Looking at some problems in higher dimensional chess by re-designing VR Parton’s Sphinx Chess. All text by Joe Joyce unless otherwise noted.
Chapter 1: A Simple Approach
This should give us Modern Sphinx Chess, but to avoid hassles, I'm calling it:
Hyperchess4
There is a link for a Zillions implementation and for an online 2-player game available with the original rules:
[http://www.chessvariants.org/3d.dir/hyperchess.html]
Introduction
Vernon Rylands Parton was one of the premier variantists of the last, or any other, century. His soaring ideas are not always easily reducible to readily-playable games however. For one example, Sphinx Chess is not quite as playable as it could be. Some fairly simple changes can be made to produce what is hopefully a better game, and which address some things Parton did only implicitly or not at all.
The rules of Sphinx Chess may be found here: http://www.chessvariants.org/parton/Cubism.txt (please scroll down to Sphinxian Chess) To save space, they will not be reproduced, merely such information as is necessary to this discussion.
I’d like to be as conservative as possible in making changes to Parton’s game. In extending chess to higher dimensions, I believe we should be conservative. The general principles I’m using are these:
• be as similar as possible to [western] chess, and its spirit [in moves and setup, for example] and to Parton’s basic concept of Sphinx Chess;
• be as simple as possible – playability is extremely important [moves are limited to linear extrapolations, for example] and the game should, as clearly as possible, be the 4D equivalent of western chess;
• keep the piece density low – let the board star as it should rather than having a clutter of pieces.
The first change is intended to give the game a more satisfying “chess” feel. The board is expanded to 4x4x4x4 from Parton’s original 4x4x3x3. Parton’s 9 little boards in a 3x3 array have become 16 little boards in a 4x4 array (the “big board”, whose 16 “big squares” are the 16 4x4 “little boards”).
Next, replace Parton’s coordinate system with a 4-digit system ranging from 1111 to 4444, which fits the new board better, and allows for easier movement representation. These 4 digits are broken into 2 pairs of coordinates, representing the location’s position on either the big board (the 4x4 array of little boards) or on the little board. The board squares are numbered this way:
41 42 43 44
31 32 33 34
21 22 23 24
11 12 13 14
The first pair of digits is the location on the big board, and the second pair of digits is the location on the little board. This allows easy calculations of moves and distances on the board, and the larger board allows us to use a more natural initial setup.
Figure 1 - Image courtesy of LL Smith.
Setup
Starting positions for white:
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 and movement
General discussion:
The pieces move the same way on the little squares and on the big squares. In general, 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 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 repeated, this time all on big squares. The other two turns are the shift from moving on little squares to moving on big squares, and the opposite shift from moving on big squares to little squares.
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 piece than the others, as it may move on both big and little squares in the same turn. (See diagram of knight moves at end.) However, these positions "mirror" each other, by switching big and little squares. 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 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, both shielding and blocking the friendly pieces from the enemy pieces at start. Without these restrictions, the game would have to have more pieces, likely far more. The main, and controlling, restriction is the elimination of most of the diagonals from the moves, including all of the “higher D” ones. There is a good reason for that, which I will try to express in simple terms.
Diagonals are evil. The best way to make the game playable is to dump most of them. There are 2D, 3D, and 4D diagonals that would let the white bishops capture 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. 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 quick count, there are 36 possible diagonals, of which 4 are used for this game.
Movement:
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. The knight in this game does not jump, it is a “bent 3-square orthogonal slider”. It must trace its path square by square through the board, turning through one of the four 90 degree 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 with the knight when actually playing the game. Players must pay close attention to how the knight is actually being moved. (See diagram of knight moves at end.)
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.
The appendix has a complete and thorough explanation of movement. Movement is shown by coordinates, by movement patterns, and for the knight, a diagram is provided. Please scroll to the bottom of the page. You can get an ugly but printable board with coordinates here: http://www.chessvariants.org/3d.dir/hyperchess_board.pdf
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.
