ChesseractChesseract is played on a four-dimensional (4D) board that measures 4x4x4x4 cells. An earlier version of Chesseract is posted on the Chess Variant Pages. My first impression of the idea of playing chess on a four-dimensional board was that it was a great idea in theory, but utterly impractical and therefore thoroughly silly. I've since begun to feel that maybe it isn't quite that silly after all. (The fact that someone actually bothered to code a Zillions of Games version may have influenced me.) For the new version I've simplified the initial piece layout and made a few other simple changes in the rules. Though the moves of the pieces are perfectly straightforward, not tricky or convoluted in any way, it seems clear that Chesseract is extremely difficult to play, owing to the challenge of envisioning all of the possible moves of a given piece in 4-space. What I'd love to see (but will probably never get around to writing) would be a computer program that wouldn't actually play the game but (much more simply) would allow you to click on any piece and see where it can move, or click on any square and see what friendly and enemy pieces protect or threaten it. The object of the game is to checkmate the enemy king, and a stalemate is a draw. The BoardConsider a cube, perhaps of clear plastic, built out of smaller cubes (let's call them cells) in a 4x4x4 arrangement. Consider four of these 4x4x4 cubes sitting side by side on a table, with chess pieces in some of the cells. Now imagine that instead of being beside one another in three-dimensional space, the four cubes are somehow "stacked" on one another in four-dimensional space, so that each of the cells in the first cube is immediately adjacent to a corresponding cell in the second cube, and so on. This is the geometrical object on which Chesseract is played. We can draw an ASCII diagram of such an object, peeling away the layers so as to represent it in only two dimensions, in the manner shown in Figures 1, 2, and so on. In general, we can refer to any two-dimensional 4x4 area of the Chesseract board as a layer. Figure 1. A 2D representation of the 4D Chesseract board. Each cell is designated uniquely by four coordinates -- a capital letter, a Roman numeral, a lower-case letter, and an Arabic numeral. For reference, cell AIIIc1 is marked with an 'x'. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| IV 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| III 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|x|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| II 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| I 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d If you like, you can imagine that the vertical sets of layers designated A, B, C, and D are the four 3D cubes made of clear plastic, and that the bottom layer of each of these cubes is labelled I, the layer above that II, and so on. However, it will probably help if you understand that since the board is a true four-dimensional object, it's entirely arbitrary which of the coordinates we consider as representing 3D cubes. They all represent 3D cubes. By "twisting" the board in 4D space, we can just as easily view it as four 3D cubes labelled I, II, III, and IV, or as four cubes labelled a, b, c, and d, or 1, 2, 3, and 4. Fortunately, it's not necessary to perform such a rotation as a routine mental exercise, much less to try to envision what is revealed about the positions of the pieces when we do so. Suffice it to say that a piece on cell BIVb2, for example, will always be on BIVb2, no matter how we choose to view the board. For convenience, we can refer to the 16-cell layers AI, AII, AIII, etc. as "quadrants," and the cells AIa1, AIa2, and so on as being contained in quadrant AI. The term "layer" refers, then, to any set of 16 cells that lie in a two-dimensional plane, and the term "quadrant" to a layer that is displayed as a group of contiguous squares in a particular diagram. While the Chesseract board is large (256 cells), in another sense it's quite compact: None of the cells are very far from one another. A piece like a rook or bishop, for example, can travel no more than three cells in a straight line before reaching the opposite side of the board. The relationships of adjacent cells on a 4D board are of four types: orthogonal, 2D diagonal, 3D diagonal, and 4D diagonal. The three types of diagonals are entirely distinct from one another. Since a 4D diagonal relationship is difficult to visualize, let's digress for a moment to talk about 4D geometry. It may be helpful to understand that the cells of the board, which we have been thinking of as little cubes of clear plastic, aren't cubes at all: They're all little tesseracts. A tesseract is the 4D equivalent of a cube. Just as a cube has 6 faces (2D squares), 12 edges, and 8 corners, a tesseract has 8 "sides" (all of which are cubes), 24 square faces, 32 edges, and 16 corners. Just as two adjacent 3D cubes in a 3D chessboard can be touching one another on a 2D face, on an edge, or at a corner, when two of the individual