5 Up Raumschach

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Raumschach, which was invented before World War I, is a classic 3D chess variant. The 5x5x5 board is an ideal size, and the additions to the FIDE set of pieces are minimal, making the game easy to learn. However, Raumschach suffers from a couple of nontrivial problems. The unicorn (which in the original version moves only on 3D diagonals) is too weak to be a useful piece, and the king is too mobile, making checkmate difficult. In addition, the piece density is less than 1/3 (40 pieces in all on a 125-cell board), which is arguably less than ideal. Richard Goode's variant of Raumschach has better unicorns, which I have adopted (see below), but his king, being restricted to single-cell moves on the 3D diagonals, is too 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 custom essay short, on the assumption that most readers will already understand 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."

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

Figure 1. The opening setup in Five Up. In the ASCII diagram, black pieces are shown as lower-case letters bracketed by asterisks. White pieces are capital letters. The A layer can be visualized as the top of the cube and the E layer the bottom (my preference), or vice-versa. The layers are positioned side by side in the ASCII diagram purely for ease in viewing or printing. Note that a slight asymmetry in the initial setup is required to put the bishops (and also the unicorns) on cells of opposite colors. The black and white positions are mirror images of one another.

5	*r*	*n*	*k*	*n*	*r*
4	*p*	*p*	*p*	*p*	*p*
3
2
1
A	a	b	c	d	e

5	*u*	*b*	*q*	*u*	*b*
4	*p*	*p*	*p*	*p*	*p*
3
2
1
B	a	b	c	d	e

5
4
3
2	P	P	P	P	P
1	B	U	Q	U	B
C	a	b	c	d	e

5
4
3
2
1
D	a	b	c	d	e

5
4
3
2	P	P	P	P	P
1	R	N	K	N	R
E	a	b	c	d	e

In addition to the pieces shown in Figure 1, each player has three pieces that are not on the board at the beginning of the game, but are added during play. The additional pieces are a wizard (W), a guard (G), and a silver unicorn (S). These pieces are placed on the board in the following manner:

Whenever a player moves one of his or her non-pawn pieces for the first time, the wizard, guard, or silver unicorn may be placed, immediately after the move, on the cell just vacated. This placement is optional, and may occur at any time during the game, provided the non-pawn piece is being moved for the first time. The wizard, guard, and silver unicorn are themselves non-pawn pieces, so a player could, for instance, move a knight and place a wizard on the knight's home cell, then on his or her next turn move the wizard and place the silver unicorn on this cell, and later move the silver unicorn and place the guard on the same cell. It's probably a good idea to enter these three pieces into play early, as doing so in the middle game, in the heat of a skirmish elsewhere on the board, may prove inconvenient.

The rook moves along unobstructed orthogonals, including verticals, the bishop along unobstructed diagonals in any plane, but not on 3D diagonals. The difference between 2D and 3D diagonals is illustrated in Figure 2. The queen combines the moves of rook and bishop. (This makes the queen a bit weaker than the original Raumschach queen, which can also use the 3D diagonals.) The knight makes a normal knight's move within any plane, as shown in Figure 3, and can pass through other pieces.

Figure 2. On a three-dimensional board, a 2D diagonal stays within a single plane. 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. The best 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. The cells marked 'x', 'o', and '.' lie on various 2D diagonals. The cells marked '*' and '+' lie on 3D diagonals.

5	O				*
4	X			*
3			*
2		*
1	*
A	a	b	c	d	e

5		O			+
4	*
3	X
2
1
B	a	b	c	d	e

5			O
4				+
3		*
2	X
1
C	a	b	c	d	e

5				O
4
3			+
2			*
1	X
D	a	b	c	d	e

5					O
4
3
2		+
1				*
E	a	b	c	d	e

Figure 3. The knight's move. (Layers D and E are shown to the right purely to make the diagram more compact.) The knight always stays within one of the three possible two-dimensional planes when moving. In this case, moves within the C layer are marked '+', moves within the 3 face are marked 'x', and moves within the c slice are marked '*'.