The Held King Rule:
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.
If a king can legally move onto the same little square as the opposing king, on any big square anywhere, the opposing king may move freely from big square to big square, but may not move off the little square it is on until the hold is broken. The little square hold may be reversed when broken, also.
Discussion: The above rule sounds excellent, but when you actually play, you find there is something akin to “opposition” a king can do to another king, which I’ve tentatively named conjunction. In this game, it blocks the opposing king from ever getting into the same 2D level or little board as the king it is trying to hold. Simply, the king is kept in whichever of the 4 central (little) squares (of the little board it is on) is diagonally adjacent to the little square the opposing king is on. This prevents the opposing king from ever getting into that little board.
The Held King Rule is more effective the larger the board is. Why larger? Because it’s easier to get onto a particular square, big or little, when there are more of them available. Ideally, you’d play on a board of minimum side 6 or larger. A 6x6x6x6 with its 1296 positions might be a little unwieldy, but a 6x6x6, with 216 locations, is a nice size to try the Held King rule for 3D.
The Neutral Pawn Move Rule:
Aka: the "this-is-not-a-backwards-pawn-move”; a blatant attempt to put some defense into the game.
One unusual feature of this game is that the pawns are already "passed". It requires both players to deliberately bring the pawns into contact. That both these statements are true illustrates somewhat the nature of a 4D game.
The movement rules state the first and third board coordinate 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 of 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 allows pawns to “drop back” one big square while simultaneously moving forward one little square. 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 ability should be restricted to a pawn's first move, and it may further be restricted to that one backwards drop for the game. I strongly recommend using this rule only this way. This move may be considered a non-capturing move, akin to a standard pawn’s initial double step, weakening it still further, probably too much.
But it may also be considered a legitimate pawn capturing move, which may be used all game long, as this move is the same as any other sideways move the pawn is allowed, being still the same number of squares from either side’s back rank at the beginning and end of the move. As such, it may both drop back 1 big square (moving one little square forward on the new little board) or drop back 1 little square by jumping up to the next big square at the same time.
The full implementation of this rule gives the pawns the same sort of 4D freedom the knights have in this game. But it turns them into pieces, and greatly changes the game away from what chess “should” be, and into a game where pawns dominate. Use of the full rule should very likely be balanced by augmenting most of the pieces in some way; certainly the pieces become victims of the pawns. These pawns are quite capable of flushing out the enemy king and harrying him to his doom.
Discussion:
Mating
Playtesting over the summer of 2010 [B. Reiniger, J. Joyce] has demonstrated that the minimum needed to force mate of a bare king is 2 bishops and a king. A queen and bishop, or 2 queens, obviously also work. The required position is shown here: http://play.chessvariants.org/pbm/play.php?game=Hyperchess+-+White+to+move+loses+1&log=joejoyce-benr-2010-269-189
Piece Values:
I do not have really good piece values, but the pieces can be ranked. The queen and knight are of the same value in this game, the most powerful pieces, and apparently a reasonable trade for each other. The knight’s power, courtesy of its 4D freedom, has increased to rival that of the queen on this board. Next in value, the equivalent of the rook in FIDE, is the bishop, a 4D dragon bishop-type piece. My original thought, that the rook was more powerful in 4D, was wrong. It no longer interdicts, and drops in power to the lowest of the pieces, attacking only 12 squares max. The king attacks from 6 to 16 squares, so is effectively more powerful in the center of the board. Finally, pawns in this game are relatively more powerful than in FIDE. And if you use all the optional rules, the pawns are almost certainly too powerful.
Alternate Representations:
A 4D board can be portrayed in many ways when it's reduced to a 2D surface. (You can cut the tesseract in different directions.) Abdul-Rahman Sibahi suggested reversing the 2 pairs of coordinates, giving the following exactly equivalent setup:
Size of board/minimum edge length:
The size of the game board has a major influence on the play, but it’s not always obvious how great this influence is in 2D. The typical 3D games range from 8x8x8 to 8x8x2 to 4x4x4. Higher-dimensional games tend to use smaller edge sizes.
