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Hyperchess4A game information page
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Kevin Pacey wrote on 2017-02-25 UTC

Joe Joyce wrote on None

... I think your minimum 2D level size is not 19 but 37, giving 1369 locations. The only game software I know that would handle that size is Vassal. But it should handle it very easily.

Something I don't understand is why Game Courier might be unable to handle, say, a 36x36 (6x6x6x6) square cell 4D variant. I once dug back in some game logs, and discovered a single log for TessChess6by6xxx, a 4D variant that does have those large dimensions. A game of it was in fact not started (I gather it was just a test diagram for a hypothetical variant; otherwise the game was said to be between Ben and you), but I tried 'Edit' followed by moving some of the pieces, as a way to practice a game against myself. The Game Courier software seemed to have no problem handling this size of variant for my purpose of use, at least (note no rules were enforced by Game Courier software, though).

One reason I earlier rejected having 37 cells to each of 37 two dimensional hexagonal boards was that in a starting position I thought of fairly recently, pawns would take 9 to 11 moves to promote, if making no captures along the way, which I felt was too many moves to make (adding to a vague concern such a big board may make a game too long, even with few pieces). However, if I could think of a smooth way to allow, say, some sort of initial triple-move by a pawn, and be able to smoothly explain how to do en passant capturing in 4D, then pawns in my setup could take 7 to 9 moves to promote (without making any captures), which wouldn't be quite so bad. I was also concerned such a large board might make the pieces too hard to see in a diagram, unless it took up more than one whole screen (say on my laptop), but a recent crude stage diagram test I tried indicates such a game may be playable in that respect, too.

If Game Courier can actually handle such a board (say without rules enforcement) and I can think of pawn rules I like and can explain easily without additional time consuming diagrams (or I cook up a different setup position), I might later press on with the idea of eventually submitting a 37 cell two dimensional hexagonal boards 4D game, adding to the 19 cell two dimensional boards 4D hexagonal game I've recently submitted.


Kevin Pacey wrote on 2017-02-13 UTC

I read your last post too late Joe, as I'm about to cut & paste a time consuming diagram of my 4D idea (i.e. still just a hexagonal shaped 4D board, with nineteen 2D boards that have 19 hex cells & are hexagonal shaped). I kind of thought you'd still dislike this version of a 4D hexagonal board due to the clumsy knight's 4D board tour issue concerning the 2D boards (hopefully I'm not breaking with a standard & strong convention for 4D variants that I'm unaware of). However, I still felt comfortable enough with it myself as it is, besides my not having to re-think everything, in making a different (and greatly larger) 4D hexagonal board-based game somehow work. It's possibly a good idea to move any further discussion of my shortly-to-be-posted diagram to my diagram testing thread, rather than continue it in the general Hyperchess4 discussion thread, if you wish.


Joe Joyce wrote on 2017-02-13 UTC

Hey, Kevin, we're saying the same thing about the N move, although I said it rather clumsily, an occupational hazard when I'm very short on sleep. Let me phrase it this way:

The board is too small if the knight cannot make a complete tour on any and every individual 2D board section presented in the game without ever leaving that 2D section until the tour is complete.

Heh, saying the N could make the tour 'comfortably' was probably not as precise as I would have liked. But however you cut it, I think your minimum 2D level size is not 19 but 37, giving 1369 locations. The only game software I know that would handle that size is Vassal. But it should handle it very easily.


Kevin Pacey wrote on 2017-02-12 UTC

Hi Joe

I think a knight in my hexagonal variant idea can reach the centre hex on the centre board, or leave it, provided I (as planned) use for knight movement rules an analogue of Hyperchess4 Knight movement (as I understand it works with square cells or 2D boards). That's since a knight can move orthogonally to a different 2D Hexagonal board that's just one or two 2D boards away orthogonally (i.e. in 4D). When a knight leaves from a centre hex, I'd have the knight arrive only two or one cells orthogonally away from the centre hex (on a cell that must be of a different basic colour than the knight started on), on the destination 2D board. Note that from the centre hex of the centre 2D board I indeed could not make an entirely BIG move, i.e. to a centre hex of a new 2D board that would be (if it were possible) just a BIG knight move away, as I think you are alluding to (besides that a knight cannot reach a centre hex by moving on the same 2D board, which I alluded to earlier too). The fact a rather 'impure' knight's tour is the only kind possible makes the variant idea a little more ugly, perhaps, but I didn't mind, in order to try to have a more reasonable 4D board size that's still hexagonal in all respects.

