Checkmate?
Checkmate
As the dimension of the board increases, it becomes increasingly difficult to checkmate the opposing king. Consider that a full-powered king (a piece which moves to any adjacent square) has $3^N-1$ possible moves when in the center of the board; that's 8 in 2D chess, 26 in 3D, 80 in 4D, 242 in 5D, etc. Even when backed into a corner, the full-powered king has $2^N-1$ moves; that's 3, 7, 15, 31, etc. Combine this with the rook's loss of blocking power, and the king seems nearly uncatchable. Several solutions have been proposed:
- Crippled King: We limit the directions of movement of the king, say to orthogonal moves only. Since then the king then cannot capture a piece immediately adjacent, perhaps we restrict only non-capturing moves, while allowing capture in all (or just more) directions.
- Bare Royals Loses: We call a baring of the opponent's king a win. Of course the queen is even more mobile than the king, so perhaps reducing an opponent's army to just K, Q should be a win.
- Fortress: As in Xiangqi, we could assign a certain block of cells as the king's fortress, which he may not leave. Depending on the size and location of the fortress, this sometimes allows draw by perpetual check when it might otherwise be escapable.
- Gladiator Kings: When a king comes to occupy some given region along with the opposing king (perhaps a common plane), the opposing king is "stuck" in that area until your king leaves the area. Perhaps both kings become stuck, and perhaps another piece may intervene and break the constraint.
- God Pieces: We introduce new pieces which are more adept at checkmating, e.g. the amazon (extended appropriately into multiple dimensions).
- Multiple Royals: Give each player more than one king, and force the players to keep all of their kings out of harm's way.
- Nothing: As slippery as a king can get in multiple dimensions, in 4x4x4x4 chess a king and queen against a lone king will be able to checkmate. I'm not sure how well this generalizes, but perhaps this is enough.
page revision: 2, last edited: 18 Mar 2010 13:43
You've got a nice list here. While it is not exhaustive, it does seem to cover a lot of ground, and it even puts my suggestion there in the middle. [Nice name for the idea - mind if I use it in my write-up? I freely confess to being naming-deficient.] The problem, of course, becomes just how effective each method is. And, since we're discussing a game, I'd like to add one more constraint, playability, as a separate consideration to look at along with the mathematics of the situation.
The first thing I'd note is that there are mates in 2D on 8x8 that apparently can't be done on a 16x16, for example, although I'd have to ask a couple experts on that sort of thing to give specific examples. They do involve variant chesspieces, but, by definition, 4D pieces are variant, and it's probable that K+Q vs K mates which work on a 4x4x4x4 will not work so easily on a 10x10x10x10, say, which gives more sidestepping room. The 4x4x4x4 is the smallest 4D board which allows more or less standard chess pieces, like the knight, to function as intended. It is exceedingly cramped for a chess board, and prevents running, so facilitating checkmate by the 4D king and queen combination.
However, that leads to the opposite problem: on such a small board, how do you prevent checkmate? A 4D queen is pretty tough; stick her on one of the 16 central locations on a 4x4x4x4, and how many other locations on the board does she not reach? So where do you set her up in the beginning? Someplace where the enemy king is not in check, and the enemy queen is not en prise?
This takes us to the entire initial setup and questions of piece density. What sort of setup is required for fully 4D pieces? Here you should, I think, consider the role of pawns in standard chess. The pawns both protect and block the power pieces, and the 2 sets of them form a pretty stable wall around which the longrange pieces work. What do fully 4D pieces require in the way of pawn protection/blockade for the game to have even such a thing as an opening phase? And what does this do to the playability?
At this point, I'm not convinced that any of the answers, including my own, is adequate. But if you look at the "gladiator kings" idea, it is actually a refinement of "cripple the king". [As an aside, couple that idea with "bare king", and you might be closer to giving achievable victory conditions.] And under the right conditions, it might work very well indeed. Consider that rule for a 3D game, for example, or on a 4x4x6x6, where it should be easier to get to the other side's "king level, because it has more destination squares. Or add a piece - or set of them - that acts as a non-royal king to hold the other side's king in place. Whether it is also held, or free, is a method of "tuning" the piece to aid game balance. Well, anyway, any general idea is subject to refinement [you can see the gladiator idea as providing transient 'fortress' areas, too, for example] in many different directions, I suppose, and that's what I was getting at in this paragraph in my usual roundabout way. [It would probably work better if I didn't leave out so many of the in-between steps…]
I'm looking forward to the next developments.
Of course no list of fixes will be exhaustive (just as there can be no complete list of variants). If you know of other options that people have supported, please add them. I suppose since, as you note, any general idea can be refined in many different ways, we should try to keep the categories fairly general. I'll ponder on combining some of these also.
I do like your comment about special pieces which hold the king, sort of a specialized Medusa/Nemesis piece.
I tend to believe that a fully armed 4D K+Q can mate a lone K on any size board, but am not positive about it.
I think that there's really almost no hope of coming up with a universal 4D game with variants tacked on; at this point the best we can hope for is a handful of nicely playable games, perhaps each with their own variants. When the semester gets around to being over, I'll hopefully have the time to play a couple of different games with you and we can work out which options seem plausible and what other tweaks we can introduce.