tesseracts within a 4D board are adjacent, they can share a cubical 3D side, a single square face, a single edge, or only a corner. On the Chesseract board, when two cells share an entire cubical "surface," they have an orthogonal relationship; when they're touching on a face, they have a 2D diagonal relationship; when they're touching along an edge, they have a 3D diagonal relationship; and when they're touching only at a corner, they have a 4D diagonal relationship. (Note that if you're visualizing the board as made up of four 3D cubes, you won't be able to "see" two cells touching one another on an entire cubical side. Those that would appear to be touching one another on square faces in a 3D "unfolded" view are actually touching along entire cubes.) From the numbers above, it should be obvious that each tesseract- shaped cell, assuming it's surrounded on all sides by other cells, will be orthogonally adjacent to 8 other cells, 2D diagonally adjacent to 24 other cells, 3D diagonally adjacent to 32 other cells, and 4D diagonally adjacent to 16. Figure 2 confirms this count. Figure 2. Orthogonal and diagonal relationships on the Chesseract board. Cell CIIIb2 (marked 'X'), is adjacent to (that is, touching) all of the other marked cells -- 80 in all. The cells that are orthogonally adjacent are marked 'o', those that lie on the same 2D diagonal are marked '2', those that lie on a 3D diagonal are marked '3', and those on a 4D diagonal are marked '4'. A moment's study should make the pattern clear. Notice that a 2D diagonal move, for instance, can be made either by moving diagonally within a given quadrant (within CIII from the b2 to cell to the c3 cell, for instance), by moving to the equivalent cell in a diagonally related quadrant (from CIIIb2 to the b2 cell in quadrant BIV, for instance), or by moving orthogonally twice, once within a quadrant and once to a new, orthogonally related quadrant (from CIIIb2 to CIVc2, for instance, by way of CIVb2). This method of visualizing moves will also be found helpful where the knight is concerned (see Figure 13). _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |4|3|4|_| |3|2|3|_| |4|3|4|_| IV 2|_|_|_|_| |3|2|3|_| |2|o|2|_| |3|2|3|_| 1|_|_|_|_| |4|3|4|_| |3|2|3|_| |4|3|4|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |3|2|3|_| |2|o|2|_| |3|2|3|_| III 2|_|_|_|_| |2|o|2|_| |o|X|o|_| |2|o|2|_| 1|_|_|_|_| |3|2|3|_| |2|o|2|_| |3|2|3|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |4|3|4|_| |3|2|3|_| |4|3|4|_| II 2|_|_|_|_| |3|2|3|_| |2|o|2|_| |3|2|3|_| 1|_|_|_|_| |4|3|4|_| |3|2|3|_| |4|3|4|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| I 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d Before we go on to discuss the movements of the pieces, one other aspect of the board may be worth noting. In a three-dimensional 4x4x4 cubical board, there are four distinct types of cells -- 8 corner cells, 24 cells that lie on an edge but are not at a corner, 24 cells that are in the center area of one of the side- faces, and 16 cells that lie entirely within the cube. On the Chesseract board, there are five distinct types -- 16 corners, 64 cells that are on an edge but not in a corner, 96 that are in the center area of one of the 2D faces, 64 that are in the center area of one of the cubical sides, and 16 that are entirely in the interior of the tesseract. The locations of these types on the two-dimensional board diagram are shown in Figure 3. This fact is significant because it affects the power of the various pieces. As on a conventional chessboard, pieces are more powerful (that is, they have more choices of other cells to which they can move) when situated at or near the center of the board. I'll leave it for the interested reader to calculate exactly how many cells each piece can reach when its move starts on each of the five types of cells. Figure 3. The five geometrically distinct types of cells on the Chesseract board. Cells marked with the same symbol are equivalent with respect to the "reach" of equivalent pieces situated on them. The center cells are marked ':::', the center cells of the cubical sides are marked 'o', the center cells of faces are marked '*', the edge cells are marked '-', and the corner cells are marked '>'. These types remain the same for all cells irrespective of the type of 3D orientation used to depict the board. While cells AIVb2, BIVa2, and BIIIa4, for instance, appear geometrically distinct from one another in this 2D representation, in the actual 4D board they are all center cells of 2D faces. BIIIa4 is NOT a corner cell, even though it appears to be one in this representation. The corner cells are those whose algebraic representation uses only the letters 'a' and 'd' (capitalized or not) and the numbers '1' or '4' (either Arabic or Roman). In order to view this diagram properly, you may need to open up your browser window as wide as it will go. (If your monitor doesn't give you enough room, try viewing the HTML source, copying the text of the diagram, pasting it into a word processor, and reducing the point size of the type.)