I'm going to check my current ideas a little more, then very soon I may post a diagram for it in my diagram testing thread, to study it even more closely at leisure.


Joe Joyce wrote on 2017-02-12 UTC

Hi, Kevin.

It's taken me a while to get back to this, but the problem is that there is no knight's tour of your proposed board. The absolute center hex is unreachable from any other hes on the board, which you can check by putting the knight on that center hex, and seeing it has no legal moves from there, on your proposed board size. You must up the size of the board, both the size and number of 2D levels, until the tour is doable within any given 2D level from any hex in that level. Then your board is big enough. Barely.


Kevin Pacey wrote on 2017-02-06 UTC

Hi Joe

I may not have what you think of as general goals, other than to say I'm trying to imagine on paper if any existing 4D square cell variants that I'm familiar with might have a feasible 4D Hexagonal analogue, using analoguous Hexagonal piece movement rules for a given 4D square cell variant. For now (for reasons I alluded to earlier, i.e. due to trial & error elimination) I'm concentrating on a 4D Hexagonal Hyperchess4 style analogue only. That is, with the non-pawn pieces moving like in Hyperchess4 as far as the 4D board goes, but using Glinski's Hexagonal Chess piece movement rules when moving the pieces on a 2D board (except for the bishops, which are allowed to shift a cell orthogonally, as in Hyperchess4). When it comes to moving between 2D boards, I'm moving the pieces like in Hyperchess4 (i.e to same cell on a different board, except for the knights at times).

I've experimented with different setups and pawn rules (right now preferring them to be close to Glinski's, even when moving between 2D boards). At the moment I have a (slightly[?] ugly) setup that's close to working, or actually does work, but I need to check it a lot to feel more sure. The first objection I had to a similar setup was that the White queen could force a possibly irritating 4D diagonal check in just two moves. I think by rearranging the pawns in that setup I may have solved that, though the setup's arguably uglier, but it is at least symmetrical, except for the K & Q positions, as is always unavoidable in a variant with just one K & Q per side. Otherwise, it has 17 pawns, with a pawn on the rear hex on three out of the four 2D boards with pawns and/or pieces for each side. That's with 9 pieces per side (an extra B compared to Hyperchess4, but currently there isn't going to be one starting out on each of the three basic hex colours). Sorry I don't seem to have clear general goals, though by just trying to fiddle with existing 4D square cell variants and make them work in 4D hexagonal, I hope to avoid a lot more trial and error (and thinking) then if I were to first brainstorm for some greater sort of general goals from scratch. One thing I could reconsider, if it seems helpful to me, would be the shape of my 2D Hexagonal boards, and/or the the whole 4D setup for these boards that I'm using, but for now I'd like these both to be hex shaped, like a Glinski's Hexagonal Chess board is.

If and when I have a setup (plus tentative rules) that seems to make for a variant that's at all feasible, I'll try to judge if it's possibly playable from start to finish in practice too, such as by the guidelines in Fergus' article on what makes for a good variant (e.g. having offence favoured somewhat over defence). At the moment I'm strongly thinking of just using that bared king confinement rule for a Hyperchess4 4D Hexagonal analogue that I wrote of earlier, rather than a 'held' rule as in Hyperchess4, to allow for 'basic' mates vs. a bare king. One day soon I may post a test diagram for it, before thinking of submitting it as a variant, though such a complex diagram takes some hours to make using the CVP Diagram Designer, and lately I've had a lot of eye strain after using my computers any length of time, so I don't want to really overdo things for a while. In the old days my father's Digital computer monitors brought home from work had tinted covers for screens, but no such luck for the often 'abominable' modern PCs I've gotten over the years, as my old man (& another old computer guy I knew) put it, compared to what they could have been had history been different (maybe Apple PCs are better? I don't know).


Joe Joyce wrote on 2017-02-06 UTC

The first problem with 4D is that there are far too many crazy diagonals. People cannot come close to visualizing them all. The second is forcing mate. There are too many ways to escape in 4D so that there need to be ways of restricting that 4D freedom to corral and mate a king. This is also a problem in 2D, but not as serious. However, on an infinite board, how many pieces are needed to mate a lone king? You need at least the king and a pair of rooks, the extra rook to provide the effect of a board edge.

