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Five Up

For playing chess on a three-dimensional board, it seems to me, the 5x5x5 board is just about the ideal size. A 4x4x4 board is so cramped that the opposing pawns have no room to maneuver before running into one another. The 6x6x6 board, in contrast, contains 216 cells, which means it would be mostly empty space unless each side were given a good-sized army -- on the order of 36 pieces each, at minimum. And let's face it: Playing 3D chess is hard enough without having to keep track of that many pieces.

Raumschach, which was invented before World War I, is a classic 3D chess variant that's played on a 5x5x5 board. The additions to the FIDE set of pieces are minimal, making Raumschach easy to learn. However, the game suffers from a couple of nontrivial defects. The unicorn (which moves only on 3D diagonals) is too weak to be a really useful piece. The Raumschach king, meanwhile (which can move one cell in any direction including 3D diagonals), is far too mobile, making checkmate difficult. Richard Goode's variant of Raumschach has more interesting unicorns; they move like knights with an added "leg" that takes the move into the third dimension. But Goode's king, being restricted to single-cell moves on the 3D diagonals, is weak and clumsy.

The changes below, which transform the game into a version I call Five Up, are intended to make Raumschach as playable as possible. I've kept the discussion fairly short, on the assumption that most readers will be able to grasp the basic dynamics of piece movement on a 3D board. Throughout, I have used the word "cell" in preference to "square," since the pieces' locations are in effect small cubes, not flat squares. The horizontal 5x5 planes can be referred to as "layers," the vertical planes that run longitudinally between the two players as "slices," and the transverse vertical planes whose edges are to the players' left and right as "faces."

Roger Norling, who I believe lives in Denmark, went to the trouble to write a Five Up implementation for the Zillions of Games engine. The Zillions engine, while not free, allows third parties to create their own games. Roger's implementation is available here. If you can't find anyone to play three-dimensional chess with, which wouldn't be too surprising, your computer will be happy to oblige.

Figure 0. A blank diagram of the Five Up board. Visualize these five grids as being stacked one atop the other. The A layer (left) can be considered the top, and the E layer (right) the bottom. Each cell is identified by a capital letter, a lower-case letter, and a number. Cell Aa1, for instance, appears at the lower left corner of the diagram. However, Aa1 is actually the top left corner on the side of the cube nearest the player who uses the white pieces.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Each player begins the game of Five Up with the usual complement of chess pieces, plus a wizard, a guard, and two added pawns. The initial setup is shown in Figure 1. The basic rules of FIDE chess apply except where noted, and a stalemate is a draw.

Figure 1. The opening setup in Five Up. In the ASCII diagram, white pieces are shown with capital letters and black pieces with lower-case letters. The black pawns are shown as 'o' rather than 'p' simply because it's hard to see the difference between 'p' and 'P' at a glance. Note that a slight asymmetry in the initial setup is required to put the bishops on cells of opposite colors. The black and white positions are mirror images of each other.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|b| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|o|o|o| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|   |P|P|P|P|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Movement and Capturing

All pieces in Five Up move as they capture, except the pawn and the king. With the exception of the knights, movement is blocked by the presence of a friendly or enemy piece on a cell through which the moving piece would have to move. A knight can pass freely through a solid wall of pieces.

Rook

The rook, whose move is perhaps the easiest to explain, moves along unobstructed orthogonals. (The term "orthogonal" refers to a line of squares or cells that is in the forward/backward direction, the side-to-side direction, or [on a 3D board] the vertical direction.) The movement of the rook is shown in Figure 2. Unlike the rook in standard chess, the Five Up rook is somewhat weaker than the knight and bishop. Like the rook in standard chess, however, the Five Up rook is equally strong whether it's positioned in the center of the board, at an edge, or in a corner. The other pieces in Five Up, like those in standard chess, have fewer options for moves when placed on an edge, and still fewer when positioned in a corner.

