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# CHESSERACT

Chesseract is a large Chess Variant by Jim Aikin. It is played on a four-dimensional (4D) board that measures 4x4x4x4 cells (see Figure 1). The object of the game is to checkmate the enemy king, and a stalemate is a draw.

While the moves of the pieces are perfectly straightforward, not tricky or convoluted in any way, it seems clear that Chesseract is virtually unplayable, owing to the difficulty of envisioning all of the possible moves of a given piece in 4-space and translating these to a conventional board diagram. This variant is presented purely for the amusement of those who are amused by such things.

The Board.

Consider a cube, perhaps of clear plastic, built out of smaller cubes (let's call them cells) in a 4x4x4 matrix. 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. We can draw a simple ASCII diagram of such an object in the manner shown in Figure 1.

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.

```        A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   | x |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 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 objects 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's necessary to 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 to any set of 16 cells that lie in a two-dimensional plane, and the term "quadrant" refers to a layer that is displayed as a group of contiguous squares in a particular diagram. (See Figures 4 and 5.)

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, we'll 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 looking at as little cubes of clear plastic, aren't cubes at all: They're all little tesseracts. For reference, 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 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. 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 perfectly 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
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 4 | 3 | 4 |   |   | 3 | 2 | 3 |   |   | 4 | 3 | 4 |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   |   |   |   |   | 3 | 2 | 3 |   |   | 2 | o | 2 |   |   | 3 | 2 | 3 |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 4 | 3 | 4 |   |   | 3 | 2 | 3 |   |   | 4 | 3 | 4 |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 3 | 2 | 3 |   |   | 2 | o | 2 |   |   | 3 | 2 | 3 |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|   |   |   |   |   | 2 | o | 2 |   |   | o | X | o |   |   | 2 | o | 2 |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 3 | 2 | 3 |   |   | 2 | o | 2 |   |   | 3 | 2 | 3 |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 4 | 3 | 4 |   |   | 3 | 2 | 3 |   |   | 4 | 3 | 4 |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   |   |   |   |   | 3 | 2 | 3 |   |   | 2 | o | 2 |   |   | 3 | 2 | 3 |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   | 4 | 3 | 4 |   |   | 3 | 2 | 3 |   |   | 4 | 3 | 4 |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 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 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).

```
A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| > | - | - | > |   | - | * | * | - |   | - | * | * | - |   | > | - | - | > | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| - | * | * | - |   | * | o | o | * |   | * | o | o | * |   | - | * | * | - | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
| - | * | * | - |   | * | o | o | * |   | * | o | o | * |   | - | * | * | - | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| > | - | - | > |   | - | * | * | - |   | - | * | * | - |   | > | - | - | > | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| - | * | * | - |   | * | 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
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| - | * | * | - |   | * | 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
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| > | - | - | > |   | - | * | * | - |   | - | * | * | - |   | > | - | - | > | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| - | * | * | - |   | * | o | o | * |   | * | o | o | * |   | - | * | * | - | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
| - | * | * | - |   | * | o | o | * |   | * | o | o | * |   | - | * | * | - | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| > | - | - | > |   | - | * | * | - |   | - | * | * | - |   | > | - | - | > | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

```

The Pieces.

Chesseract is played by two players, each of whom has an army consisting of a king (K), a queen (Q), a minstrel (M), two rooks (R), two unicorns (U), two wizards (W), four bishops (B), four knights (N), and 21 pawns (P). The opening layout is shown in Figure 4; an alternate way of viewing the same layout is shown in Figure 5. As in FIDE chess, all pieces capture as they move, by moving onto the cell occupied by an enemy piece, except pawns, which make non-capturing moves orthogonally and capture diagonally (see below).

FIGURE 4: The opening layout in Chesseract. The black pieces are indicated by lower-case letters braced by asterisks.

```
A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |*p*|   |   |   |*p*|*b*|   |   |*p*|*r*|*b*|   |*p*|*m*|*q*|*k*| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*n*|   |*p*|*w*|*n*|*u*| 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |*p*|*w*|*u*| 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|*p*| 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*n*|   |   |*p*|*n*|*b*| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*r*| 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
| P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*| 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| B | P |   |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*b*| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*| 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
| R | P |   |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| B | N | P |   |   | N | P |   |   |   | P |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P | P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| U | W | P |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
| U | N | W | P |   | N | P |   |   |   | P |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| K | Q | M | P |   | B | R | P |   |   | B | P |   |   |   | P |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
a   b   c   d       a   b   c   d       a   b   c   d       a   b   c   d

```

Figure 5 shows an alternate method 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 3 is now shown 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. For instance, in Figure 4 its not immediately obvious that the opposing wizards face one another on 2D diagonals, but in Figure 5 the relationship is clearly seen on the b3 and c2 quadrants.

FIGURE 5: The layout in Figure 4 can also be viewed in this manner.