I've dabbled in 3D, but find it vastly confusing, and harder to play than Hype. I don't know what  the minimum requirement for forced mate is, nor what it takes to force mate on an infinite 3D board. The mating player has to attack 27 squares on 3 adjacent 2D boards with every attacker being defended or out of the king's capture range. For a fully 4D king, 81 squares in a 3x3 array of 2D boards have to be attacked with every attacker being defended  or out of the king's capture range.

There seem to be only 2 options, restrict the king, and/or make other pieces powerful enough to capture a 3D or 4D king. Super-powered pieces bring problems of their own, especially in higher-D chess. They are impossible to guard against unless you clog the board with blocking pieces. And pieces with broad movement ability promote chaos in higher-D games. You essentially cannot predict the game state even a couple turns hence. Since the very powerful pieces act like missiles dropping out of the sky onto the target, you need area defense pieces, the equivalent of anti-missile batteries, much shorter range than the missile pieces, but invulnerable to missiles and able to shield neighbors, too. The "Missile Command" comment I made a while back discusses the idea. You'd need something to block the unblockable, a Neutralizer piece. Grin, I think I'd rather work out those ideas on a large 2D board!

The other way is to restrict the pieces in some significant way(s) without totally nerfing the 4D effects. That's basically the way I went here in Hype, along with the tiny board. But the held king rules allow the individual 2D boards to be any size - specifically longer than 5 squares/side - and still allow forced mate with K + 2 of the bishops and/or queen(s) against the lone king. Even on an infinite 2D board, you wouldn't need to add another rook to mate.

It’s a lot later than I wish it was, so I’ll just ask what your general goals are. Your 361 cell board should be big enough, but hex boards limit the number of simple pieces you can have. And my brain just stopped working here.  grin, I guess that makes this a cliffhanger.


Kevin Pacey wrote on 2017-02-04 UTC

Hi Joe

I had in mind nineteen 19 cell hex-shaped 2D hexagonal boards (peak width 5 cells, or in other words all 6 sides are 3 cells long), all arranged to form a hex-shaped 4D board. I even practised making one, using the Diagram Designer of CVP, though I had the same colouring scheme on all the 2D boards (to allow for facilitating 4D diagonal moves, I'd have to use a complex colouring scheme for the 4D board's 2D hexagonal boards, switching one of the colours about every 3rd board - that would have taken a couple more hours to do, I suppose). On 2D boards such as I described, knights move to 4 cells max. if staying on the same 2D board, or just to 3 cells (or to 0, if on the centre hex, oddly). Bigger 2D hex boards would be too unwieldy, I figured. Still, 361 cells in total is what I had for the 4D board, which is already a lot.

The problem I had (as I alluded to earlier) was trying to then visualize any half-decent setup where a knight couldn't check a king (or attack a queen, or undefended pawns) by force in one or two moves. That was using Hyperchess4 pieces & their movement rules as correspondingly on a 4D hexagonal board as I could (pawns excepted, perhaps) - i.e. an infeasible or unattractive sort of variant always resulting. The problems were even worse before that, when I tried using Balloon movement capable type pieces like in TessChess, or my 4*Chess variant. I'm pretty sure right now that with the 4D board dimensions I selected, the problems I mentioned will always arise regardless of the setup, at least as long as knights move as I wish them to (like for our 4D square cell boards).

I was just curious if a 4D hexagonal variant was feasible, and perhaps playable. At least 3D hexagonal variants have been done by other people & myself (looking at the 2D boards in my recent Hexagonal Raumschach variant CVP page might help a little in visualizing a smaller 2D hex shaped hexagonal board with its peak width = 5 cells). The only other sort of 4D hexagonal variant I've thought of would be a four 2D boards version of Alice Chess (e.g. using Glinski's Hexagonal Chess setup on one 2D board), but it wasn't anywhere near what I had in mind originally, and it has some issues too, though maybe they're not so bad. In any case, 4D variants never seem to win popularity contests, so I don't mind giving it a rest too much, at least for now. :)


Joe Joyce wrote on 2017-02-04 UTC

Geometry was what got me into chess variants, specifically trying to understand 4D for a math course. The connectivity of 4D gives remarkable freedom to the 'normal' chess pieces, and encourages higher-D analogs of the standard pieces, so the "balloon" is a 4D 'bishop'. The 4D 'queen' is often a monster, attacking around a quarter of the 4x4x4x4 board. The knight is the one piece that has a "standard" move in 4D, on square boards. Visualization of moves and counter-moves is something of a problem. (Btw, thanks, you are right about the minimum queen move in hype being 18 squares. Nice catch, sorry 'bout the typo.)