Figure 2. The rook on cell Bb3 can move to any of the cells marked 'x'. By moving vertically, it can reach Ab3 (which is directly above Bb3) or move downward through the C and D layers to Eb3, which is on the bottom layer of the board.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |_|x|_|_|_| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|_|   |_|x|_|_|_| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|x|_|_|_|   |x|R|x|x|x| 3 |_|x|_|_|_|   |_|x|_|_|_| 3 |_|x|_|_|_|
2 |_|_|_|_|_|   |_|x|_|_|_| 2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|
1 |_|_|_|_|_|   |_|x|_|_|_| 1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Bishop

The bishop moves along unobstructed diagonals in any plane, but not on 3D diagonals. The difference between 2D and 3D diagonals is illustrated in Figure 3a.

Figure 3a. On a three-dimensional board, a 2D diagonal stays within a single plane (either a layer, a face, or a slice). If we picture the cells as being colored alternately white and black, as on a conventional chessboard, 2D diagonals always stay on cells of the same color. One way to visualize 3D diagonals is by seeing the individual cells as little cubes. When moving on a 3D diagonal, a piece passes between two cells that are joined only at the corners, not along the edges. Another way to look at 3D diagonals is to visualize moving the piece on a 2D diagonal on one of the layers and then moving up or down to an adjacent layer. The cells marked 'x', 'o', and '.' lie on various 2D diagonals. The cells marked '*' and '+' lie on 3D diagonals.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _  
5 |o|_|_|_|.|   |_|o|_|_|+| 5 |_|_|o|_|_|   |_|_|_|o|_| 5 |_|_|_|_|o| 
4 |x|_|_|.|_|   |*|_|_|_|_| 4 |_|_|_|+|_|   |_|_|_|_|_| 4 |_|_|_|_|_| 
3 |_|_|.|_|_|   |x|_|_|_|_| 3 |_|*|_|_|_|   |_|_|+|_|_| 3 |_|_|_|_|_| 
2 |_|.|_|_|_|   |_|_|_|_|_| 2 |x|_|_|_|_|   |_|_|*|_|_| 2 |_|+|_|_|_| 
1 |.|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |x|_|_|_|_| 1 |_|_|_|*|_| 
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e  

Figure 3b. As on a conventional chessboard, the cells in a Five Up board can be considered to be of alternate colors. The coloring has no significance except as a visual aid, but as in standard chess, bishops will always be confined to cells of a single color. The black cells are marked 'x' here.
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |x|_|x|_|x|   |_|x|_|x|_| 5 |x|_|x|_|x|   |_|x|_|x|_| 5 |x|_|x|_|x|
4 |_|x|_|x|_|   |x|_|x|_|x| 4 |_|x|_|x|_|   |x|_|x|_|x| 4 |_|x|_|x|_|
3 |x|_|x|_|x|   |_|x|_|x|_| 3 |x|_|x|_|x|   |_|x|_|x|_| 3 |x|_|x|_|x|
2 |_|x|_|x|_|   |x|_|x|_|x| 2 |_|x|_|x|_|   |x|_|x|_|x| 2 |_|x|_|x|_|
1 |x|_|x|_|x|   |_|x|_|x|_| 1 |x|_|x|_|x|   |_|x|_|x|_| 1 |x|_|x|_|x|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Queen

The queen combines the moves of rook and bishop. (This makes the queen weaker than the original Raumschach queen, which could also use the 3D diagonals.) The movement of the queen is illustrated in Figure 4.