Hi, Ben. Hope the semester goes successfully for you, certainly far more successfully than my recent attempts at visualization. When I looked at the [empty] board, I could see that the fully 4D K+Q can mate the fully 4D K alone. And I will also get around to reversing coloring on half the "levels" in Hyperchess [aka: hype]. Just don't know when. Once we start pushing pieces, I guess. That ought to be fun. Your goal of a handful of nicely playable games is far from modest, you know. I like that.
I don't see any more possibility of coming up with a universal 4D [locus of] game[s] than doing that for 2D. I keep trying to say we should play Jim Aikin's Chesseract, but I'm afraid of that game, so instead of telling you we really need to play Chesseract, at least a little [playtest, better], I'm going to suggest we try a game of Walkers and Jumpers to look at a [mostly] very small 4D board, a 3x3x3x7, along with the game or playtest of hype. Chesseract uses all those neat 4D moves. It also uses a strange setup with lots of blocking pieces, to keep the game from starting off as a slaughter.
Now, I see games where the power pieces can get anywhere in 2 moves as "chaotic" in the sense that you cannot project successfully into the future to any depth, and the game state can swing greatly from turn to turn. Other games, like Chieftain Chess, cannot be calculated to much depth, but are at most semi-chaotic, because the distances on the gameboard are "great" compared to the maximum distances the pieces can move in a turn. So the game state will not generally change by very much from turn to turn, giving a more stable game. These 2 types of games can be liked by different people, and I suspect that it's very possible a human would have a better chance against the computer in a game of Chieftain [aka: chief] than in something like Carlos Cetina's Sissa chess variants.
The reason I'm "afraid" of Chesseract is that it appears chaotic to me, and in that type of game, I tend to get lost in the intricacies of the various lines of play 3-4 turns out, and cannot foresee a clear end to far too many moves in such a game. I get brain-lock. In hype, I can offset the chaos the board allows by covering relatively few "squares" in the immediate vicinity of the king, even if I can't project past 3 or so turns at all consistently. So I can slow down the pace of the change of state of the game, and manage to provide reasonably good defenses - with the hype pieces and rules. Since hype and chesseract are played on the same board, and the knight [a key piece in hype] in each game is almost identical, it should be at least somewhat instructive to play both games side-by-side. There, I got it out, as the third game I suggest we play[test].
The last 2 games I will suggest we might want to look at are 3D Great Shatranj and Raumschach. These are both 3D, not 4D, but the basic principles of the 2 games appear to me to be similar to hype and chesseract. So we have a sort of simplified model of 4D chess with them. Again, looking at these 2 games might give insights useful in investigating playable 4D chess.
So we have
Walkers and Jumpers
Hyperchess
Chesseract
Great Shatranj
Raumschach
TessChess
(I had to throw my own in the mix)
How do we go about setting these up in GameCourier? Do we just keep a window open and move pieces about on our own boards, taking moves via correspondence? Or is there some way of saving a position?
Hi, Ben. Hope things are going/have gone well with school. Following are playable versions of 4 of the 6 games listed above [helped all no end 3 of them were mine]. Jim Aikin's Chesseract is not listed as a preset. We could make a private preset, if you wish. But I gave a link to the rules - try reading them… lol, I thought my hype rules were bad! Next, there is a link for Charles Gilman's Redistribution 3d Chess. Finally, there's a link on how to put together the simplest possible presets. Hope this helps. Joe :-D
[http://play.chessvariants.org/pbm/play.php?game%3DWalkers+and+Jumpers%26settings%3Ddefault]
[http://play.chessvariants.org/pbm/play.php?game%3DJoe+Joyce+Hyperchess%26settings%3DJoe+Joyce+Hyperchess+]
[http://play.chessvariants.org/pbm/play.php?game%3D3D+Sideways+Great+Shatranj%26settings%3D3D+SGtS+1]
[http://www.chessvariants.org/index/msdisplay.php?itemid=MPraumschach]
[http://www.chessvariants.org/large.dir/contest/chesseract.html]
[http://play.chessvariants.org/pbm/play.php?game%3DRedistribution+3d+chess%26settings%3DRed3d]
Of course I'll help with the preset, walk you through it. In a week, you'll be better at 'em than I. Believe me, if I could do this, just about anyone could.
[http://chessvariants.wikidot.com/preset-primer]
Hi, Ben. I did a re-write of the rules here in the wiki, and I think the rules set is better. It was done after more games were played, and incorporates my latest thinking on this subject in reasonably understandable prose. [http://chessvariants.wikidot.com/hype:a-study-in-bs] Should you find any problem using it, let me know, so we can fix it.
One of the questions I have is about the first move advantage here; it could be significant, because the board is so small. The question of actual mate needs to be looked at, also.
Don't know how many other 4D CVs are floating around, but I bet I got one of the best minimalist ones. On the other hand, Gavin Walker Smith did a whole 3D game, Prince, that uses pieces that move in 2 dimensions at once, and the extension to 3D, 4D or higher is obvious. Larry Smith also talks about things that are more powerful than my pieces, by a lot. The two of them, not related, worked together a bit on Prince.