Taking it to hex boards... well, consider that a 4x4x4x4 square board is the smallest one where the knight has freedom/ability to move in all 4 directions from every square on the board. What is the analogous hex board size/shape? To an extent, this depends on the pieces you are using, and their hex footprints. What do you want to achieve?


Kevin Pacey wrote on 2017-02-02 UTC

I'm up way too late, but I was thinking about the 'held' rule once again. It occurred to me that an interesting version (or possible extension) of it might be that a king is permanently confined (or 'held', in a way) to the 2D board (or alternatively, rank & file coordinates) upon which it is located (by defender's choice, as is indicated by his next king move) if and when it is 'bared'. That would allow for all sorts of basic mates (including K+Q vs.K for starters), or permit K+just 1 Pawn vs. K to win quite often, yet still allow for possible draws due to insufficient mating material. I thought of extending this to similarly permanently confining a king if its last pawn was captured (to allow e.g. K+Q+P vs.K+B to be an easy win, presumably), but that seemed too drastic, and might harm the chances of an otherwise superior side winning at times, e.g. with K+Q (no pawns) vs. K+P.

My idea of a Hexagonal version of Hyperchess4 is still looking problematical at the moment, but at least thinking about it led to my latest thoughts on the 'held' rule. [edit: It appears that on the 4D Hexagonal board I have in mind, there's no good setup that doesn't allow a knight to attack something way too fast.]

[edit: I corrected my value estimate for a bishop & a queen in Hyperchess4 (they were wrong due to math errors). I now have them as B=3.2 and Q=3.5. Note that a B's 'average' number of moves (14) is fairly close to a R's (always 12, regardless of its location on an empty 4D board), and I didn't wish to put a R below =3 (since it can cope at times with 3 passed pawns on an empty board in my 4*Chess variant, given its pawn movement rules). Unless I'm wrong, the minimum number of cells a queen can reach in Hyperchess4 is actually 18, so a Q can reach 20 cells if averaging min. & max. (as compared to a knight's 'average' 18 cells). Note a B (like a R, and the other piece types) seems usually unable to eventually win a single passed pawn on an empty board in Hyperchess4, if I understand how these move right. As is evident, I'm not making much of a numerical distinction between a minor piece and a major one with my estimated piece values for Hyperchess4, which might be justified in one abstract way by the preceding sentence. Also, more vitally perhaps, bear in mind that the only major piece in my 4*Chess variant is the Q, which has an average of about 65 moves on an empty board in that game (there I value it = 16, giving a king a fighting value = 10 with it reaching about 47 cells on average), compared to a Q's reaching 20 cells on average in Hyperchess4; this all might be something of a case of comparing apples to oranges, though.]


Joe Joyce wrote on 2017-02-02 UTC

Piece values. Something I was always terrible at. Experience with Hype will let me argue some qualitative values. Pawns can move to 6 squares max, min 4, 7 - 6 on the initial move Rooks always move to 12. I'd argue rooks are worth roughly twice what pawns are. Bishops (B+W) are 18 max, 10 min, and 2 can mate, with king. Queens are 22 max, 16 min, and 2 can mate, with king. Or Q + B + K can mate. Rooks and knights don't have the proper footprint to force mate from anywhere on an open board. Knights, however, have a rather amazing mobility on a 4x4x4x4 board. The N has a min if 12 squares, and a max of 24 squares, and can deliver some nasty forks. I see it as clearly more valuable than a bishop, and while its minimum is less than the queen's, is maximum is greater. I'd put it closer to the queen than to the bishop in value, and it must certainly be considered a major piece despite not being able to force mate with a pair.


Kevin Pacey wrote on 2017-02-02 UTC

I've added an edit to my last comment.