Let’s look at the lengths of each dimension (edge). The typical chess board is 8x8 or a bit larger, so no dimension has a side less than 8 in length. This allows for all the movement capabilities of FIDE. As the board shrinks, the game gets less chesslike, because the pieces no longer have room to maneuver. You can’t hold a track and field event in a broom closet. Let’s turn this around and look at the smallest sizes.
If the edge is 2, then on a 6D board, the positions can be defined [as above, for 4D] by 6 digits, ranging from 111111 to 222222. And that demonstrates exactly what the problem is, because 111111 -> 222222 is exactly 1 6D diagonal step. Every single position on this board is geometrically next to (touching) every other possible position. Now our rook has become a wazir, and our bishop a ferz, because they can move only 1 square maximum in any direction, or in the case of the ferz-bishop, any combination of directions. And the knight is nowhere.
Consider an edge of 3, and a 4D board. From the center location, 2222, it’s one 4D diagonal step to 1111 or 3333. But the sliders, R and B, have a potential of sliding 2 in a move. How about the knight? From the center location, a “standard” knight has no moves. A knight on an edge square can make an awkward tour around the edge squares, but never reach the center.
Let’s look at this a little closer. A 2D bishop moves “diagonally” through the corner of the cell it’s in to an “adjacent” (touching at the corner) cell, and this is characterized as a series of (1,1) moves on the 2D board. Most extrapolations of chess to higher dimension also increase the power of the “bishop” pieces to move “diagonally”, thus the 3D bishop has a (1,1,1) step that goes through one of the 8 corners of the cube in which the 3D bishop starts. Just so would a 4D bishop have a (1,1,1,1) step, and a 6D “bishop” would have a (1,1,1,1,1,1) step, and thus could step from one “corner” of a 2x2x2x2x2x2 to the other in a single turn. And, if it’s given all the “higher-D” diagonal moves, there will only be 6 locations on a 6D board of side 2 that it cannot reach in 1 move. If the piece starts at (1,1,1,1,1,1), then the only spots it could not step to in 1 turn are the 6 locations one orthogonal step away from (1,1,1,1,1,1), namely (2,1,1,1,1,1); (1,2,1,1,1,1); (1,1,2,1,1,1); (1,1,1,2,1,1); (1,1,1,1,2,1); and (1,1,1,1,1,2). Of course these locations can be accessed on the following turn.
The absolute minimum size that allows all the pieces to have reasonably chesslike movement in 4D is 4x4x4x4. The movement is minimal, as no piece can possibly move more than 3 squares in any direction, and that’s only from edge squares. But it unchains the knight, making it a major piece. The knight has full freedom of the 4x4x4x4 board: it may start on any location and move legally through one or more steps to any other location, the first "square" board size large enough for this to be true. It is still quite small; a better minimum edge size would be 6 [even for fair colorbound pieces], but a 6x6x6x6 has 1296 locations, and this might be a bit unwieldy.
Acknowledgements and Confession:
At this point, I have to confess that I designed this game, mostly in the early 1970s, completely unaware of VR Parton or even of chess variants in general, though I was aware that shogi, Ultima, and Jetan existed. When I got on line in 2003, I started working on it again, and finished in 2004, when it was first posted at chessvariants.org.
Larry Smith was the one who introduced me to Sphinx Chess in a comment he made about the posting, and he was also kind enough to make a zillions implementation under the name "Hyperchess".
Dennis Joyce and Abdul-Rahman Sibahi playtested it with me. Both Jeremy Good and Abdul-Rahman Sibahi made presets for this game, found at play.chessvariants.org. Further playtesting to determine the minimum material for a forced mate was conducted by Ben Reiniger and the author in 2010.