Figure 4. The queen on Cc3 can reach any of the cells marked 'x'.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|x|_|_|   |_|_|_|_|_| 5 |x|_|x|_|x|   |_|_|_|_|_| 5 |_|_|x|_|_|
4 |_|_|_|_|_|   |_|_|x|_|_| 4 |_|x|x|x|_|   |_|_|x|_|_| 4 |_|_|_|_|_|
3 |x|_|x|_|x|   |_|x|x|x|_| 3 |x|x|Q|x|x|   |_|x|x|x|_| 3 |x|_|x|_|x|
2 |_|_|_|_|_|   |_|_|x|_|_| 2 |_|x|x|x|_|   |_|_|x|_|_| 2 |_|_|_|_|_|
1 |_|_|x|_|_|   |_|_|_|_|_| 1 |x|_|x|_|x|   |_|_|_|_|_| 1 |_|_|x|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Knight

The knight makes a normal knight's move within any plane, as shown in Figure 5, and can pass through other pieces. Another way to visualize the movement of the knight is that its movement begins with a two-cell orthogonal vector in any of the six directions (left, right, forward, backward, up, or down) and then bends at a right angle in any of four directions.

Figure 5. The knight's move. The knight always stays within one of the three possible two-dimensional planes when moving. In this diagram, moves within the C layer are marked '+', moves within the 3 face are marked 'x', and moves within the c slice are marked '*'.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _      _ _ _ _ _ 
5 |_|_|_|_|_|   |_|_|*|_|_| 5 |_|+|_|+|_|   |_|_|*|_|_|  5 |_|_|_|_|_|
4 |_|_|*|_|_|   |_|_|_|_|_| 4 |+|_|_|_|+|   |_|_|_|_|_|  4 |_|_|*|_|_|
3 |_|x|_|x|_|   |x|_|_|_|x| 3 |_|_|N|_|_|   |x|_|_|_|x|  3 |_|x|_|x|_|
2 |_|_|*|_|_|   |_|_|_|_|_| 2 |+|_|_|_|+|   |_|_|_|_|_|  2 |_|_|*|_|_|
1 |_|_|_|_|_|   |_|_|*|_|_| 1 |_|+|_|+|_|   |_|_|*|_|_|  1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e   E  a b c d e 

Pawn

The pawn moves one cell either vertically or horizontally, always proceeding toward the enemy's "home row" (the row on which the enemy king begins the game). The pawn captures by moving toward the enemy's home row diagonally. The diagonal move can be either in a horizontal or a vertical plane; this gives the pawn, except when it's at the edge of the board, two possible non-capturing moves and five possible cells where it can capture, as shown in Figure 6. Pawns are promoted to any piece (except a king) on reaching the enemy's home row.

The pawns that begin the game on the A or E layer can make an initial double move; those that begin on the D and B layers can't. (The reason for the double standard is to prevent any pawns from reaching the C3 row on their first move.) The initial two-cell move can be horizontal, vertical, or in an L-shaped trajectory. For the A pawns, this L-shaped double move would take them from the A4 row to the B3 row. The L-shaped vector can be conceived of as passing through either of the intermediate cells (for instance, from Aa4 to Ba3 by way of either Aa3 or Ba4). The L-shaped move is blocked only if both of the intermediate cells are occupied. Capturing en passant can take place immediately after an initial double move. If it's an L-shaped move, and if both intermediate cells are vacant, leaving it ambiguous which of them the pawn passed through, the pawn can be captured en passant on either of the intermediate cells.

Figure 6. The white pawn shown here, which is advancing toward the A5 row, can make capturing moves to the cells marked 'x' and non-capturing moves to the cells marked '+'.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _  
5 |_|_|_|_|_| 5 |_|_|_|_|_| 5 |_|_|_|_|_| 5 |_|_|_|_|_| 5 |_|_|_|_|_| 
4 |_|_|_|_|_| 4 |_|x|_|_|_| 4 |x|+|x|_|_| 4 |_|_|_|_|_| 4 |_|_|_|_|_| 
3 |_|_|_|_|_| 3 |x|+|x|_|_| 3 |_|P|_|_|_| 3 |_|_|_|_|_| 3 |_|_|_|_|_| 
2 |_|_|_|_|_| 2 |_|_|_|_|_| 2 |_|_|_|_|_| 2 |_|_|_|_|_| 2 |_|_|_|_|_| 
1 |_|_|_|_|_| 1 |_|_|_|_|_| 1 |_|_|_|_|_| 1 |_|_|_|_|_| 1 |_|_|_|_|_| 
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e  

King

The king can move or capture to an adjacent cell in any of the six orthogonal directions. In addition, it can capture (but not make a non-capturing move) by moving exactly one cell along a 3D diagonal, as shown in Figure 7. The king can't make 2D diagonal moves. Note that the king can approach and threaten any enemy piece except a guard or the enemy king.