Kevin Pacey wrote on 2017-02-01 UTC

Regarding piece values for Hyperchess4, the rook & knight move the same way as in my 4*Chess game, where I gave R=3 and N=3.4. For now I'll give P=1, still (though they move very differently in the two games). In Hyperchess4 after my own calculations I'd tentatively estimate a K's fighting value=2, with B=3.2 and Q=3.5. An abstract way I can try to justify have a Q ranked slightly more than a N is that a Q moves to slightly more cells on average on an empty board, with a Q's relative long-range movement capability, on such a 4D board, roughly offsetting the N's leaping ability.

[edit: If I understand Pawn movement rules correctly, upon further reflection it's plain a P is worth more than 1 in Hyperchess4, if, say, a R=3 as I estimate. It's not yet clear to me how to evaluate fairly closely what a P is worth in this game, except it seems a R can normally (at best) only keep a single passed pawn from promoting on an empty board, and is never able to win it (the side with the P could try to repeat the position, eventually, if no other pieces are ever in the picture). So, for now I'd guess a P would be worth between 1 and 3 pawns, with a rook still being worth more due to its greater mobility, in my eyes. Perhaps P=1.5 would not be too bad an estimate, unlike my initial P=1 assumption.]

At the moment I'm playing around with the idea of a Hexagonal version of Hyperchess4. It might never be at all popular, but maybe someone could later make use of the board diagram for it if I ever made a preset for such a game. One thing that's difficult is deciding on the pawn rules, assuming such a variant could ever be feasible.


Kevin Pacey wrote on 2016-03-04 UTC
I agree it's an interesting (but kludgy) optional rule worth noting, with 96 2D slices as you've described, Joe.

I'd note that if there's ever a rules enforcing preset for Hyperchess4 on Game Courier (and if or if not it would also use the optional rule as you've described), then it might be feasible & desirable to warn a player whenever his king could legally be held next turn (by the opponent's king) if he does nothing about it, much like a player can be warned if he is in check.

Joe Joyce wrote on 2016-03-04 UTC
Hey, Kevin, it's apparently dangerous for me to reply to your comments - lost 2 so far on this one, from power failures. 

First, I do like the idea of holding to a 3D Row or Column. Just for starters, it's halfway between full freedom of the board and being held on a 4x4 2D plane. And I've done games (though I don't know if any yet survive) with kings facing queen analogs/replacements. So I think that makes an excellent optional rules set - swap 1 K&Q's positions and use 3D holds as well as 2D holds.

As I was thinking about how to write this part to say the idea is interesting but too kludgy, I realized that you could get the same effect you suggested with easier visualization by using the 16 2D slices that have the same Column and file number and the 16 2D slices that have the same Row and rank numbers. This takes the number of 2D slices you can hold the king on from 32 to 64, which might be a little overkill. But once you're there, you could then try Row and file plus Column and rank, which if I'm doing this right, adds another 16 + 16 2D slices, for 96 different 2D "levels". While I would be hesitant to try playing with that many 2D planes available, this is certainly worth a write-up in the notes, in my opinion. What's your opinion?

Kevin Pacey wrote on 2016-02-28 UTC
Hi Joe

Fwiw there are two player equal army chess variants I've seen with each side having the K & Q switched in the setup position so as to be each opposite to the other side's Q & K respectively, so at least the idea would not be without precedent. I don't yet see that holding a lone enemy king on just 1 of the 4 4D board coordinates would suffice to work on its own, in order to be able to force a mate (say with just 1 Q). Perhaps the superior side's king needs to be allowed to further extend the hold onto 2 (or later even possibly 3) coordinates, if the lone king can be forced onto a cell from where it can be held by one more coordinate (once held on just 1 4D board coordinate), except I don't yet see how the lone king can be thus done in by force.

My suggestion that I wrote of earlier at least would not have the two K's already in a hold situation if using the current setup position, since one of the types of holds I suggested adding would be the Column & rank coordinates (i.e. I was careful not to say 'file' instead of rank); the other type of hold I suggested adding was Row & file. Still, the 2 kings (if my suggestion were adopted as it was) would not be too far from one being able to hold the other, by at least one of the kings advancing rank by rank while still staying on the same Column as each others' position in the existing setup. In this sense my suggestion made me a little uncomfortable. In another sense it did as well, in that it is something of a considerable 'kludge'. Hope you can make use of the suggestion somehow after all, even if the idea is indeed to be modified.