Abdul-Rahman Sibahi proposed the extension of the held king rule to occupying the same little squares. This aids significantly in mating.
Dropped Rule
The following rule is no longer in effect, it has been dropped as of Sept. 2010:
Pawn Protection:
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.
Appendices for Chapter 1
(You can get an ugly but printable board here:
http://www.chessvariants.org/3d.dir/hyperchess_board.pdf )
Taming The Slippery King
Page 116 Variant Chess 61 July 2009
Review by John Beasley, editor
A fundamental difficulty in multidimensional chess is that of confining and mating the king. Joe Joyce’s fourdimensional Hyperchess [Joyce] uses a method which appears novel.
The first reference in the literature to four-dimensional chess appears to have been in a paper by Dawson in the December 1926 issue of the Chess Amateur. This presented three problems, one on a 3x3x3x3 board and the other two on a 4x4x4x4, and employed the natural analogues of
two-dimensional rules: the rook moved in one dimension, the bishop simultaneously in two, a “unicorn” in three, and a “balloon” in four. The queen moved as R+Bi+U+Ba, the king similarly but one step only.
No initial array was specified and the variant seems never to have become more than a vehicle for these three problems, but in principle it would appear to have been playable. In particular, K + Q v K was a win, so there was a reasonable chance of converting a modest middle-game advantage into a win by playing for the ending and promoting a pawn. (If king and queen are allowed to move in any number of dimensions simultaneously, K + Q v K is a win on a board of size up to 5x5x5x… in any number of dimensions. The stronger side simply moves its king to the central cell, and lets the queen do the rest. However, mating a king other than by pushing it to a side face and plonking a queen directly in front of it is almost impossible.)
Parton, in his Sphinx Chess (Chessical Cubism, 1971), adopted a quite different approach. He set up a 3x3 array of 4x4 boards, and a piece was allowed to move either within a board or to the same square on another board. So, supposing the boards to be numbered from A1 to C3, a bishop on square c4 on board A1 could move to squares b3-a2 and d3 on board A1, and to square c4 on
boards B2 and C3 (always supposing any intermediate squares to be empty). This gave a more restricted bishop, queen, and king than the 1926 rules, while there was no analogue of the unicorn and balloon at all. The knight was replaced by a “centaura” which moved like a knight on a board and like a queen between them. But it was now impossible for K + Q to force mate against a bare king, and even Parton’s suggested restriction of the kings to boards B1 and B3 didn’t help. Presumably in compensation, Parton stated that perpetual check could be claimed as a win.
Joe Joyce’s approach is basically that of Parton, though with a 4x4 array of boards and many differences of detail. “Diagonals are evil,” he writes. “The only way to make the game playable was to dump most of them.” It has to be said that an unfortunate error in the 1926 paper bears him out, one of the problems being meaningless because the Black king is already standing in check from a unicorn. Not only was this overlooked by Dawson, but Anthony Dickins appears not to have noticed it when quoting the problem in A Guide to Fairy Chess (1967/69).
To control the king and make mate possible, Joe introduces the idea of a “held king”: when a player moves his king to the same 4x4 board as is already occupied by his opponent’s king, his opponent’s king (though not his own) is “held” and cannot leave that board. If his own king leaves the board and his opponent’s king follows it to its new board, it is now his own king which is “held”.
There are many other differences from Parton’s game and indeed from what happens in normal chess, and although these differences may seem rather arbitrary they were apparently inspired by practical experience. In particular, the bishop can make a one-step rook move to change colour,
the knight slides two-and-one without jumping, and the pawn is allowed a sideways move and captures with its normal move. The full rules can be found on the Chess Variants web site
<www.chessvariants.org> and there is a Zillions implementation. For those who prefer playing against people, Joe is <mjjoyce3> at <verizon.net> and would welcome opponents.
