### Power of Sliders

In standard chess, we're happy with the notion that the bishop is colorbound, and that the rook has the very convenient power of blocking a king's movement. Things become more "interesting" in multiple dimensions. Here I'll work with all possible diagonals for sliders.

##### Rook

We quickly see the rook lose power. In N-dimensional chess, it takes a rook N moves to reach from one square to another. More devastating, the rook no longer provides the blocking power it once had. On the other hand, it does of course continue to be able to reach every square on the board. The rook has $2N$ directions of movement when he is unobstructed by edges or pieces.

##### Bishop

Perhaps somewhat surprisingly, a properly colored board will still keep the bishop on one of two colors, regardless of N. The bishop has $2N(N-1)$ directions of movement when unobstructed. Notice that this is considerably larger than that for the rook when N becomes large.

##### Unicorn

In 3D chess, the board can be colored with four colors so that a unicorn is limited to squares of one color. In higher dimensional chess, this is no longer the case, and the unicorn can reach any square. The unicorn has $(4/3)(N)(N-1)(N-2)$ directions of movement.

##### Balloon

In 4D chess, the board may be colored with 8 (!) colors so that a balloon is restricted to squares of one color. In higher dimensional spaces, the balloon reaches half the board. The balloon has $(2/3)(N)(N-1)(N-2)(N-3)$ directions of movement.

##### Everything

Outside of the more common sliders listed above, there are of course infinitely many others, although they will see increasingly less actual gameplay. I believe the following characterize an n-dimensional slider in N-dimensional space:

• If n=N, then the board can be colored by $2^{n-1}$ colors;

• if n<N and n is even, the board can be colored by $2$ colors; and

• if n<N and n is odd, the slider can reach any square on the board.

• The slider has $\displaystyle \frac{ N! \cdot 2^n}{n! (N-n)!}$ directions of movement.

So it seems that even-dimensional sliders and sliders of the highest/lowest dimensions have a disadvantage.

### Power of Leapers

As far as I can tell, most multidimensional games have used only one leaper, usually some sort of extension of the knight. An asymmetric leaper like the knight will benefit more from extra dimensions than a symmetric leaper. For instance, if each piece is unrestricted by the edges of the board or friendly pieces, the number of moves available are:

N |
wazir |
knight |
ferz |
(1,1,1) |
(1,2,3) |

2 | 4 | 8 | 4 | - | - |

3 | 6 | 24 | 12 | 8 | 48 |

4 | 8 | 48 | 24 | 32 | 192 |

5 | 10 | 80 | 40 | 80 | 480 |

6 | 12 | 120 | 60 | 160 | 960 |

Hi, Ben:

Just off the top of my head, VR Parton, LL Smith, and Dan Troyka have all done higher than 4D chess variants, Dan's 6D on a board with only 64 "squares", 2x2x2x2x2x2.

Have you had the opportunity to play a 4D game? Even the simple ones are likely very instructive. Pushing pieces gives you a much better feel for the "nuances" of multi-D design. But simpler is better, I believe, to start off. It gives you some feel for the game and its potential. On the other hand, Aiken's Chesseract has been played a lot, unlike my Hyperchess, and his game is notably more complex than mine. So I could be a minority opinion.

Joe

ReplyOptionsI might have heard of one or more of those 5+ dim games, but don't recall them at the moment. I'm having trouble finding them; do you have links to those games?

I've played my own 4D game a fair bit, and I agree that actually playing is the best way to start to actually figure out how it works. I agree that simplicity is good, but I really like leaving as much of the higher dimensional symmetry intact as possible. I remember liking Chesseract in that regard, but I think the number of new ideas (turning rooks, sticky pawns, etc.) could be a turnoff to new players. On the other hand, it does counter the weakness of the rook in 4D chess that this article discusses…

ReplyOptionsThere are at least 2 schools of thought on 4D chess [as the best human-playable examples for nD chess with n>2] and how it should be played. I freely admit to being in the minority on this question, but I've looked at it for a while, and think I offer a good game for 4D, because I let the board star, rather than the pieces. As much as possible, I tried to restrict Hyperchess to the pieces and spirit of chess. Trying to figure out how totally new higher-D pieces actually work on a bizarro board that you don't understand just might be overkill.

