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# Hyperchess

## Introduction

Hyperchess: a four-dimensional chess variant that closely parallels traditional, or "standard" chess. It uses the standard chess pieces, and as much as possible, the pieces keep their standard moves.

## Board

The board is 4x4x4x4; 256 cells in a 16x16 array. Lines and shades of color divide the board into 16 "Big Squares". Each of the Big Squares has 16 "Little Squares" in it. The big squares are numbered 11 to 44. The little squares are also numbered 11 to 44.

The pattern for both is:
______________
| 41 | 42 | 43 | 44 |
| 31 | 32 | 33 | 34 |
| 21 | 22 | 23 | 24 |
| 11 | 12 | 13 | 14 |

See the attached PDF for the complete board.

Each of the 256 cells has a 4 digit position number,  from 1111 to 4444. The first 2 digits are the big square; the last 2, the little square.

Looked at one way, each big square is a quarter-size standard chess board. Pieces move on this mini-board like normal chess pieces, some with slightly different moves. Looked at another way, the 16 big squares together are a large-scale quarter-size chess board. Moves occur on this economy-sized mini-board in the same way they occur on the little boards.

## Setup

Starting positions for white: (Refer to the PDF for the board.)
2112 - P        2113 - P       2212 - P      2213 - P      2312 - P      2313 - P      2412 - P      2413 - P
1112 - R        1113 - N      1212 - B      1213 - Q     1312 - K     1313 - B      1412 - N     1413 - R

Starting positions for black mirror those for white:
3142 - P      3143 - P      3242 - P      3243 - P      3342 - P      3343 - P      3442 - P      3443 - P
4142 - R      4143 - N     4242 - B      4243 - Q     4342 - K     4343 - B      4442 - N      4443 - R

## Pieces

The pieces move the same way on the little squares and on the big squares. If a piece can move in a particular pattern within a big square, it may move in that exact pattern between big squares. Example: If a bishop is in big square 11, on little square 11, it can move to little squares 22, 33, or 44 in big square 11. Or it can move to little square 11 in big squares 22, 33, or 44.

There are knight moves that involve change of position both within and between levels. These moves mirror each other, switching big and little squares.

There are four straight-line directions (rook moves) and four main diagonals (bishop moves) on the board, defined by the big squares and little squares.

We can also define four 90o turns on the board.  These are what the knight uses in its move. Two are the obvious turns, the turn the knight makes during its' standard L-shaped move, all on little squares, and the same turn, all on big squares. The other two turns are the shift from moving on  little squares to big squares, and the opposite shift from moving on big squares to little squares.

The king may move one square in any direction. It may move to any of the (up to) 8 adjacent little squares in its' starting big square. It may also move to the same little square in any of the (up to) 8 adjacent big squares. Kings may neither move into nor be left in check.

The queen combines the moves of the rook and the bishop.

The rook may move as far as possible in a straight line in any one of the 4 perpendicular directions. That is, it moves only through the sides of squares, big or little. It moves on the little squares like a standard rook. Or, it  may move as far as possible in a straight line along the big squares, landing on its' original little square each step of the way.

The bishop may move as far as possible in any one of the 4 main diagonal directions. It may move as a normal bishop on the little squares. Or it may move as far as possible in a diagonal line along the big squares. It must land on the same little square in each big square along the way. Or it may sidestep one square, big or little, moving like a rook for 1 square only. This gives the two bishops the ability to cover all four colors.

The knight moves exactly 3 squares using its' standard L-shaped move. It cannot move diagonally. It must trace its' path square by square through the board, turning through one of the four 90o angles, after either the first or second square. It must change direction one time only during its' move. It may move just on little squares, just on big squares, or it may switch between them during its' move. It is the only piece that may move on both little and big squares in the same turn. Warning. It is relatively easy to make two changes of direction.

Pawns may move one square forward, toward the opposing player's back rank, or one square sideways, left or right. The one square may be little or big. They may never move backwards: closer to one's own rear rank. They may never move diagonally.

Pawn Promotion - Pawns are the only pieces that promote. When they reach the opponent's back rank, they are queened as usual.

Following is a listing of all possible moves for each of the pieces:

K starts on 2222. It may move to  squares 2211, 2212, 2213, 2221, 2223, 2231, 2232, 2233, 1122, 1222, 1322, 2122, 2322, 3122, 3222, or 3322.

R starts on 1111. It may move to 1112, 1113, 1114; 1121, 1131, 1141; 1211, 1311, 1411; 2111, 3111, or 4111.

B starts on 2222. It may move to squares 2211, 2213, 2231, 2233, 2244; 1122, 1322, 3122, 3322, 4422; 2212, 2221, 2223, 2232; 1222, 2122, 2322, or 3222.

N starts on 2222. It may move to 2241, 2243, 2234, 2214, 1242, 1224, 2142, 2124, 3242, 3224, 2342, 2324, 4221, 4232, 4223, 4212, 2421, 2432, 2423, 2412, 4122, 4322, 3422, or 1422.

White P starts on 2212. It may move to 2211, 2222, 2213; 2112, 3212, or 2312.