A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| > | - | - | > | | - | * | * | - | | - | * | * | - | | > | - | - | > | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ IV
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| > | - | - | > | | - | * | * | - | | - | * | * | - | | > | - | - | > | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| * | o | o | * | | o |:::|:::| o | | o |:::|:::| o | | * | o | o | * | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ III
| * | o | o | * | | o |:::|:::| o | | o |:::|:::| o | | * | o | o | * | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| * | o | o | * | | o |:::|:::| o | | o |:::|:::| o | | * | o | o | * | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ II
| * | o | o | * | | o |:::|:::| o | | o |:::|:::| o | | * | o | o | * | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| > | - | - | > | | - | * | * | - | | - | * | * | - | | > | - | - | > | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ I
| - | * | * | - | | * | o | o | * | | * | o | o | * | | - | * | * | - | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| > | - | - | > | | - | * | * | - | | - | * | * | - | | > | - | - | > | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
MovementTo shorten the game, which would otherwise be quite long with so many pieces spread across so many cells, to say nothing of the amount of time needed to insure that any given move won't lead to disaster due to an ambush in the fourth dimension, Chesseract is played as a double-move variant. The player with the white pieces makes a single initial move; in all subsequent turns, including black's first turn, each player moves two pieces. The following restrictions apply:
The PiecesChesseract is played by two players, each of whom has an army
consisting of a king (K), a queen (Q), a dragon (D), a minstrel (M), four rooks
(R), two each of unicorns (U), wizards (W), bishops (B), and
knights (N), and 16 pawns (P). The opening layout is shown in
Figure 4. (There are five other ways of viewing the same layout; they're all shown in Because the opening layouts of the opposing pieces are so close to one another, Chesseract requires a Non-Aggression Pact Rule: No piece that has not yet been moved can capture an enemy piece that has likewise not yet been moved. The Non-Aggression Pact does not prevent the capture of pawns. Figure 4. The opening layout in Chesseract. The black pieces are indicated by lower-case letters braced by asterisks. Two things are worth noting about this layout. First, while the piece density is 25%, compared to 50% in standard chess, each piece has far more potential moves than its equivalent in standard chess, so the tactical density is higher than it might at first appear. Second, all of the pieces in each player's rear layer can move (and be attacked) even before any pawns have been moved, because the pawn layer covers the rear layer from only one side, not from the other side or diagonally. Both of these observations will make more sense after we look at how the various pieces move.
A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
4 | | | | | | | | | | | | | | | |*r*|*u*|*w*|*r*| 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
3 | | | | | | | | | | | | | | | |*b*|*k*|*m*|*b*| 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ IV
2 | | | | | | | | | | | | | | | |*n*|*d*|*q*|*n*| 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
1 | | | | | | | | | | | | | | | |*r*|*u*|*w*|*r*| 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
4 | | | | | | | | | | | | | | | |*p*|*p*|*p*|*p*| 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
3 | | | | | | | | | | | | | | | |*p*|*p*|*p*|*p*| 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ III
2 | | | | | | | | | | | | | | | |*p*|*p*|*p*|*p*| 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
1 | | | | | | | | | | | | | | | |*p*|*p*|*p*|*p*| 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
4 | P | P | P | P | | | | | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
3 | P | P | P | P | | | | | | | | | | | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ II
2 | P | P | P | P | | | | | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
1 | P | P | P | P | | | | | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
4 | R | U | W | R | | | | | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
3 | B | K | M | B | | | | | | | | | | | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ I
2 | N | D | Q | N | | | | | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
1 | R | U | W | R | | | | | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
Figures 5a and 5b show two alternate methods of viewing the same layout. All that has changed is that we've flipped the board inside-out, in a manner of speaking, so as to view a different set of quadrants conveniently. All of the pieces are on the same cells as before, but any piece that was on, for instance, a local cell b1 within any quadrant in Figure 4 is shown in Figure 5a as located in quadrant b1. This type of rearrangement would be useful if anybody were to actually try to play Chesseract, as it brings to light vectors of movement and threat that might otherwise be overlooked. Writing a computer program to enable a given board position to be viewed in any of six different ways would be very simple. If we consider what might be displayed in the upper left quadrant of the diagram, the six main diagrams become easier to list: This quadrant can display the AI layer, the Aa layer, the A1 layer, the Ia layer, the I1 layer, or the a1 layer. Figure 5a. The layout in Figure 4 can also be viewed in either of the ways shown below. The positions of the pieces are exactly the same as before; only our perspective has changed. Changing the view makes clear factors that might otherwise be overlooked, such as the fact that in the opening layout the opposing bishops and queens threaten one another along 2D diagonals. This is why the Non-Aggression Pact is needed. In the same way, opposing rooks would threaten one another after the pawn in front of either rook was moved aside.