Figure 7. The black king, on Bd4, can move or capture on any of the cells marked '+'. It can also capture, but not make non-capturing moves, to any of the cells marked 'x'.
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|x|_|x|   |_|_|_|+|_| 5 |_|_|x|_|x|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|+|_|   |_|_|+|k|+| 4 |_|_|_|+|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|x|_|x|   |_|_|_|+|_| 3 |_|_|x|_|x|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Also worth noting: In an endgame position against a lone king, the player who has the advantage can force the opposing king back to an edge using only his or her own king. This is done by approaching the other king on a 2D diagonal, as shown below. Black is preparing a checkmate, but there's no need to bring the queen into play just yet. The white king (on Cc3) is blocked from moving up or forward, because those two cells (marked '+') are threatened by the black king using its orthogonal move. The lateral moves (marked '*') are on the black king's 3D diagonals, so they're not safe either. The white king can only move backward to Cc2 or down to Dc3 (marked '.'), after which black will repeat the position by moving to either Bc3 or Cc4.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _  
5 |q|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_| 
4 |_|_|_|_|_|   |_|_|k|_|_| 4 |_|_|+|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_| 
3 |_|_|_|_|_|   |_|_|+|_|_| 3 |_|*|K|*|_|   |_|_|.|_|_| 3 |_|_|_|_|_| 
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|.|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_| 
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_| 
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e  

Castling. In a castling move, the king and rook simply exchange positions. They must be on the same orthogonal (row, file, or column) and the cells between them must be empty. The king cannot castle through check, into check, or to get out of check. Each player can castle only once during the course of a game. The player can castle at any point in the game, even after both king and rook have moved.

Guard

The guard moves one cell in any direction, including orthogonals and both 2D and 3D diagonals, as shown in Figure 8. From an interior cell, it can reach 26 other cells, which makes it a powerful piece in spite of its limited mobility.

Figure 8. The guard on Dc2 can move or capture to any of the cells marked 'x'.
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|x|x|x|_|   |_|x|x|x|_| 3 |_|x|x|x|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|x|x|x|_|   |_|x|G|x|_| 2 |_|x|x|x|_|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|x|x|x|_|   |_|x|x|x|_| 1 |_|x|x|x|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Wizard

The wizard moves any number of unobstructed cells along 2D or 3D diagonals, as shown in Figure 9. Unlike the bishops, the wizard can reach any cell on the board, because the 3D diagonal movement allows it to switch from black cells to white ones.

Referring back to the opening setup shown in Figure 1, you'll note that when the pawns on Bb4 and Db2 have been moved, the two wizards face one another on a diagonal. If you dislike trading powerful pieces in the opening, this is a situation to be wary of.

Figure 9. The wizard on Bb2 can reach any of the cells marked 'x' by moving along 2D diagonals, and any of the cells marked '*' by moving along 3D diagonals.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |_|_|_|_|x| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|x|_|_|*|
4 |_|_|_|_|_|   |_|_|_|x|_| 4 |_|_|_|_|_|   |_|x|_|*|_| 4 |_|_|_|_|_|
3 |*|x|*|_|_|   |x|_|x|_|_| 3 |*|x|*|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |x|_|x|_|_|   |_|w|_|_|_| 2 |x|_|x|_|_|   |_|_|_|x|_| 2 |_|_|_|_|x|
1 |*|x|*|_|_|   |x|_|x|_|_| 1 |*|x|*|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Right-Sized Armies

In an earlier version of Five Up, I gave each player another three pieces, which were "in hand" at the beginning of the game and were placed in play subsequently. I resorted to this awkward, unchesslike expedient out of a fear that the piece density on the Five Up board was too low. In classic chess, the game begins with half of the squares occupied by pieces, but in Five Up, with 40 pieces in all on 125 cells, less than 1/3 of the cells are initially occupied.