Black P starts on 2212. It may move to 2211, 2213; 2112, 2312, or it may move to 1212 and, on that square in the opponent's back rank, it must promote.

Movement in terms of board location - the 4 position numbers of each cell:

The king moves by changing any one of its' position numbers by plus or minus one, or each digit of the first (big square) or second (little square) pair of position numbers independently by plus or minus one.

The rook moves by changing any one of its' position numbers, always in the same "direction" (plus or minus), as far as it can.

The bishop moves by changing either its' first or second pair of position numbers, each number of the pair simultaneously and independently, always in the same direction, plus or minus, as far as it can; or by changing any one of its' position numbers by plus or minus one.

The knight moves by changing any one of its' four position numbers by plus or minus one, and another of its' position numbers by plus or minus two. The +/- 1 change must be made either before or after the +/- 2 change. The +/- 1 change cannot be made in the middle of the +/- 2 change, as this is two changes of direction.

The white pawn moves by changing any one of its' four position numbers by one. The first and third position numbers may only stay the same or increase. The second and fourth position numbers may increase, decrease or stay the same.

The black pawn moves by changing any one of its' four position numbers by one. The first and third position numbers may only stay the same or decrease. The second and fourth position numbers may increase, decrease, or stay the same.

## Rules

All pieces capture by landing on and removing enemy pieces. No piece may jump over another piece. This includes the knight. Pieces may not cross a big square boundary line while moving only from little square to little square during a turn. Checkmates and draws are as in standard chess.

White moves first.

A king may hold the other player's king in one big square. When a player moves the king into the same big square as the opposing player's king, the opposing player's king is "held". The opposing player's king keeps its' moves on the little squares, but may not move out of the big square both kings are in. Only the opposing player's king is held. The player who created the hold may freely move the king out of the big square the enemy king is held in, even to get out of check, or to give a discovered check. This breaks the hold. Otherwise, both kings stay in the same big square until one of them is checkmated. The held king only needs to be checkmated in the big square it's in, it can't leave. The holding king, to be mated, must have all its' allowed big square destinations guarded also.

The formerly held king may immediately follow the other king from the big square, reversing the hold.

Optional Rule: Pawn Protection:

Pawns are allowed to go to the aid of their "held" king. When a player's king is held, that player may move pawns from any of the up to 8 adjacent big squares into the big square where the king is held, landing on the same little square started upon, capturing any opposing piece on that square. This allows pawns to move from big square to big square backwards, diagonally, and diagonally backwards. Basically, when a king loses the ability to move to any adjacent big square, that king's pawns which are close enough get that ability, and only those pawns. The pawns do not change the way they move on little squares - they cannot move diagonally or backwards on those little squares. A pawn not in a big square next to the held king's big square could, using its' regular move, move into any one of the big squares next to the held king, then on the next turn, use the pawn protection rule to move into the same big square as the held king. This is offered to balance the king hold rule.

## Playing Tips

Some things are not dealt with in the game. One is perpetual check. Here is the rule: Don't do it. Another is castling. Haven't seen any useful way to do it. There are others.

Without some kind of restriction on the king's move, it is extremely difficult to get checkmate. Checks are relatively easy, especially when a number of pieces have been taken, and pawns have promoted. Checkmates are quite different. A king and three queens against a bare king is a draw without either a king restriction or queens getting more moves.  BK @ 3333; WK @ 1122, WQs @ 1111, 1112, 1121; either moves first, is a draw.

## Notes and History for Hyperchess

A strong, aggressive player can destroy a more timid, defensive player. The game is too wide open to start with a defensive posture. Black might do well to mirror the bulk of white's moves in the beginning of the game if white is aggressive - attempting to defend white's moves rather than attacking strongly as black can be a serious mistake. (I learned this the hard way - my opponent queened 4 pawns in one game, before he stopped bothering. He had 3 left.)

Optional Rule: The "This-is-not-a-backwards" pawn move rule. [A blatant attempt to put some defense into the game.] The rules state the first and third digits can never decrease for white pawns, and can never increase for black pawns. The pawns move by changing one of their four position numbers, and must never go "backward". Allow the pawns to change both their first and third position numbers, one increasing and the other decreasing, as a move. This is, in effect, a 'neutral' move, neither forward nor backward. This would allow pawns to 'drop back' one big square while simultaneously moving forward one little square, or vice versa. For example, on its' first move, the king's pawn could move to the little square directly in front of the king. Thus, the white king's pawn could move from 2312 to 1322, and the black, from 3342 to 4332. This need not be restricted to a pawn's first move. This move may be considered a non-capturing move. [This optional rule, unlike the "pawn protection" optional rule, is not currently implemented in Zillions.]