Still, there are unavoidable compromises when you actually try to play the game. How you handle mate, for example, is a key element of the design of your 4D game. The point of chess is the capture of the king, after all; if you cannot capture the king, you don't have much of a game… Size of board is also an important parameter - too big, and it becomes effectively unplayable; too small, and what you have is not chess because there isn't room for chesspieces to move normally.

I've looked at the idea of using 1 set of pieces that move on 1 of 2 different boards, either the "Big" or "Little" squares, owner's choice, each turn, calling it Chess on 2 Boards [I'm not all that good with names ;-)], but haven't managed a decent game of it yet.

Went to my bookmarks to get the link for some higher-D stuff, and found it was in Geocities, which is defunct. So went to my 4D article here, and checked this, which is still good: http://www.chessvariants.org/parton/Cubism.txt That should get you started with Parton.

Enjoy,

Joe

Perhaps the problem with expecting people to take more quickly to simpler games is that, if anyone is willing to even look at 4D chess, then they're probably expecting something complicated. I've been trying to garner support for my game among friends, and some will play from time to time, but a lot get scared off just by the idea. I have tried to do the same as you, wanting to reserve a lot of the standard chess feel, but with a couple of new dimensions. I suppose the main difference between us is that I really want as much symmetry as possible (probably because of my math studies).

Surely mate is the biggest problem to deal with. We have a lot of options floating around (many I first heard of on the Taiga page here), but all of them seem to lack something; perhaps there isn't one ideal solution. I might write up something on that for here, but am afraid I'd get a bit biased. Size is certainly another issue, but I don't think it's a terribly big one. Once you've settled into the idea of wanting a certain size board, you're probably pretty happy with how that allows/restricts movement.

Another big issue is how exactly you want the pieces to move. I think the most straightforward way is as I've outlined in this article, with perhaps the addition of some combined pieces and/or odd leapers, but this isn't the only way. I've seen one person try to retain the rook's king-blocking power by drastically extending it's (and every other piece's) movement into the third dimension (I don't recall the name of that game now, but it's on CV).

ReplyOptionsShall we set up a couple of boards and push pieces a little? I admit to being curious about your initial setup, not to mention how the game would play. I find my game rather straightforward, but that's with a little bit of practice, and a lot of looking at possible moves. I believe you were looking at a 4x4x4x4? We could put together a preset to play with, but it won't rules-check, 'cause I can't do that. [What kind of internet connection do you have? the Alfaerie: Many piece set is rather large…] And we could examine the kinds and placement of pieces. So if you're interested, let me know. Enjoy.

ReplyOptionsSounds good! I'm not sure how the preset stuff works, so you'll have to give me some sort of intro. My connection is decent, so I can load the Alfaerie: Many, but I don't expect we would need nearly that large of a set.

(I've been trying to get a rule-checking program together for my game, but that seems particularly difficult for a multi-dim game, trying to translate the movesets into 2d moves for the computer to understand…)

ReplyOptionsHey, Ben. I got 2 versions of a 4x4x4x4 board, the first by Jeremy Good, the second by Abdul-Rahman Sibahi:

[http://play.chessvariants.org/pbm/play.php?game%3DChess+on+Two+Boards%26settings%3DC02B]

[http://play.chessvariants.org/pbm/play.php?game%3DJoe+Joyce+Hyperchess%26settings%3DJoe+Joyce+Hyperchess+]

All my stuff is non-rules-checking, so you can just move pieces in and out of the preset rather easily. To begin to learn about presets, you can go here:

[http://chessvariants.wikidot.com/preset-primer]

The Co2B game is one of my busts. It does have the original hype board done by Jeremy, though. So if you prefer that to A-RS's version, which is the current blue-green hype preset, we can just edit the Co2B preset to put the hype pieces in.

ReplyOptionsFor symmetry's sake, I like the Co2B setup a bit better. If we were to ever play a game like your Walkers and Jumpers, where pieces see it as 2D also, the other setup would obviously be preferable. For both I recommend swapping some colors, so that squares an orthogonal move away are opposite dark/light.

Classes are getting more into full swing, so I might lag a bit.

ReplyOptionsGood luck with classes.

I tried the color swap for some board squares, and found it jarring and annoying - the black and white squares didn't look right. So I went to the boards you see now, for the general light-dark pattern. And if you do make that change, you throw off the bishops - they change color sometimes, and sometimes not… It's aesthetic, I guess.

Any questions with the preset, please ask.

ReplyOptions