a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | |*r*| | | | |*u*| | | | |*w*| | | | |*r*| IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III | | | |*p*| | | | |*p*| | | | |*p*| | | | |*p*| III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ 4
II | P | | | | | P | | | | | P | | | | | P | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | R | | | | | U | | | | | W | | | | | R | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
A B C D A B C D A B C D A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | |*b*| | | | |*k*| | | | |*m*| | | | |*b*| IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III | | | |*p*| | | | |*p*| | | | |*p*| | | | |*p*| III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ 3
II | P | | | | | P | | | | | P | | | | | P | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | B | | | | | K | | | | | M | | | | | B | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
A B C D A B C D A B C D A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | |*n*| | | | |*d*| | | | |*q*| | | | |*n*| IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III | | | |*p*| | | | |*p*| | | | |*p*| | | | |*p*| III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ 2
II | P | | | | | P | | | | | P | | | | | P | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | N | | | | | D | | | | | Q | | | | | N | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
A B C D A B C D A B C D A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | |*r*| | | | |*u*| | | | |*w*| | | | |*r*| IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III | | | |*p*| | | | |*p*| | | | |*p*| | | | |*p*| III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ 1
II | P | | | | | P | | | | | P | | | | | P | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | R | | | | | U | | | | | W | | | | | R | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
A B C D A B C D A B C D A B C D
Figure 5b. Here's yet another way of looking at the same opening setup. Again, the positions of the pieces haven't changed; all that has changed is how we're looking at them.
1 2 3 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV |*r*|*u*|*w*|*r*| |*n*|*d*|*q*|*n*| |*b*|*k*|*m*|*b*| |*r*|*u*|*w*|*r*| IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III|*p*|*p*|*p*|*p*| |*p*|*p*|*p*|*p*| |*p*|*p*|*p*|*p*| |*p*|*p*|*p*|*p*|III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ D
II | | | | | | | | | | | | | | | | | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | | | | | | | | | | | | | | | | | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | | | | | | | | | | | | | | | | | | IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III| | | | | | | | | | | | | | | | | | | |III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ C
II | | | | | | | | | | | | | | | | | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | | | | | | | | | | | | | | | | | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | | | | | | | | | | | | | | | | | | IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III| | | | | | | | | | | | | | | | | | | |III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ B
II | | | | | | | | | | | | | | | | | | | | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | | | | | | | | | | | | | | | | | | | | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
IV | | | | | | | | | | | | | | | | | | | | IV
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
III| | | | | | | | | | | | | | | | | | | |III
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ A
II | P | P | P | P | | P | P | P | P | | P | P | P | P | | P | P | P | P | II
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
I | R | U | W | R | | N | D | Q | N | | B | K | M | B | | R | U | W | R | I
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
The PiecesThe King. The Chesseract king can move exactly one cell orthogonally in any direction, and cannot move diagonally; see Figure 6. From a center cell, the king can reach eight other cells; when backed into a corner, he can reach only four other cells. Figure 6. The king can move or capture to any orthogonally adjacent cell. The cells available to the white king (K) are marked here 'o', and the cells available to the black king (k) are marked '*'. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|*|_| |_|_|_|_| IV 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|*|_| |_|_|_|_| 3|_|_|_|_| |_|_|*|_| |_|*|k|*| |_|_|*|_| III 2|_|_|_|_| |_|_|_|_| |_|_|*|_| |_|_|_|_| 1|_|o|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|*|_| |_|_|_|_| II 2|_|o|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|o|K|o|_| |_|o|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| I 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|o|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d The Pawn. Notions of "forward" and "backward" don't make much sense on the Chesseract board. Accordingly, pawns exhibit no directional bias in their movement. Because the board is not long, an initial move of more than one cell makes little sense (which also obviates the need for an en passant rule). And because there is no "rear rank," pawns never promote. A pawn can make a non-capturing move of one square orthogonally in any direction, and can capture on any 2D diagonal, as shown in Figure 7. Pawns cannot move or capture on the 3D or 4D diagonals. Figure 7. The pawn's movement. The white pawn (P) on CIIIc2 can make a non-capturing move to any of the cells marked 'o', and can capture an enemy piece on any of the cells marked 'x', all of which are adjacent to CIIIc2 on 2D diagonals. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|x|_| |_|_|_|_| IV 2|_|_|_|_| |_|_|x|_| |_|x|o|x| |_|_|x|_| 1|_|_|_|_| |_|_|_|_| |_|_|x|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|x|_| |_|x|o|x| |_|_|x|_| III 2|_|_|_|_| |_|x|o|x| |_|o|P|o| |_|x|o|x| 1|_|_|_|_| |_|_|x|_| |_|x|o|x| |_|_|x|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|x|_| |_|_|_|_| II 2|_|_|_|_| |_|_|x|_| |_|x|o|x| |_|_|x|_| 1|_|_|_|_| |_|_|_|_| |_|_|x|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| I 2|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d As Figure 7 makes clear, the Chesseract pawn is a powerful piece. Starting in the center of the board, it can capture enemy pieces on any of 24 different cells -- as many as the knight. To weaken the pawn, and also to make the game a bit more chesslike, I've added the Sticky Pawn Rule. The Sticky Pawn Rule: Whenever two pawns belonging to the opposing armies occupy orthogonally adjacent cells, neither pawn can make a non-capturing move. The "stuck" pawns can move only by capturing (2D diagonally). You'll note that this rule leaves the door open to situations in which several opposing pawns are all caught in the same "sticky" chain. Such a chain can even branch, as shown in Figure 8. Figure 8. A sticky pawn chain. None of the pawns shown can make a non-capturing move, because they're all orthogonally adjacent to opposing pawns. However, the black pawn on CIIc2 can capture the white queen on CIIIc1. If it does so, the white pawn on CIIc1 will be free to move, but the white pawn on CIIc3 will still be stuck (to the black pawn on BIIc3) and the white pawn on CIIb2 will still be stuck (to the black pawn on DIIc2).
A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ IV
| | | | | | | | | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ III
| | | | | | | | | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | Q | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | |*p*| | | | | P | | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ II
| | | | | | | | | | | | P |*p*| | | P |*p*| | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | P | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | P | | | | |*p*| | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | P |*p*| | | | | P | | | | | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ I
| | | | | | | | | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
The Knight. Like its standard chess counterpart, the knight moves in an L shape -- exactly two cells in one direction, followed by a single cell at right angles to the first two. Its move is not obstructed by pieces on the cells through which it passes. (You could say it makes a 5-dimensional jump "over" them, but if that makes you nervous, just assume it passes through them.) The knight is slightly less powerful than the bishop, with only 24 possible destinations from the center of the board, but it's more maneuverable than the bishop. Figure 9. The knight's move. The knight on BIIIb2 can reach any of the cells marked 'o'. Note that certain of its moves can be visualized as being from one quadrant to the equivalent cell in a quadrant a knight-move away from the starting quadrant (from BIIIb2 here to the b2 cell in quadrant DII or AI). _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|o|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| IV 2|_|_|_|_| |_|_|_|o| |_|_|_|_| |_|o|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|o| |_|_|_|_| |_|o|_|_| III 2|_|_|_|o| |_|N|_|_| |_|_|_|o| |o|_|o|_| 