```
a                   b                   c                   d

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |*p*|   |   |   |*p*|*m*|   |   |*p*|*r*|*q*|   |*p*|*b*|*b*|*k*| IV
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*n*|   |   |*p*|*n*|*b*| III
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+     4
|   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*b*| II
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P |   |   |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |*p*| I
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |*p*|   |   |   |   |*w*|   |   |   |*p*|*n*|   |   |*p*|*n*|*u*| IV
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |   |*p*|*r*| III
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+     3
| P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*| II
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| U | P |   |   |   | W |   |   |   |   | P |   |   |   |   |   |   |   |   | I
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |*p*|   |   |   |   |*w*|   |   |   |*p*|*u*| IV
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |*p*| III
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+     2
| R | P |   |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |   | II
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| U | N | P |   |   | N | P |   |   |   | W |   |   |   |   | P |   |   |   | I
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| P |   |   |   |   |   |   |   |   |   |   |   |   |*p*|   |   |   |   |*p*| IV
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| B | P |   |   |   | P |   |   |   |   |   |   |   |   |   |   |   |   |   | III
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+     1
| B | N | P |   |   | N | P |   |   |   | P |   |   |   |   |   |   |   |   | II
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| K | B | B | P |   | Q | R | P |   |   | M | P |   |   |   | P |   |   |   | I
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
A   B   C   D       A   B   C   D       A   B   C   D       A   B   C   D

```

THE 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

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   | = |   |   |   | = |*k*| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | = | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   |   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | = | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|   |   | o |   |   |   | o | K | o |   |   |   | o |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   |   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
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 cannot 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 BIIIb2 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 BIIIb2 on 2D diagonals.

```
A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   | X |   |   |   | X | o | X |   |   |   | X |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | X |   |   |   | X | o | X |   |   |   | X |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
| X | o | X |   |   | o | P | o |   |   | X | o | X |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | X |   |   |   | X | o | X |   |   |   | X |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   | X |   |   |   | X | o | X |   |   |   | X |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 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 on 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 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 less powerful than the pawn, which seemed undesirable. So the Chesseract rook 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 hits an edge of the board, it can turn at a right angle in any direction and proceed orthogonally across unobstructed cells as before. Only one right-angle turn is permitted per move. The available moves for a Chesseract rook are shown in Figure 9. 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 enemy pieces in other quadrants.

FIGURE 9: The move of the rook (R). For purposes of visual clarity, the cells that it can reach in the initial leg of its move are marked 'o', the cells it can reach by turning at a right angle after it reaches an edge are marked '^', and the cells that it can reach (after turning) by moving along either of two pathways are marked 'X'. In notating a move, it may be necessary to be clear about what vector the rook follows, if one of its possible paths is blocked by another piece.

```
A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   | X |   |   |   | ^ | o | ^ | X |   |   | ^ |   |   |   |   | X |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | ^ |   |   |   | X | o | ^ | X |   |   | ^ |   |   |   |   | ^ |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | ^ |   |   |   | ^ | o |   | ^ |   |   |   |   |   |   |   | ^ |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
| ^ | o | ^ | ^ |   | o | R | o | o |   | ^ | o |   | ^ |   | ^ | o | ^ | ^ | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | ^ |   |   |   | X | o | ^ | X |   |   | ^ |   |   |   |   | ^ |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   | ^ |   |   |   | ^ | o |   | ^ |   |   |   |   |   |   |   | ^ |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | X |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
|   | X |   |   |   | ^ | o | ^ | X |   |   | ^ |   |   |   |   | X |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | ^ |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
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, which 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, the enemy piece can't make a capturing move, not even to remove a check on its king. The minstrel can pacify 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).

THE BISHOP. The bishop moves on 2D diagonals, exactly like its FIDE chess equivalent. 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 10, 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 10: 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 WIZARD. A wizard is rather like a bishop, except that it uses the 3D and 4D diagonals exclusively, and not the 2D diagonals. The wizard is the only piece in Chesseract that actually makes a four-dimensional move. 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 11: 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

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| o |   | o |   |   |   |   |   |   |   | o |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
| o |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o | o | o |   |   | o |   | o |   |   | o | o | o | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|   |   |   |   |   |   | o |   | o |   |   |   |   |   |   |   | o |   | o | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o | o | o |   |   | o |   | o |   |   | o | o | o | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| o |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   | o |   |   |   |   |   |   |   | o |   | o | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   |   |   |   |   |   |   |   |   |   |   |   | W |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   | o |   |   |   |   |   |   |   | o |   | o | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o | o | o |   |   | o |   | o |   |   | o | o | o | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
|   |   |   |   |   |   | o |   | o |   |   |   |   |   |   |   | o |   | o | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o | o | o |   |   | o |   | o |   |   | o | o | o | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
a   b   c   d       a   b   c   d       a   b   c   d       a   b   c   d

```

THE QUEEN. As you'd expect, the Chesseract queen moves like either a (Chesseract) rook or like a bishop. This makes it, again as you'd expect, the most powerful piece on the board. From a center cell in an empty board, it can reach only six more cells (66 in all) than the rook, but it can reach many cells by two different vectors, which makes its attack more difficult to block than the rook's.