Joe Joyce wrote on 2016-02-27 UTC
Hi, Kevin. Already lost a long comment and have been/will be busy over the next few days so let me answer briefly here , then when I have time, look at the board and push pieces. Grin, that's a lot easier than trying to visualize it in my head when I have a few quiet minutes. But the mental gymnastics gave me this much, using the 1111 - 4444 coordinate system: 

The current rules state a king is held when on a square with either the same first 2 digits or the same last 2 digits. (11xx - 44xx or xx11 - xx44)

The kings start with the second and fourth digits the same. (x3x2) This precludes using every 2D slice of the board, given the current set-up. Switching 1 K-Q pair's start squares would make all 4 digits different, allowing all 2D planes to be used, but that's a true headache. The easiest effective hold is just hold on big or little square - real easy to see.

That's what I got last night while falling asleep. It seemed to me then that your idea wouldn't work, but in the shower just now, I realized it could possibly work. You move your king onto a square with the same first digit - makes no real difference what the other 3 are for this purpose - to hold the opponent's king on 1 of the 4 3D rows of 2D levels. Trying to use the 4 3D columns this way - 2nd digit same - could really mess up the game, I think, since the kings start in the same column. And switching the K and Q for 1 side doesn't seem as satisfactory here, though it could work. In fact, except for the game starting with the kings restricted in movement, it would be okay. I don't like that, aesthetically, I guess. Doesn't mean it wouldn't work fine. I think you've just added a paragraph to the game write-up. This is certainly worth optional rule status.

Kevin Pacey wrote on 2016-02-26 UTC
Hi Joe

In trying to think up a way to allow a K & Q to force mate vs. a lone K in Hyperchess4, or to possibly allow other 'basic' mates as in standard chess, I came up with the idea of further extending the "Hold" rule to include situations where the kings occupy a cell on the same "column" of 4x4 2D little boards and also share the same rank with each other, or situations where the kings occupy a cell on the same "row" of little boards and share the same file. It would then also be possible to hold an enemy king on 3 coordinates (i.e. Column, file, Row or rank), as opposed to just 2 coordinates, which could in turn lead to situations where a king that's being held on just 2 coordinates might be allowed to 'reverse the hold' by holding the other king by moving onto a cell having a 3rd coordinate that they would then share (tentatively speaking, I wouldn't advise to have the rule that the hold is actually reversed though). I haven't thought through all of the implications of this idea, but I didn't wish to reject it out of hand in case I'm at least close to being onto something at all interesting (if so, you might let me know at some point).

Note that I rejected the idea of extending the "Hold" rule to the kings sharing any 2 co-ordinates since for one thing the two kings would then be in a hold situation right at the start of a game.

Kevin Pacey wrote on 2016-02-25 UTC
Joe Joyce wrote: "...Just as in FIDE chess there is a condition called "opposition" in K+P vs K endings which prevents the pawn from successfully promoting, the same sort of thing happens when you try to get your king onto the same 2D level in Hype. The opponent's king stays as close to your king as possible, in both a neighboring big and little square, preventing your king from ever actually getting on the same board as your opponent's king to hold it to a specific 2D level..."

Thanks Joe. What you explained about an analogy in Hyperchess4 to the opposition of Ks in standard chess (an analogy given by you elsewhere too, I think) I independently finally realized (or it sunk in) today, and I meant to make an edit to my previous post to that effect, but I saw that you beat me to it. I found that even with the lone king in a corner cell of some little 2D board, it can never be smoked out by just two pieces (K & Q), say if they are placed on adjacent cells to it, but on an adjacent little board, since there would be always be one of the three adjacent cells available to the lone K on the same little board as its corner cell.

Joe Joyce wrote on 2016-02-25 UTC
Actually, it's a pretty simple explanation. Just as in FIDE chess there is a condition called "opposition" in K+P vs K endings which prevents the pawn from successfully promoting, the same sort of thing happens when you try to get your king onto the same 2D level in Hype. The opponent's king stays as close to your king as possible, in both a neighboring big and little square, preventing your king from ever actually getting on the same board as your opponent's king to hold it to a specific 2D level. The position shown is the minimum force needed to force your king's way onto the same 2D level as the opponent's king to get the hold. Once held, yes, even a rook and king deliver mate. Hope this answers your question adequately.