+ + + +
Email from John Beasley to Joe Joyce, 10/15/2009:
"Greetings once more, and your latest to hand. You are very welcome to quote the review, but use it intact, please. I appreciate the editorial constraints, but if it were to be significantly shortened I would want to approve the result to make sure that the editing had not involved unintentional distortion, and at the moment I do not have the time to give proper attention to such things."
MOVEMENT
a) in terms of board location using 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.
b) in terms of coordinates:
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. See diagram at end.
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.
KNIGHT MOVEMENT DIAGRAM
The knight, starting at N, location 2222, may move to the 24 locations marked "O", provided there is a clear path to that location. There are always 2 possible paths to any destination.
3. Sample game
This is not the best example of play possible, but it does provide a chance to see how the game works. It was an email game with a 3 weeks/move time limit, on the chessvariants.org server. To view it there, use this URL:
http://play.chessvariants.org/pbm/play.php?game=Hyperchess&log=sibahi-joejoyce-2006-333-477
You may also print out the board PDF, which has the 4 digit co-ordinate numbers used below in each square, and play/analyze along.
1 P 2231-3231 P 3234-3233
2 P 2221-3221 B 4334-3234
3 P 2321-3321 P 3224-3223
4 P 2331-3331 P 3324-4323 Black makes the basic "neutral pawn move".
5 B 1221-1243 P 3134-3133
6 N 1421-1222 N 4424-4412
7 N 1131-1332 P 3223-3323
8 B 1331-2231 P 3124-3134
9 P 2431-2331 R 4434-4433
10 B 2231-2242 B 4224-2424
11 N 1222-2224 check! K 4324-4224 White's knight forks the king,
queen, a rook, and a pawn
12 P 3221-4221 R 4124-2124
13 N 2224-3222 check! K 4224-4214
14 P 3231-4231 P 3334-4333
15 N 1332-2334 Q 4234-2434
16 N 2334-2322 N 4134-3334
17 P 4231-4232 B 3234-3212
18 N 3222-3342 N 3334-1333 check!
19 K 1321-1221 N 1333-1231 NxQ an even exchange; first capture of game
20 R 1431-1231 RxN B 3212-2312
21 N 2322-4222 check! N 4412-4222 NxN
22 P 4221-4222 PxN P 3323-3333
23 R 1231-4231 Q 2434-2234
24 B 1243-2243 Q 2234-1234
25 K 1221-2221 P 3333-2333
26 N 3342-1332 B 2312-1212 White's knight forks the queen and bishop
27 N 1332-1234 NxQ X 1332-1234 XxN An error: black mistyped the move,
and moved a blank square onto the white knight,
erasing it from the board - neither player noticed,
apparently from the combination of game complexity,
3 weeks/turn time limits, and the expectations of both
players that the N would be gone
28 P 3331-4331 B 2424-2324
29 B 2243-2234 P 2333-1333
30 B 2242-1342 P 1333-1233
31 P 2131-1132 B 1212-1112
32 B 1342-2242 B 1112-1121 BxR
33 K 2221-1121 KxR P 1233-1333
34 P 1132-1232 B 2324-1232
35 B 2234-3134 BxP P 3133-2133
36 B 2242-2142 P 2133-2233
37 B 2142-2124 BxR P 2233-1233
38 P 1232-1233 PxP P 1333-1332
39 R 4231-1231 K 4214-4224
40 P 2421-1422 B 1324-1224 Sadly, Black's mind is more on college than
this game over the past few turns
41 B 2124-1224 BxB Resign
Chapter 2: Going All The Way
TessChess
Introduction
In January 2010, Joe Joyce contacted Ben Reiniger after Joe found Ben's Yahoo!Answers discussion of 4D chess. Ben's goal was to design a workable fully 4D chess game. To that end, the pieces are logically extrapolated into higher dimensions. Discussion and playtesting by Ben and Joe developed a choice of starting setups which vary in the exact placement of the 8 pieces.side, and also vary the number of pawns/side, from 8 to 12.
Rules
[http://chessvariants.wikidot.com/tesschess]
[[/code]]