FIGURE 12. The queen's move. Cells that it can reach from BIIIc3 moving only like a rook are marked 'x', cells that it can reach moving only like a bishop are marked 'o', and cells that it can reach in either manner are marked '@'. In a single move, the queen can reach all of the cells in each of its current layers, but none of the cells that lie in none of its current layers.

```
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 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 let's not think about that.) 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 13: 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

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
|   |   |   |   |   |   |   |   | o |   |   |   |   |   |   |   | o |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   | o |   |   |   | o |   | o |   |   |   | o |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   | o |   |   |   |   |   |   |   | o |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|   |   |   | o |   |   | N |   |   |   |   |   |   | o |   | o |   | o |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   | o |   |   |   |   |   |   |   | o |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
|   |   |   |   |   |   |   |   | o |   |   |   |   |   |   |   | o |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   |   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
|   | o |   |   |   | o |   | o |   |   |   | o |   |   |   |   |   |   |   | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |   |   |   |   |   | o |   |   |   |   |   |   |   |   |   |   |   |   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
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 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. Each player begins the game with two unicorns, which are positioned on cells of opposite colors.

FIGURE 14: The unicorn's move. The board has been "colored" with the symbol ':::' to indicate the dark cells. The unicorn can only reach cells of the same color as the cell its move starts on.

```
A                   B                   C                   D

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |:::|   |:::|   |:::| o |:::|   |   | o |:::| o |:::|   |:::| o |:::|   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::| o |:::|   |   |   |:::|   |:::|   |:::|   |:::| o |   |   |:::|   |:::| 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    IV
| o |:::| o |:::|   |:::|   |:::| o |   |   |:::|   |:::|   |:::|   |:::| o | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::| o |:::|   |   |   |:::|   |:::|   |:::|   |:::| o |   |   |:::|   |:::| 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::|   |:::|   |   | o |:::| o |:::|   |:::|   |:::|   |   | o |:::| o |:::| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| o |:::| o |:::|   |:::|   |:::| o |   |   |:::|   |:::|   |:::|   |:::| o | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    III
|:::|   |:::|   |   |   |:::|   |:::|   |:::| U |:::|   |   |   |:::|   |:::| 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
| o |:::| o |:::|   |:::|   |:::| o |   |   |:::|   |:::|   |:::|   |:::| o | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |:::|   |:::|   |:::| o |:::|   |   | o |:::| o |:::|   |:::| o |:::|   | 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::| o |:::|   |   |   |:::|   |:::|   |:::|   |:::| o |   |   |:::|   |:::| 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    II
| o |:::| o |:::|   |:::|   |:::| o |   |   |:::|   |:::|   |:::|   |:::| o | 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::| o |:::|   |   |   |:::|   |:::|   |:::|   |:::| o |   |   |:::|   |:::| 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|:::|   |:::|   |   |   |:::|   |:::|   |:::|   |:::|   |   |   |:::|   |:::| 4
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |:::|   |:::|   |:::| o |:::|   |   | o |:::| o |:::|   |:::| o |:::|   | 3
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+    I
|:::|   |:::|   |   | o |:::| o |:::|   |:::|   |:::|   |   | o |:::| o |:::| 2
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
|   |:::|   |:::|   |:::| o |:::|   |   | o |:::| o |:::|   |:::| o |:::|   | 1
+---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+

```

VARIANT. To 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 can be played with each player in each turn making two legal moves rather than one. The following restrictions apply:

(1) The same piece may not be moved twice in the same turn, unless only one piece can make a legal move, in which case it must be moved twice, if possible. In this situation the second move by the piece may bring it back to its starting cell.

(2) If the first move by the only movable piece opens up the possibility of another piece moving, that other piece must be moved to complete the turn: A double move by one piece is not allowed if a move by another piece is made possible by the first move. However, the player is not required to make the first move in such a way as to open up the possibility of moving a different piece, if it is possible to do so: The player may choose to move the only movable piece in such a way that it will still be the only movable piece.

(3) If a player has only one piece that can make a legal move, and if, after it moves, the player has no legal moves remaining, but is not in check, the game is a draw.

(4) If a player has only one piece that can make a legal move, and if its first move checks the enemy king, and if the first move does not open up the possibility of another of the same player's pieces moving, the second move by the same piece may be used to actually capture the enemy king, ending the game with a win for the player who had only one movable piece.

(5) If a player's king is in check, either the first or the second move (or a combination of the two moves) may be used to remove the check.

(6) The first of the two moves may place the player's own king in check, as long as the second move removes the check. However, a first move that produces check cannot be made by moving the king itself.

## Play It!

Use Zillions of Games to play this game! If you have Zillions of Games installed, you can download this game and play it.

Written by Jim Aikin. HTML conversion by David Howe.
This variant is an entry in the 1999 Large Variant contest.

WWW page created: January 8, 1999. Last modified: May 8, 1999. ﻿