1|_|_|_|_| |_|_|_|o| |_|_|_|_| |_|o|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|o|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| II 2|_|_|_|_| |_|_|_|o| |_|_|_|_| |_|o|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|o|_|_| |_|_|_|_| |_|_|_|_| I 2|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| 1|_|_|_|_| |_|o|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d The Unicorn. The unicorn's move is borrowed from a 3D chess variant developed by Richard Goode; it's also used in my Five Up variant. It makes a knight's move, as shown above, followed by another (compulsory) single cell at right angles to both of the first two legs of the 'L'. This allows it to reach twice as many cells (48) as the knight. Like the knight, the unicorn can pass through intervening pieces. Like the bishop, however, a given unicorn can reach only half of the cells on the board, being always confined to the light or dark cells. At the start of the game, each player's two unicorns are positioned on cells of opposite colors. Figure 10. All of the cells that can be reached by the unicorn on CIIIb2 are marked 'o'. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|o|_|_| |o|_|o|_| |_|o|_|_| 3|_|o|_|_| |_|_|_|_| |_|_|_|o| |_|_|_|_| IV 2|o|_|o|_| |_|_|_|o| |_|_|_|_| |_|_|_|o| 1|_|o|_|_| |_|_|_|_| |_|_|_|o| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |o|_|o|_| |_|_|_|_| |o|_|o|_| 3|o|_|o|_| |_|_|_|o| |_|_|_|_| |_|_|_|o| III 2|_|_|_|_| |_|_|_|_| |_|U|_|_| |_|_|_|_| 1|o|_|o|_| |_|_|_|o| |_|_|_|_| |_|_|_|o| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|o|_|_| |o|_|o|_| |_|o|_|_| 3|_|o|_|_| |_|_|_|_| |_|_|_|o| |_|_|_|_| II 2|o|_|o|_| |_|_|_|o| |_|_|_|_| |_|_|_|o| 1|_|o|_|_| |_|_|_|_| |_|_|_|o| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|o|_| |o|_|o|_| |_|o|_|_| I 2|_|_|_|_| |_|o|_|o| |_|_|_|_| |o|_|o|_| 1|_|_|_|_| |_|_|o|_| |o|_|o|_| |_|o|_|_| a b c d a b c d a b c d a b c d The Rook. On an empty 4x4x4x4 board, a rook that moved in a strictly orthogonal manner would be able to move to only 12 different cells. This would make it far less powerful than the pawn, which seemed undesirable. So I've beefed up the Chesseract rook. It can move along unobstructed orthogonals, as expected, and cannot jump over pieces that lie in its path (only the knights and unicorns can pass through obstructing pieces) -- but when the rook's path is obstructed, eitehr by the edge of the board or by a friendly piece, it can turn at a right angle in any direction and proceed orthogonally across unobstructed cells as before. A rook can make only one right-angle turn in a given move. It can't change direction when its path is blocked by an enemy piece, because the enemy piece is not actually an obstruction; it could be captured by the rook instead. However, when an enemy piece has been rendered uncapturable by the minstrel (see below), it becomes an obstruction, so the rook can change direction on encountering it. The movement possibilities for a Chesseract rook are shown in Figure 11. Starting from a central cell on an empty board, the rook can reach 60 different cells. Also worth noting: There are many configurations in which a pair of rooks can protect one another while both also threaten an enemy piece in another quadrant. Figure 11. The possible moves of the rook (R) on BIIIb2. To make the diagram clearer, the cells that it can reach during the initial leg of its move are marked 'o', the cells on which it can turn at right angles are marked '+', the cells it can reach after the turn are marked '*', and the cells that it can reach (after turning) by moving along either of two pathways are marked 'x'. The pawn at DIIIb2 allows the rook to reach two cells in the CIII quadrant that it couldn't reach otherwise. (In notating a rook move in the game record, it may be helpful to be clear about what vector the rook follows, if one of its possible paths is blocked by another piece.) _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|_|_|_| |_|x|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|*|_|_| |_|_|_|_| |_|_|_|_| IV 2|_|x|_|_| |x|+|*|x| |_|*|_|_| |_|*|_|_| 1|_|_|_|_| |_|x|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|x|_|_| |x|+|*|x| |_|x|_|_| |_|_|_|_| 3|_|*|_|_| |_|o|_|*| |_|*|_|_| |_|_|_|_| III 2|x|+|*|x| |+|R|o|+| |x|+|*|x| |_|P|_|_| 1|_|x|_|_| |x|+|*|x| |_|x|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|*|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| II 2|_|*|_|_| |*|o|_|*| |_|*|_|_| |_|*|_|_| 