 5.4 Pawns, the Rook at 8.2 Pawns and the Queen at 13.2 Pawns. I'd note that Zillions of Games does not suggest a fighting value for a King in Alice Chess (though it cannot be exchanged), but it would be less than in 4D Quasi-Alice Chess, due to a king's augmented capturing capabilities in the latter variant. In accordance with Zillions of Games' piece values for Alice Chess given thus far, I'd suppose they might, if asked, elect to give an Alice Chess King a fighting value of (Bishop+Rook)/2 = 6.8 Pawns.

Getting back to 4D Quasi-Alice Chess, I'd give the King in this variant a tentative fighting value of (Chess King)+(Chess King)*('Average' number of boards a 4D Quasi-Alice Chess King can transfer to per move in a whole game)/(Number of boards a 4D Quasi-Alice Chess King does not start a move on)+Pawn = (4)+(4)*((3+1)/2)/(3)+1 = 4+2.7+1 approx. = 7.7 Pawns approx.

Note that another way to get a value estimate for a 4D Quasi-Alice Chess King might be to take the average value of an Alice Chess King & an Alice Chess Rook, which are two pieces that a 4D Quasi-Alice Chess King should be somewhere in between in value, and whether my values or Zillions of games' values for Alice Chess pieces are used, either way yields an average value which is quite close to my estimate for the value of a 4D Quasi-Alice Chess King.

As far as the rules of 4D Quasi-Alice Chess go regarding the kings, the main idea is to allow more 'basic' (and thus forcible) checkmates than might otherwise be the case, if the rules were to be even closer to those of standard (3D) Alice Chess as played on just two boards. I've convinced myself that K & Q vs. K and K & R vs. K are still basic mates in 4D Quasi-Alice Chess, and that K & N & 2 opposite-coloured bishops vs. K (or K & 2Ns & B vs. K) suffices to mate, too. If indeed so, this would at least be a slight improvement on the number of basic mates available in Circular Chess (where at the least K & R can't normally mate vs. K).

In favour of Circular Chess as a variant is that its terrain has an interesting effect on the play, e.g. of the pawns (also note that K & P vs. K in wins in that variant, due to there being no possible stalemate). Returning to 4D Quasi-Alice Chess, one might think that it may be nice that it's a 4D variant, but what more tangible compensating feature(s) of it makes up for it having less basic mates than Alice Chess, in comparing these two variants?

Well, the dynamic effect that either king's 2D board transference (when moved) can have on the play in 4D Quasi-Alice Chess could prove an interesting & positive feature for any number of players (a feature that wouldn't change Alice Chess if applied to it, due to it having only 2 boards). That's aside from the rather less tangible enjoyment one might derive from the fact that a number of pieces might end up on the same corresponding squares during a game. Plus, a king's ability to capture a piece when transferring might come into play from time to time (besides allowing the number of basic mates that this game has), though this feature admittedly might be added to Alice Chess in an effort to create a slight variant on it.

Note that more often than in Alice Chess, the rules of 4D Quasi-Alice Chess may make it impossible to defend a piece (or pawn) against capture by an enemy piece, which might be able to then transfer to another 2D board where it is at no risk of being recaptured in turn. That's other than by defending the threatened piece with a friendly king, or by moving it to another 2D board. This note applies especially if the two kings happen to be on the same 2D board, as they are at the start of a game. On the plus side, games might be shorter on average than they could be otherwise.

One difference 4D Quasi-Alice Chess has from Alice Chess, right from the start of the opening, is that if White immediately moves a rook's pawn (transferring it to any empty 2D board) it would seem he is assured of being at the least able to safely capture Black's same rook's pawn. That's unlike in Alice Chess, where it could be protected by moving the appropriate knight to bishop's three, so as to in effect defend the rook's pawn from the only available alternate 2D board in Alice Chess, which the White rook would have no choice but to transfer to if capturing the rook's pawn next.

However, in 4D Quasi-Alice Chess White isn't really winning a pawn for nothing by force this way, since for one thing Black can instantly retaliate by moving his rook's pawn on the other flank in reply, but this not clearly necessary (and might not be best either). For if Black instead moves, say, a centre pawn, and White immediately follows through with his 'threat', i.e. captures the the enemy rook's pawn with his rook, transferring it to any apparently safe 2D board, Black at once moves the closest knight to bishop's three, transferring it to the same 2D board as the lone White rook, so as to attack it. At that point the White rook would have no safe square to flee to, since by the rules it must transfer to the 2D board that both kings are on. If the rook (for example) retreats to anywhere on the same file, followed by transferring, it can be captured by the Black rook on the same file, which would then transfer to true safety on another 2D board. On the other hand, observe that if White refrains from capturing the pawn with his rook on move 2, and moves his other rook's pawn instead, Black may not wish to develop either knight, at the least for some time, and the evaluation of this sort of 'opening' requires further thought, or testing.

Note further that in comparison to Alice Chess, in 4D Quasi-Alice Chess bishops are still always colour-bound (like in standard chess), but knights are not necessarily colour-bound for an entire game if they are located on any particular 2D board at some point (Directed Alice Chess III is somewhat similar in this respect).

Note that with Directed Alice Chess III by Joe Joyce (an Alice Chess variant played using just three 2D boards), a way to make a 4D Alice Chess variant may have already been invented in disguise. That's since a bishop may be able to make a single square diagonal step move (forward or backward) when going onto each of 3 different 2D boards in single steps, as done on 3 consecutive turns, with the bishop possibly finishing on its starting 2D board, but also possibly just one diagonal step from its original square on that starting board. This is not possible in a typical 3D chess variant (e.g. 5x5x5 Raumschach) with, say, a bishop that moves in standard 3D chess fashion, as the bishop could not even return to its starting 2D board if visiting three of the 2D boards in single steps in 3 consecutive turns. However, it could be possible in a 4D variant (i.e. having 4 or more 2D boards, normally, although at least one of the corner 2D boards might be voluntarily excluded), with a 'standard' 4D bishop still changing just 2 (of now 4) co-ordinates.

It might also seem to be possible if using 3 somehow otherwise interconnected 2D boards than in typical 3D chess (as could be the case for Directed Alice Chess III), except that then I do not see how a typical bishop (changing 2 co-ordinates as it moves) could possibly finish on its starting 2D board being just one diagonal step away from the square it started on, after visiting 3 different 2D boards in 3 consecutive moves (in the single step fashion I described above) without the interconnection being in effect 4 dimensional. Perhaps a math wizard might try to explain it to a layman like me, if my conclusion is wrong. If I am not wrong, at least with this line of reasoning, then a similar sort of reasoning can be made to claim that 4D Quasi-Alice Chess is indeed a 4 dimensional chess variant, or at least can rather be like one if and when a situation arises where each king is (initially) on a different 2D board and, say, a bishop is on yet a third 2D board, as still can be seen even after reading the rules of 4D Quasi-Alice Chess further below regarding the effect that the placement of the kings can have on what 2D board(s) pieces can transfer to. This is all to say that 4D Quasi-Alice Chess indeed is effectively a 4D 8x8x2x2 setup, and not a 3D 8x8x4 setup in disguise.

Here's a link to Alice Chess, as described on chessvariants.com:

A similar link, to Directed Alice Chess III:

Last revised by Kevin Pacey.