5
4			*
3		X		X
2			*
1
A	a	b	c	d	e

5			*
4
3	X				X
2
1			*
B	a	b	c	d	e

5			*
4
3	X				X
2
1			*
D	a	b	c	d	e

5
4			*
3		X		X
2			*
1
E	a	b	c	d	e

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 4. 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 double 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 can pass through either of the intermediate cells (for instance, from Aa4 to Ba3 by way of either Aa3 or Ba4), and the L-shaped move is blocked only if both of them are occupied. Capturing en passant can take place during the 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 4. The white pawn shown here, which is advancing upward (toward the A layer) can make capturing moves to the cells marked 'x' and non-capturing moves to the cells marked '+'.

5
4
3
2
1
A	a	b	c	d	e

5
4		X
3	X	+	X
2
1
B	a	b	c	d	e

5
4	X	+	X
3		P
2
1
C	a	b	c	d	e

5
4
3
2
1
E	a	b	c	d	e

The king can move or capture to an adjacent cell in any of the six orthogonal directions. It can't make 2D diagonal moves. It can capture (but not make a non-capturing move) by moving one cell along a 3D diagonal. As a result, the king can approach and threaten any enemy piece except a guard.

Also worth noting: In an endgame position against a lone king, the playerwho 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
4
3
2
1
A	a	b	c	d	e

5
4		X
3	X	+	X
2
1
B	a	b	c	d	e

5
4	X	+	X
3		P
2
1
C	a	b	c	d	e

5
4
3
2
1
D	a	b	c	d	e

5
4
3
2
1
E	a	b	c	d	e

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

An optional rule: The king can castle at any point in the game, even if
both king and rook have moved. 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 or to get out of check.

The unicorn moves like a 3D version of the knight. Its move, which may be a
bit hard to grasp at first, can best be visualized as a knight's move in
any plane followed by a one-cell sideways move perpendicular to that plane
(see Figure 5). If you like, you can think of that 3D "dogleg" as being the
unicorn's horn. Like the knight, the unicorn can pass through other pieces.
Like the bishop, a given unicorn is restricted to cells of one color, and
can reach only half of the cells on the board. Thus, while it can reach as
many cells as the knight when starting from the center of the board, the
unicorn is somewhat weaker than the knight. Starting from a corner cell,
for instance, the unicorn can reach only three other cells to the knight's
six.

Figure 5. The unicorn, shown here on Cc3, moves like a 3D version of the
knight, allowing it to reach any of the cells marked '*'. Note that none of
the unicorn's possible destinations are in the same planes as its present
position.

5
4
3
2
1
A	a	b	c	d	e

5
4		X
3	X	+	X
2
1
B	a	b	c	d	e

5
4	X	+	X
3		P
2
1
C	a	b	c	d	e

5
4
3
2
1
D	a	b	c	d	e

5
4
3
2
1
E	a	b	c	d	e

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

The silver unicorn moves like a unicorn, but it can also move any number of
cells along unobstructed 3D (but not 2D) diagonals, as shown in Figure 6.

Figure 6. The silver unicorn's move. For variety, let's look at it from an
asymmetrical position near the lower left corner. The cells the silver
unicorn can reach with a unicorn-type move are marked 'u', and the cells it
can reach via a 3D diagonal are marked 'd'.

5
4
3
2
1
A	a	b	c	d	e

5
4		X
3	X	+	X
2
1
B	a	b	c	d	e

5
4	X	+	X
3		P
2
1
C	a	b	c	d	e

5
4
3
2
1
D	a	b	c	d	e

5
4
3
2
1
E	a	b	c	d	e

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

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

The wizard moves any number of unobstructed cells along 2D or 3D diagonals,
as shown in Figure 7. Unlike the bishops but like the silver unicorn, the
wizard can reach any cell on the board, because the 3D diagonal movement
allows it to switch from black cells to white ones.

Figure 7. The wizard on Bb2 can reach any of the cells marked 'x' by moving
along 2D diagonals, and any of the cells marked '*' along 3D diagonals.