1|_|_|_|_| |_|*|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|x|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|*|_|_| |_|_|_|_| |_|_|_|_| I 2|_|x|_|_| |x|+|*|x| |_|x|_|_| |_|*|_|_| 1|_|_|_|_| |_|x|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d The Minstrel. The minstrel moves like a Chesseract rook. It's a weak piece -- a noncombatant, in fact. It can neither capture nor be captured during the course of the game. The minstrel has only one talent: If it's sitting orthogonally adjacent to an enemy piece, that enemy piece can neither capture other pieces, nor itself be captured. The enemy piece is not allowed to make a capturing move even to remove a check on its own king. The minstrel can pacify (and simultaneously protect) up to eight enemy pieces in this manner (though it can pacify no more than seven immediately after its own move, because it has to have arrived on its present cell by moving in an orthogonal manner). Pieces that have been pacified by the minstrel cannot check the minstrel's king, so the king is free to move onto or remain on a cell that would otherwise be threatened by the enemy piece. If the king is in such a position, it is of course illegal to move the minstrel. The Bishop. The bishop moves on 2D diagonals, exactly like its equivalent in standard chess. It cannot make use of the 3D or 4D diagonals. If we consider that Chesseract cells are colored in an alternating light-and-dark pattern, as shown in Figure 12, the bishops are always restricted to cells of a single color. From a central cell in an empty board, the bishop can reach 30 other cells. Each player begins the game with four bishops, two of each cell color. Figure 12. The move of the bishop, shown on a board whose cells have been "colored" to make the 2D diagonals easier to see. The bishop on CIIc2 can move to any of the cells marked 'O'. These are all light-colored cells.
A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| |:::| |:::| |:::| | | |:::| O |:::| |:::| |:::| | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| |:::| |:::| |:::| | | |:::| |:::| 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ IV
| |:::| O |:::| |:::| |:::| | | O |:::| |:::| |:::| |:::| | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| |:::| |:::| |:::| | | |:::| |:::| 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| |:::| |:::| |:::| | | |:::| |:::| 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| |:::| |:::| |:::| | | |:::| O |:::| |:::| |:::| | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ III
|:::| |:::| | | |:::| O |:::| |:::| O |:::| O | | |:::| O |:::| 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| |:::| |:::| |:::| | | |:::| O |:::| |:::| |:::| | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| O |:::| |:::| |:::| | | O |:::| |:::| |:::| |:::| | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| O |:::| |:::| O |:::| O | | |:::| O |:::| 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ II
| O |:::| |:::| |:::| O |:::| O | | |:::| B |:::| |:::| O |:::| O | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| O |:::| |:::| O |:::| O | | |:::| O |:::| 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
|:::| |:::| | | |:::| |:::| |:::| |:::| | | |:::| |:::| 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| |:::| |:::| |:::| | | |:::| O |:::| |:::| |:::| | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ I
|:::| |:::| | | |:::| O |:::| |:::| O |:::| O | | |:::| O |:::| 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| |:::| |:::| |:::| |:::| | | |:::| O |:::| |:::| |:::| | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
The Queen. As you'd expect, the Chesseract queen moves like either a Chesseract rook or like a Chesseract bishop. This makes the queen the most powerful piece on the board except the dragon. From a center cell in an empty board, the queen can reach only six more cells (66 in all) than the rook, but she can reach many of these cells by two different vectors, which makes her attack more difficult to block than the rook's. Figure 13. The queen's move. Cells that she can reach from BIIIc3 moving only like a rook are marked 'x', cells that she can reach moving only like a bishop are marked 'o', and cells that she can reach in either manner are marked '@'. In a single move from a central cell, the queen can reach all of the cells in each of her current layers, but none of the cells that lie in none of her current layers. (From an edge cell, she will be unable to reach one or two cells in her current layers that are a knight's-move away.)