Ben Reiniger helped me find this position by playing the lone king as I looked over various possibilities. So let me here publicly thank him for his patience, as he didn't have my certainty there actually was a solution.

And finally, there is no requirement for the king which initiates the hold to stay on the same 2D level. This only applies to the king being held. So the lone king, being the one doing the holding, cannot be forced anywhere in particular other than to any safe square.

Kevin Pacey wrote on 2016-02-25 UTC
Joe Joyce wrote earlier:
...
"Checkmate is one of the trickier parts of higher-dimensional chess. The standard method for K + Q vs K in 4D is to move the king to the/a middle square, then move your queen between your king and the enemy king, pinning it against the edge of the board. But this only works if your individual "little" boards are no bigger than 5x5. My method, by restricting diagonal moves greatly and introducing the "held king" concept, will work on any size (rectangular) "little board". It allows forced mate with K and any 2 of the Q and pair of Bs vs. a lone K. And it took a little help, as Abdul-Rahman Sibahi gave me the final piece of the hold rule - that it works on the individual matching squares in each little board. A version of the hold rules can be applied to any higher than 2D variant, though it might well have to be tweaked to fit each higher dimension."

With the final version of the "hold" rule now in effect, at this point I don't quite get why just a K & Q do not suffice to force a win against a lone K. Hopefully the answer won't need to be too lengthy or involve a hard-to-post diagram, but any short answer may depend on if one thing was considered in your previous playtesting. That is, the idea of actually allowing the superior side's K to step onto a cell where it could be immediately held by the lone K. If not, I'm wondering if the following method would then suffice to win (assuming I've got the rules right and my visualizing of the 4D board is correct). There would be two cases once the superior side's K has allowed himself to be held (if he is not then held by the lone K, he will proceed to hold the lone K himself), namely:

Case 1) If the superior side's K is held on an individual matching square on a different little board, play the Q to a safe individual matching square as the Ks are on, i.e. on a third little board. At that point the Q plus K combo (with the Q sticking to individual matching squares on little boards that match those of the Ks) would seem to be able to eventually force the lone K to an edge (as opposed to central) little board (unless the lone K voluntarily released its hold early), and in a sort of checkmate-lite by the Q will force the lone K in having to move out of check (to an adjacent square, still on its edge little board) to abandon its hold on the superior side's K. At that point the superior side's K can hold the lone K, now with each K to be on the same individual matching square (but a different one than before), and then the Q can be brought to a safe individual matching square as the Ks are now on, at which point the process just described can be essentially repeated, with the superior side's K never releasing its hold, and instead of the Q & K combo ultimately delivering checkmate-lite on an edge little board, it would actually now ultimately deliver a real checkmate;

Case 2) In this second case, the superior side's K is held on the same little board by the lone K. Bring the Q to a safe square on the same little board as the Ks are on, then play to drive the lone K to an edge cell in order to deliver a checkmate-lite (if the lone K does not first voluntarily leave the little board that the Ks are on). At that point the lone K must go to an adjacent little board, and the superior side's K can follow him to that same little board, holding him there (again never intending to release the hold). At that point the Q can be brought to a safe square on the same little board as the Ks, then the Q & K combo would ultimately proceed to deliver a real checkmate on the new little board that the Ks are on.


If the above method actually works, then I'm wondering if K & R vs. K etc. are now to be considered basic mates too, as in chess, and whether for this purpose bishops need to have the enhanced little board movement rules that they do in Hyperchess4, i.e. so that a pair of them are able to force mate.

Kevin Pacey wrote on 2016-02-25 UTC
Joe Joyce wrote earlier:
...
"I like this design [Hyperchess4] because it is humanly playable, looks like chess, and gives people the feel of 4D and the pieces the freedom to move through 4D space without overwhelming the players with 3D and 4D diagonals, or very many of the available 2D diagonals, for that matter. Most who design 4D chess variants using the 2D layout of 2D boards to represent 4D space give the pieces moves that are based on a 4D space that is 2D x 2D = 4D. I've found this creates a totally chaotic game, where the state of the board cannot be reasonably projected even 2 or 3 turns into the future. Pieces move so freely that in a couple moves, they can be anywhere on the board, generally by many paths which can't all possibly be guarded in 2 or 3 moves. So I get my 4D by basing movement on a 2D + 2D = 4D concept. It gives you the same 4D game space, but it restricts movements to fairly easily visualizable, fairly simple patterns that players can project 2 - 3 moves into the future. While it is fast-moving, it is not chaotic."
...