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

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, a unicorn is worth no more than 3, perhaps slightly less. The bishop and
knight are worth more than the unicorn, perhaps 4. The rook would then be
worth 5. The value of a hypothetical piece that could move only on 3D
diagonals would be about 2, which makes the silver unicorn worth 5 and the
wizard 6. The guard is probably worth 6 or a little more, and the queen 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 row (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 except
the rooks and the kings 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 are marked '+' (for the
corner bishop) and '*' (for the inside bishop):

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

Note that the bishops threaten, from their opening squares, opposing pawns
in the king layer. The same is true of the queen, which threatens the pawn
directly in front of the opposing king. B-Cd1 may not be an especially good
opening move for white, because black can respond N-Bd3, threatening the
bishop. On the other hand, trading bishops and knights in the opening is
not unusual. And B-Ca1 as an opening move poses no such difficulty.

Next, consider that the knights can each move, from their starting
positions, to six different unoccupied cells rather than to two, as in
conventional chess. The black Ad5 knight can reach any of the cells marked
'.' in the diagram below. The inside unicorn, though a less powerful piece
on the whole, has even more options for its first move -- the seven squares
marked '*' below for the white unicorn. The corner unicorn can reach only
four vacant cells from its starting position, but that's still twice as
many cells as a knight can reach in conventional chess. On two of the cells
in the C layer the unicorn will be subject to immediate capture, but U-Ec3
is likely to be playable as an opening.

_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|.|_|.|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|b| 5 |_|_|_|_|_|
4 |o|o|o|o|o| 4 |_|_|_|_|_|
3 |_|_|_|.|_| 3 |_|_|_|_|_|
2 |_|_|*|_|*| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |U|B|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|.|_|.| 5 |_|_|_|_|_|
4 |_|_|_|.|_| 4 |_|_|_|_|_|
3 |_|_|*|_|*| 3 |_|_|*|_|*|
2 |_|*|_|_|_| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |R|N|K|N|R|
C 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, 11 unicorn moves, 13 bishop moves,
and 10 queen moves, or 71 moves in all, compared to the 18 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 will have as
many as 20 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 best, but it's not hard to see that some of them are
bad. Consider this fool's mate:

1. P-Cc2 P-Cc4
2. B-De4/G P-Bd3?
3. N-Db3/S P-Cc3??
4. SxBe4++
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|_|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|b| 5 |_|_|_|_|_|
4 |o|o|_|o|S| 4 |_|_|_|_|B|
3 |_|_|_|o|_| 3 |_|N|_|_|_|
2 |_|_|_|_|_| 2 |P|P|_|P|P|
1 |_|_|_|_|_| 1 |U|G|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|o|_|_| 3 |_|_|_|_|_|
2 |_|_|P|_|_| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |R|_|K|N|R|
C a b c d e E a b c d e

The silver unicorn's 3D diagonal move allows it, from the left-side
knight's starting cell, to threaten the corner bishop's pawn, which black
has foolishly left unprotected. Black could easily have prevented this
disaster by leaving the knight's pawn where it was -- or, for that matter,
with 3. ... U-Ac3, since in that position the unicorn protects the pawn. Or
with 3. ... P-Bc3, which opens up an escape cell for the king.

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

Another topic in opening theory is how to introduce the wizard, guard, and
silver unicorn into play. It may be a bad idea to drop anything where it
would block the rooks' vertical development. The silver unicorn and wizard
should be dropped with a view to how their 3D diagonal moves can best
attack the enemy's home position. Drop the guard to the left or right of
the queen (or on the queen's cell after the queen moves), where it can
protect as many pawns as possible and also advance into the center.