On further reflection, however, I've realized that piece density is too crude a measure of the strategic complexity of the game layout. A better measure might be called "potential coverage density," which is to say, the number of squares on the board that can potentially be moved to by the complement of pieces.

Without getting too rigorous about it, we can easily arrive at a rough figure for the potential coverage density offered by a given piece complement on a given board. Placing each piece in turn on the center of a vacant board, we count the number of squares it can move to or capture on. After adding up the values for all of the pieces that are on the board at the beginning of the game, we divide by the total number of squares on the board to determine the coverage density.

In standard chess, the king can cover 8 squares, the queen 27, each rook 14, each bishop 13, each knight 8, and each pawn 3. The total coverage factor for the two armies is therefore 258 squares, giving a potential coverage density of a little more than 4.0 on a 64-square board.

In Five Up, the king can cover 14 cells, the queen 36, the guard 26, the wizard 40, each rook 12, each bishop 24, each knight 24, and each pawn 7. Thus the total coverage factor for the two armies is 612, giving a potential coverage density of almost 4.9 on a 125-cell board. Because of this, it's clear that the preponderance of empty cells on the board doesn't make Five Up too simple. On the contrary: Other things being equal, it will be almost 20% harder in Five up than in standard chess to find a safe square to which to move a given piece, and almost 20% easier to protect a threatened piece.

Piece Values

Let's make some educated guesses about the likely value of pieces, in order to get a handle on the desirability of exchanges. Assuming a pawn is worth 1, the rook is worth no more than 3. (As noted earlier, the mobility of the rook is not reduced when it's located on an edge or in the corner. This increases its value slightly compared to the other pieces.) The bishop and knight, being more powerful than the rook, are worth perhaps 5 each. The value of a hypothetical piece that could move only on 3D diagonals would be about 2, which makes the the wizard worth 7. The guard is probably worth 6, and the queen about 9.

Opening Strategies

The possibilities for openings on the Five Up board (as on any three-dimensional board) are inevitably more complex than those in conventional chess. To begin with, since the A and E layer pawns are permitted L-shaped opening moves, two different pawns can reach each of the squares in the 3 inner rows (B3 for black, D3 for white). Also, the opposing armies are simply closer together. The inner row of pawns on each side can capture, from their starting positions, any enemy piece foolish enough to venture onto the middle row of the board (C3); this situation has no counterpart on a two-dimensional board.

Even more significant is the fact that all of the non-pawn pieces in the B and D layers can move before any pawns are moved, a possibility offered to knights alone in conventional chess. In the diagram below, the squares to which the black bishops can be developed without moving a pawn are marked '+' (for the corner bishop) and '*' (for the inside bishop):

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r| 5 |n|b|q|n|b| 5 |_|_|*|+|_| 5 |_|_|+|*|_| 5 |_|+|_|_|*|
4 |o|o|o|o|o| 4 |o|o|o|o|o| 4 |_|*|_|_|+| 4 |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_| 3 |_|_|_|_|_| 3 |_|*|_|_|+| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |_|_|_|_|_| 2 |_|_|_|_|_| 2 |P|P|P|P|P| 2 |P|*|P|P|+|
1 |_|_|_|_|_| 1 |_|_|_|_|_| 1 |_|_|_|_|_| 1 |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

You'll note that the bishops threaten, from their opening squares, opposing pawns in the king layer. Fortunately, the pawns are well protected. The queen, in the same way, threatens the pawn directly in front of the opposing king. B-Bc1 may not be an especially good opening move for white, because black can respond N-Bb3, threatening the bishop. Likewise, black can respond to B-Bd1 with ... N-Bc3. On the other hand, there's a long and distinguished tradition of openings in which bishops and knights are traded.

