On the implementation of multidimensional chess.
The creation of Cartesian 3D chess is exceedingly simple, being little more than an array of 8 standard chess boards arranged vertically. Modification of the rules is as simple as allowing movement along the Z axis, following the same restrictions as movement along the traditional X and Y axes. This is an exercise so simple as to be deemed trivial.
A more complex exercise, however, comes in the addition of a more abstract axis, which we will cal the T axis to represent time. The concept of moving chess pieces forward and backward in time may seem impossible given the current advances in physics, but with some careful thought it can be done. For simplicity, I will discuss the addition of a T axis to a traditional 2D game of chess.
In terms of computer programming, a chess board is very much like a static, 2-dimensional array of 64 elements (8, 8). The addition of a T axis would require shifting instead to a 3-dimensional dynamic array where the first two indices are static (X, Y) and the third index is dynamic. At the start of the game, the state of the 2D board would be stored across the X and Y indices at T = 0. After the first move, the state of the board would again be saved, but as T = 1, so on and so on. Maintaining this much data would, unfortunately, necessitate the game to be played through a computer. The use of pen-and-paper bookkeeping would only be practical in long-term games such as "chess by mail".
Now that we have found a way of visualizing and tracking the T axis, the next step is to modify the rules to account for motion along this new axis. Unlike the trivial case of the Z axis, players are unable to directly move their pieces through time (though miniaturized time machines would make this far easier). Instead, we must make do by first deciding to visualize time as a line. Then we must decide that any influence we may have on the past changes it instantly, creating an entirely new future. Essentially, altering the past cancels the present and everything in between, replacing it with the present as it would be after being impacted by our changes in the past. In terms of our dynamic array model, this means that moving a piece back in time means resetting the board to how it was at the target point in the past, discarding every time since, and continuing from there.
For the sake of continuity (as anyone who's watched any science fiction should know) a few special rules must be added regarding time travel, assuming that a piece has been sent back in time:
1. If the original piece is lost, so is its duplicate from the future. If a piece was captured in the past, then we couldn't have sent it back from the future.
2. When the game again reaches the T where the piece was sent back from, the past piece is removed. A confusing concept, but it simply means that once you reach the point where you sent the piece back in time, you must do so again, regardless of anything else. This could result in an important piece being taken out of play at a very inconvenient time.
3. The King cannot travel through time. Allowing the King to travel through time would make it far too easy for the king to vanish into the future, drawing out the game.
As for sending a piece into the future, the piece is simply taken out of play for a period of time, until the targeted T is reached.
Next come the rules regarding how individual pieces can travel through time. This is simple to do, as we can apply the same rules to the T axis as already apply to the X and Y axes. For example, the Rook can move up to 8 units along X or Y, so it would be able to do the same along T. Similarly, the Bishop can travel up to 8 units, but must travel the same number of units in any two dimensions, in this case meaning any combination of the X, Y, and T axes. The Queen, however, is a dangerous piece indeed. At first thought, it seems that the Queen should be able to travel up to 8 units along any number of axes, but this is far too overpowered, like a cruise missile in a time machine. As such, another rule must be added:
4. The Queen may not move long more than two axes at a time.
4. The Queen may only move a maximum of 16 units, summed between movement along all axes.
Personally, I prefer the second version, as the first version makes the queen no different than the Bishop.
Another balance issue comes into play regarding pieces traveling through time and capturing opposing pieces. In theory, the game could be ended in two moves of the Queen, traveling a couple moves into the past and into the space of the opposing King. To remedy this:
5. No piece may make a capture in the same move as traveling backwards in time.
Traveling forwards is less of an issue, since it would be more difficult to attack a piece that has not yet moved into a space, making it no easier than any other traditional strategy. The issue with traveling forwards in time, however, if what to do if the target space is occupied in the future. The solution is rather simple:
6. If a piece travels forward in time and arrives in a space occupied by an opposing piece, it the arriving piece is captured. If the space is occupied by a friendly piece, however, the arriving piece is placed in an adjacent open space, chosen by the opponent.
As a result, a player would have to be careful not to allow a targeted space to be blocked, lest they give their opponent an advantage by allowing them to choose the new location of the piece.