Interesting. By my estimate, in a hypothetical worst case position, one side in a game Hyperchess4 might have around 200 legal moves available, if I've got the rules correctly. My earliest effort (4*Chess) to produce a 4D variant is based on a 4D space that is indeed 2D x 2D = 4D, and by contrast it might have around 600 legal moves available in such a worst case, so that a computer (or possibly a human player) could be able to look about 4 ply ahead in Hyperchess4 for every 3 ply looked ahead in playing my variant, I estimate. However, that's without taking into account the possible human difficulty with visualizing moves in 4D chess that you've noted.

Joe Joyce wrote on 2016-02-20 UTC
Thanks for the comment, Kevin. Checkmate is one of the trickier parts of higher-dimensional chess. The standard method for K + Q vs K in 4D is to move the king to the/a middle square, then move your queen between your king and the enemy king, pinning it against the edge of the board. But this only works if your individual "little" boards are no bigger than 5x5. My method, by restricting diagonal moves greatly and introducing the "held king" concept, will work on any size (rectangular) "little board". It allows forced mate with K and any 2 of the Q and pair of Bs vs. a lone K. And it took a little help, as Abdul-Rahman Sibahi gave me the final piece of the hold rule - that it works on the individual matching squares in each little board. A version of the hold rules can be applied to any higher than 2D variant, though it might well have to be tweaked to fit each higher dimension. 

I like this design because it is humanly playable, looks like chess, and gives people the feel of 4D and the pieces the freedom to move through 4D space without overwhelming the players with 3D and 4D diagonals, or very many of the available 2D diagonals, for that matter. Most who design 4D chess variants using the 2D layout of 2D boards to represent 4D space give the pieces moves that are based on a 4D space that is 2D x 2D = 4D. I've found this creates a totally chaotic game, where the state of the board cannot be reasonably projected even 2 or 3 turns into the future. Pieces move so freely that in a couple moves, they can be anywhere on the board, generally by many paths which can't all possibly be guarded in 2 or 3 moves. So I get my 4D by basing movement on a 2D + 2D = 4D concept. It gives you the same 4D game space, but it restricts movements to fairly easily visualizable, fairly simple patterns that players can project 2 - 3 moves into the future. While it is fast-moving, it is not chaotic. 

Almost nobody plays it, or ever has. Grin, that's the common fate of most variants. But getting it to work, finally, taught me something about chess design. For one thing, your pawns are there as much or more to protect your opponent's pieces as yours. And the checkmate problem in higher-D chess is merely a symptom of the chaos problem in higher-D chesses. The attempt to reduce chaos is part of what led me to short-range pieces. Restricting board size only limits the number of chaotic states that can occur, bringing it down from the all-but-infinite to a smaller but still effectively infinite number in terms of human lifetimes, as individuals or as a species.

Kevin Pacey wrote on 2016-02-18 UTCGood ★★★★
If I understand the up-to-date rules of Hyperchess4 right, a K & two Bs can force mate vs. a lone K. It is pleasing that this is the case in Hyperchess4, since in the classic Raumschach (a mere 3D, as opposed to 4D, variant) there doesn't seem to be any forcible 'basic' mate in the case of K & any combination of 2 pieces (excluding Qs or Ps) vs. a lone K. 

On the other hand, it could take getting used to that a K & Q can't seem to force mate vs. a lone K in Hyperchess4. Plus, as in most (if not all!) 3 or 4D variants, it seems that a King & Pawn vs. King wouldn't win nearly as often as it would in standard chess (in the case of Hyperchess4, I suppose it would quite rarely be a win, given that a K & Q cannot seem to generally force mate, at least unless the lone K is already "Held" by the superior side's K in the same 'big square').

Still, given that at least some 2-piece combinations can force mate vs. lone K in Hyperchess4, all it might take is to be just 2 pawns ahead in material to often win a game with good play thereafter, and this does not necessarily require that the opponent always play that badly before that stage. On this count alone, I'd think that Hyperchess4 is a pretty decent 4D variant.

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