Next, consider that the inside knight can each move, from its starting position, to seven different unoccupied cells rather than to two, as in conventional chess, while the corner knight can reach five cells. The white Dd1 (inside) knight can reach any of the cells marked '.' in the diagram below, while the Da1 (corner) knight can reach any of the cells marked '+'.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|g|k|w|r|   |n|b|q|n|b| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|o|o|o| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |+|_|_|.|_|   |_|+|.|_|.| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |+|_|_|.|_| 2 |_|_|_|_|_|   |P|P|P|P|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|+|.|_|.| 1 |_|.|+|_|_|   |N|B|Q|N|B| 1 |R|G|K|W|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

If we add it all up, we find that each side starts the game with 25 possible pawn moves, 12 knight moves, 12 bishop moves, and 10 queen moves, or 59 moves in all, compared to the 18 first moves available on a conventional chessboard. To be sure, only 5 or 6 of those 18 are ever played, but if the same proportion holds for Five Up, white may have more than a dozen usable opening moves, and black an equal number of responses. It would take a lot more study than I've done so far to figure out which of these moves are the most playable.

A principle probably worth bearing in mind is that in the opening position several rows of easily accessible cells are already behind the enemy's pawn position, and thus can't be protected by pawns. Most of these cells are guarded in the opening position, but three of them aren't -- and the movement of pieces toward the center is likely to leave more of them unguarded. In the diagram below, the unguarded cells behind black's pawn shield are marked '*'. It's not hard for black to guard these cells if need be -- for example, moving the Ba5 knight allows the rook beneath it to attack Ea5. Even so, the ease with which each player can slip pieces behind the other's pawn defenses is a feature of the three-dimensional board.

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|b| 5 |_|_|_|_|*|   |_|_|_|_|_| 5 |*|_|_|*|_|
4 |o|o|o|o|o|   |o|o|o|o|o| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|   |P|P|P|P|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Next, let's look briefly at some pawn formations that can arise in the center. The simplest and most natural pawn opening may prove to be this one:


1. P-Cc2     P-Cc4
2. P-Cc3
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|b| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|_|o|o| 4 |_|_|o|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|P|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|_|_|   |P|P|_|P|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

White has staked a claim to the very center of the board. The importance of commanding the center may turn out to be greater on a 3D board, because everything is closer to the center. So what is black to do about white's positional superiority? If black wants to attack immediately, there are three ways to do it with pawns -- by moving a pawn to Bc4, Cb4, or Cd4. B-Ce5 and N-Cd5 also attack it. The black queen can attack it by moving to Ec5, but this would probably be premature.

Cb4 and Cd4 are nearly equivalent pawn moves, but not quite. Cd4 opens a diagonal for the corner bishop. So let's start there. White can respond by advancing the Cc3 pawn to Bc3, but this move most likely gains nothing, so white will either support the pawn or capture the black pawn attacking it. The same three moves available to black for attack are available to white for support.


2. ...     P-Cd4
3. P-Cd2
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|b| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|_|_|o| 4 |_|_|o|o|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|P|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|_|P|_|   |P|P|_|_|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

At this point, the possibilities begin to multiply. Black could, for instance, advance the Ab4 pawn to Bb3, attacking the white pawn again from above, or move the Bb4 pawn to Cb4, attacking it from the other side. Moving the king pawn out to Bc3 blocks the white pawn, making it an easier target. P-Dd4 moves past it, using the Cc4 pawn to support an incursion onto the D layer; black can later move the A layer pawn up to Cd4, renewing the attack. Just as likely, black will treat the pawn position as stable for the moment, and seize the initiative by developing a piece. The Bb5 bishop can't move out along the Bc4-Ba2 diagonal, because all of the cells on that diagonal are subject to capture by white pawns, but B-Ca5, B-Bd2, N-Cb5, and N-Cc5 all look playable.

White may find B-Dc2 useful. In this position the bishop protects both center pawns -- and P-Bc3 will be a discovered check.