A B C D
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | @ | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | @ | | | x | @ | x | @ | | | | @ | | | | | x | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ IV
| | | | | | | | @ | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | x | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | @ | | | x | @ | x | @ | | | | @ | | | | | x | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| x | @ | x | @ | | x | x | Q | x | | x | o | x | @ | | @ | x | x | x | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ III
| | | @ | | | x | o | x | @ | | | | o | | | | | x | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | x | | | @ | x | x | x | | | | x | | | | | @ | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | @ | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | @ | | | x | o | x | @ | | | | o | | | | | x | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ II
| | | | | | | | o | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | x | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | x | | | | | | | | | | | | 4
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | x | | | @ | x | x | x | | | | x | | | | | @ | | 3
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ I
| | | | | | | | x | | | | | | | | | | | | 2
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | @ | | | | | | | | | | | | 1
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
a b c d a b c d a b c d a b c d
The Wizard. A wizard is rather like a bishop, except that it uses the 3D and 4D diagonals exclusively, and not the 2D diagonals. While more powerful than a bishop, being able to reach 44 other cells from a cell in the center, the wizard is the opposite of a queen in that it can never threaten a piece in its own layer. Figure 14. The wizard moves any number of unobstructed cells along 3D and 4D diagonals, from CIIc2 here to any of the cells marked 'o'. Note that it reaches AIIa4 by passing through BIIb3, and would be obstructed by a piece on this cell. Similarly, it reaches AIVa4 by passing through BIIIb3, and so on. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|o|_|o|_| |_|_|_|_| |o|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| IV 2|o|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|o|o|o| |_|o|_|o| |_|o|o|o| III 2|_|_|_|_| |_|o|_|o| |_|_|_|_| |_|o|_|o| 1|_|_|_|_| |_|o|o|o| |_|o|_|o| |_|o|o|o| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|o|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|o|_|o| |_|_|_|_| |_|o|_|o| II 2|_|_|_|_| |_|_|_|_| |_|_|W|_| |_|_|_|_| 1|_|_|_|_| |_|o|_|o| |_|_|_|_| |_|o|_|o| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| 3|_|_|_|_| |_|o|o|o| |_|o|_|o| |o|o|o|_| I 2|_|_|_|_| |_|o|_|o| |_|_|_|_| |o|_|o|_| 1|_|_|_|_| |_|o|o|o| |_|o|_|o| |o|o|o|_| a b c d a b c d a b c d a b c d The Dragon. The dragon is a monstrously powerful piece -- more powerful than the queen. It combines the moves (appropriately) of the wizard and the unicorn. From the center of the board, a dragon can reach 92 (44 + 48) of the other cells, as illustrated in Figure 15. Like the wizard and the unicorn, it can never threaten or protect pieces in its own layer. The dragon is obstructed by pieces in its path when moving along diagonals, but not when moving as a unicorn. Figure 15. The dragon on BIIb3 can reach 92 other cells. Those that it can reach using a wizard-type 3D or 4D diagonal are marked 'x', and those that it can reach using a unicorn-type 3D knight-move are marked 'o'. _ _A_ _ _ _B_ _ _ _C_ _ _ _D_ _ 4|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| 3|o|_|o|_| |_|_|_|_| |o|_|o|_| |_|_|_|x| IV 2|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| 1|_|_|_|_| |_|_|_|x| |_|_|_|_| |_|x|_|x| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|x|x|x|_| |x|_|x|o| |x|x|x|_| |_|o|_|_| 3|x|_|x|o| |_|_|_|_| |x|_|x|o| |o|_|o|_| III 2|x|x|x|_| |x|_|x|o| |x|x|x|_| |_|o|_|_| 1|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|x|_|x|o| |_|_|_|_| |x|_|x|o| |o|_|o|_| 3|_|_|_|_| |_|D|_|_| |_|_|_|_| |_|_|_|_| II 2|x|_|x|o| |_|_|_|_| |x|_|x|o| |o|_|o|_| 1|o|_|o|_| |_|_|_|_| |o|_|o|_| |_|_|_|x| a b c d a b c d a b c d a b c d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4|x|x|x|_| |x|_|x|o| |x|x|x|_| |_|o|_|_| 3|x|_|x|o| |_|_|_|_| |x|_|x|o| |o|_|o|_| I 2|x|x|x|_| |x|_|x|o| |x|x|x|_| |_|o|_|_| 1|_|o|_|_| |o|_|o|_| |_|o|_|_| |_|_|_|_| a b c d a b c d a b c d a b c d |
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