Hyperchess started as only the board, some 3 decades ago; as an attempt to understand the 4th dimension. About 4D - Hyperchess is in the ChessVariants 3D chess section, and I figure it gets people to look, so I'm not objecting too strenuously, but... the game started as a piece of "4D" graph paper. I drew a "double grid" 4x4x4x4 bit of graph paper, numbered it from 1111 to 4444, and decided it would make a great chess board. Got a degree a long time ago, so I have at least a nodding acquaintance with the concepts of relativity and time as our fourth dimension, but the Euclidian treatment of "pure space" is so much easier on the head, comparatively. Anyway, it was obvious the 4x4x4x4 bit of graph paper would make a great chess board, so I started fitting pieces and moves to it. I quickly caught the "let's add a whole bunch of pieces, some with slightly different moves" disease. I remember 16 pawns all the way across the board at one point, and then 16 more. It was always a fascinating idea, but it was never humanly playable, so it kicked around for a couple of decades, then I got lucky. [As an aside, it seems that most variant players look for a whole bunch of different pieces on a relatively small board. Am I doing something wrong??? Is there another chess game with a starting piece density of 12.5%, or lower? Sphinx chess (VR Parton) starts at 22%.

My next game has fewer cells and more pieces, I swear. It's called "Walkers and Jumpers", and has a starting piece density of 19% - 36 total pieces on 189 squares.] What I wanted was a simple, easy-to-play game. I think I got it, after maybe 30 years. I honestly believe the set-up is elegant. It basically came to me in a flash of inspiration, about 5 years ago. The set-up gave me the pieces for the game, as they were right there in front of me. So, I got the pieces and starting positions, but did not yet have the movement rules. The basic moves fell out of the board configuration pretty quickly, but the rule that made it a game was the king hold rule. This actually allows checkmate, which helps a game out all no end. It took me many months, and an assist from my playtester, to get the hold rule. During this time, my son finally got me on line, and I found Chesseract, which was discouraging in tone, and ChessVariants, which was moderately encouraging.

The only person who responded to my email was Tony Quintanilla. Thus, this site has my chess games. I never heard of V.R. Parton until Larry Smith mentioned him in Mr. Smith's most kind comment on my game. Designing Hyperchess was almost purely an intellectual exercise, done with no knowledge of the world of chess variants. I only played 2 games before publication, one before and one after the king hold rule. As I can't play chess against myself, once the game 'gelled', I had to publish to get it played. Since then, I've played several more, and gotten promises to play... (The previous 8 sentences were my disclaimer.)

I think what I'm saying is I designed the game backwards, but it was always an attempt to use 4 spatial dimensions for chess. This is not a popular position; few believe in 4 spatial dimensions. When I try to see my hype board as a 2D board that may or may not adequately represent some three-dimensionality, I can do it, but it hurts my head. I follow others' arguments on dimensions (ref: Yahoo 3D Chess). I don't agree with them. I can appreciate people feeling this way (it makes sense when looked at from a certain angle), but I can't think that way. All I get are vague feelings, that I then have to translate painstakingly and rigorously into a Euclidian 4D system. Flat (2D) chess has a bishop that is just about perfect, as long as the board is relatively square, that moves in 2D. Hyperchess has a bishop that works very well in its' "traditional" (ie: diagonal) move mode. And, if you won't grant me 4D, you gotta admit that my bishops move in 2D and 2D, and they're different 2Ds. It's admittedly limited, but, just as 2+2=4, 2D + 2D = 4D, in effect, if not actuality.

Big and little square moves have no effect on each other. If you move in the little squares, you're still in the same big square, and if you move on the big squares, you're still in the same little square. The knight is a more complicated case, as it may move on both big and little squares in the same turn, but the positions "mirror" each other. In some ways, you can say there are only 16 squares on the board - they're just in 16 different places at the same time. The next game I have sent in uses both 2D-moving pieces and 4D-moving pieces, on a 9x21 or 3x3x3x7 board. It uses both a 4D numbering system and the "standard" alphanumeric chess notation, depending on which piece moves how.

Some rules were not used but could have been. For example, "kinging" a pawn. All promoted pawns would gain the moves of the king. This limited promotion was dropped before a good way to ensure a reasonable number of checkmates was found; the "king hold" rule is new. Every attempt to deal with the king's ability to move out of check before the king hold rule was flawed in some way. Most of the attempts involved making other pieces more powerful. These did not work and a different solution was used, but "queening" was not changed back to "kinging". This would be worthwhile if too many pawns get queened in a game - having 3 - 4 queens per side would appear to unbalance the game.

Diagonals are evil. The only way to make the game playable was to dump most of the diagonals. There are 2D, 3D, and 4D diagonals* that would let the white bishops take their opposite numbers on the first move, and the queen could either take the black queen or put the black king in check on the first move.

The key to this variant design is restriction. The types of moves, the numbers of pieces, and the strengths of the sides were all deliberately restricted. This is exactly why black and white can get away with only 8 pieces and 8 pawns each. The pawns just barely do their main job, shielding the friendly pieces from the enemy pieces at the start.

Without these restrictions, the game would have to have more pieces.

*2D diagonals are changes in any 2 of the 4 position numbers, 3D diagonals change any 3 of the 4 position numbers, and the 4D diagonals would let you go from 1111 - 2222 - 3333 - 4444 as a bishop-type move. It would also let the Q @ 1213 -> 2322 -> 3431 checking K @ 4342. By my count, there are 36 possible diagonals, of which 4 are used. The right four, of course.

Enjoy the game. Comments are welcome.

## Credits

Design: Joe Joyce     Playtesting: Dennis Joyce   Board: Peter Joyce