White has needed two moves to get a pawn to Cc3, which gives black the chance to gain a tempo by developing the first non-pawn piece. What if white decides to forget about Cc3 for the time being in order to keep the tempo? B-Ca1 prevents black from taking Cc3, at least temporarily. The opening might proceed something like this:

1. P-Cc2        P-Cc4
2. B-Ca1        P-Cd4
3. N-Bc1        B-Bd4
4. P-Dc3        N-Cb5
5. P(Dd2)-Dd3   R-De5
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|_|   |n|b|q|_|_| 5 |_|n|_|_|_|   |_|_|_|_|r| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|_|b|o| 4 |_|_|o|o|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|   |_|_|P|P|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|_|P|_|_|   |P|P|_|_|P| 2 |P|P|_|P|P|
1 |_|_|_|_|_|   |_|_|N|_|_| 1 |B|_|_|_|_|   |N|_|Q|_|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Opening pawn gambits may not work well on the Five Up board. In an opening gambit in conventional chess, one player offers a pawn as a sacrifice in order to draw an enemy pawn away from the center. This is more difficult to do here. For one thing, gambit pawns that are pushed toward the center tend to be protected by other pawns, so they can't be captured without fear of immediate reprisal. Also, it takes two moves to bring the opposing pawns in contact with one another, which makes a gambit harder to set up. Nevertheless, let's look at a possible gambit:

1. Cc2     Cd4
2. Cb2     Cd3
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|b| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|o|_|o| 4 |_|_|_|_|_|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|o|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|P|P|_|_|   |P|_|_|P|P| 2 |P|P|P|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

Black offers a pawn. Let's see if the maneuver gains anything. White could decline the gambit with 3. Cc3, which would tempt black to capture the center pawn with the queen pawn (3. ... PxCc3. 4. PxCc3). But if white takes the gambit pawn, the sequence below might follow. After black follows up with P-Ce4, the white pawn can advance no further without being captured, so white must either defend it or take the second pawn. Let's accept the new gambit, and see what happens.

3. PxCd3     Ce4
4. PxCe4     BxCe4
5. Cc2
   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |r|w|k|g|r|   |n|b|q|n|_| 5 |_|_|_|_|_|   |_|_|_|_|_| 5 |_|_|_|_|_|
4 |o|o|o|o|o|   |o|o|o|_|_| 4 |_|_|_|_|b|   |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_|   |_|_|_|_|_| 2 |_|P|P|_|_|   |P|_|_|P|P| 2 |P|P|_|P|P|
1 |_|_|_|_|_|   |_|_|_|_|_| 1 |_|_|_|_|_|   |N|B|Q|N|B| 1 |R|W|K|G|R|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

White's center doesn't seem to have suffered; if anything, the two white pawns dominate the position. So the gambit would appear to be a failure. The central problem, it seems to me, is that the pawns (in this case, the Dc2 and Ec2 pawns) are doubled on the slices (the vertical fore-and-aft planes). Two pawns can reach Cc2. Pulling the first one away to the edge of the board is of little value, because the other can step in to fill the gap. Black would have to sacrifice another pawn to get rid of this replacement, and that would require two more moves, which would leave black far behind in development.

I'm sure this quick survey doesn't even begin to cover the topic of openings. But if you're interested in playing Five Up, I hope I've suggested some lines of investigation that might be worth pursuing.

Variants of Variants

Hypercylindrical Board. There's no particular reason to restrict a 3D playing area to a Euclidean 3D space. One of the interesting things one could do with the Five Up board is warp it so that its sides join one another. The simplest way of doing this, and one that should make for good game play, joins the a and e faces in a simple four-dimensional hypercylinder. This is an exact higher-dimensional analog of the board in Cylindrical Chess. A rook, for instance, that leaves the playing area by traveling leftward from Cb1 through Ca1 reenters at Ce1. This is not a very interesting trajectory for a rook, but it gets more interesting when the cylindrical move is made by a knight. Here, for instance, are the possible moves of a knight starting on Ca3 on a hypercylindrical board:

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |x|_|_|_|_| 5 |_|x|_|_|x|   |x|_|_|_|_| 5 |_|_|_|_|_|
4 |x|_|_|_|_|   |_|_|_|_|_| 4 |_|_|x|x|_|   |_|_|_|_|_| 4 |x|_|_|_|_|
3 |_|x|_|_|x|   |_|_|x|x|_| 3 |N|_|_|_|_|   |_|_|x|x|_| 3 |_|x|_|_|x|
2 |x|_|_|_|_|   |_|_|_|_|_| 2 |_|_|x|x|_|   |_|_|_|_|_| 2 |x|_|_|_|_|
1 |_|_|_|_|_|   |x|_|_|_|_| 1 |_|x|_|_|x|   |x|_|_|_|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 

We could go further, linking the A and E layers and the 1 and 5 faces to form a hypertorus. This is a fascinating four-dimensional object -- unlike a 3D torus, it has two holes. It has no edges: All 125 cells are equally "in the center of the board." There is probably no practical way to play chess on such a board, however, because it's not possible to get the opposing armies far enough apart at the beginning of the game to prevent their attacking one another. We might consider a non-combatant rule, however, in which no piece can capture or be captured until after it has moved. Such a rule is an essential part of Uncertainty.

Riding on a Unicorn. In this variant of Five Up, each player has one additional piece -- a unicorn. The unicorn has some unusual properties. Its move consists of a knight-move (in any plane) followed by an additional one-cell "leg" at right angles to the rest of the movement vector. A unicorn moves like this:

   _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _     _ _ _ _ _ 
5 |_|_|_|_|_|   |_|x|_|x|_| 5 |_|_|_|_|_|   |_|x|_|x|_| 5 |_|_|_|_|_|
4 |_|x|_|x|_|   |x|_|_|_|x| 4 |_|_|_|_|_|   |x|_|_|_|x| 4 |_|x|_|x|_|
3 |_|_|_|_|_|   |_|_|_|_|_| 3 |_|_|U|_|_|   |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|x|_|x|_|   |x|_|_|_|x| 2 |_|_|_|_|_|   |x|_|_|_|x| 2 |_|x|_|x|_|
1 |_|_|_|_|_|   |_|x|_|x|_| 1 |_|_|_|_|_|   |_|x|_|x|_| 1 |_|_|_|_|_|
A  a b c d e  B  a b c d e  C  a b c d e  D  a b c d e  E  a b c d e 
Four factors make the unicorn unusual:
  1. It can occupy the same cell as any friendly piece (or pawn) except the king.
  2. If it's situated on a cell by itself, it can't capture enemy pieces nor check the enemy king.
  3. When the unicorn moves, a piece with which it is sharing a cell can "ride" along with it to its destination cell, provided that cell is vacant, or is occupied by an enemy piece.
  4. When a piece sharing a cell with the unicorn moves, the unicorn can ride along with that piece, adopting the other piece's normal mode of movement (or capture).

When the unicorn and another piece move together, the action is considered to be a single move. The unicorn can move into or out of a cell occupied by a friendly piece, or between the cells occupied by two friendly pieces. Such a move takes up the player's entire turn: You can't move the unicorn onto or off of a cell occupied by another piece and then move the other piece in the same turn.

When the unicorn is sharing a move with another piece, the pair can capture enemy pieces and check the enemy king using either the other piece's movement vector or the unicorn's movement vector.

Like the knight, the unicorn can freely pass through a wall of pieces when moving. However, when the unicorn and another piece are moving as a single unit, this ability only applies if the two are using the unicorn's move (or the knight's move, if the other piece is a knight). If the unicorn is sharing a cell with a rook, for example, and the two move together using the rook's movement vector, they can't pass through other pieces.

When a piece sharing a cell with a unicorn is captured, both it and the unicorn are captured. When the unicorn is on a cell by itself, it can be captured by enemy pieces in the normal way.

The unicorn begins the game on the same cell as the wizard.


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