Next, let's look briefly at some pawn formations that can arise in the
center. The simplest and most natural pawn opening is as follows:

1. Cc2 Cc4
2. Cc3
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|b| 5 |_|_|_|_|_|
4 |o|o|_|o|o| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |P|P|_|P|P|
1 |_|_|_|_|_| 1 |U|B|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|o|_|_| 4 |_|_|_|_|_|
3 |_|_|P|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |R|N|K|N|R|
C 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? Broadly speaking, black can either attack the white
pawn or ignore it. 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-Ca5
also attacks it. Capturing the center pawn with the unicorn would be
pointless at the moment, since a white unicorn is guarding it. Unlike the
situation on a conventional chessboard, here the knights are not in a
position to exert any immediate pressure on the center cell: It would take
a knight two moves to do so (for instance, N-Aa3 followed by N-Ca4). The
queen can attack the pawn 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
with 3. Bc3, but this 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. ... Cd4
3. Cd2
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|b| 5 |_|_|_|_|_|
4 |o|o|_|_|o| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |P|P|_|_|P|
1 |_|_|_|_|_| 1 |U|B|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|o|o|_| 4 |_|_|_|_|_|
3 |_|_|P|_|_| 3 |_|_|_|_|_|
2 |_|_|_|P|_| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |R|N|K|N|R|
C a b c d e E a b c d e

At this point, the possibilities begin to multiply. Black could play, for
instance, Bb3, attacking the white pawn again from above, or Cb4, attacking
it from the flank. Moving the king pawn to Bc3 blocks the white pawn,
making it an easier target. P-Dd2 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-Be2
diagonal, but B-Ca5, B-Bd2, U-Ac3, N-Ca5, and N-Cc5 all look playable.

White may find B-Dc2 useful. In this position the bishop protects both
center pawns -- and if black later advances the king pawn, P-Bc3 becomes 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. Cc2 Cc4
2. B-Ca1/G Cd4
3. N-Ea3/S B-Bd4/G (guarding Ba4)
4. Dc3
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|g| 5 |_|_|_|_|_|
4 |o|o|_|b|o| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|P|_|_|
2 |_|_|_|_|_| 2 |P|P|_|P|P|
1 |_|_|_|_|_| 1 |U|G|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|o|o|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |N|_|_|_|_|
2 |_|_|P|_|_| 2 |P|P|P|P|P|
1 |B|_|_|_|_| 1 |R|S|K|N|R|
C 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 safely be captured. 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. Cd2 Cd3
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|b| 5 |_|_|_|_|_|
4 |o|o|o|_|o| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |P|P|_|_|P|
1 |_|_|_|_|_| 1 |U|B|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|_| 4 |_|_|_|_|_|
3 |_|_|_|o|_| 3 |_|_|_|_|_|
2 |_|_|P|P|_| 2 |P|P|P|P|P|
1 |_|_|_|_|_| 1 |R|N|K|N|R|
C 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 BxC34/W
5. Cc2
_ _ _ _ _
5 |r|n|k|n|r|
4 |o|o|o|o|o|
3 |_|_|_|_|_|
2 |_|_|_|_|_|
1 |_|_|_|_|_|
A a b c d e
_ _ _ _ _ _ _ _ _ _
5 |u|b|q|u|w| 5 |_|_|_|_|_|
4 |o|o|o|_|_| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|_|_|_| 2 |P|P|_|_|P|
1 |_|_|_|_|_| 1 |U|B|Q|U|B|
B a b c d e D a b c d e
_ _ _ _ _ _ _ _ _ _
5 |_|_|_|_|_| 5 |_|_|_|_|_|
4 |_|_|_|_|b| 4 |_|_|_|_|_|
3 |_|_|_|_|_| 3 |_|_|_|_|_|
2 |_|_|P|P|_| 2 |P|P|_|P|P|
1 |_|_|_|_|_| 1 |R|N|K|N|R|
C a b c d e E a b c d e

White's center position doesn't seem to have suffered; if anything, the two
white pawns dominate the position. So the gambit would appear to be a
dismal failure. The central problem, it seems to me, is that the pawns (in
this case, the Dc2 and Ec2 pawns) are doubled on 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. But if
you're interested in playing Five Up, I hope I've suggested some lines of
investigation that might be worth pursuing.

Board Geometry: A Footnote

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 or unicorn. Here, for instance, are the possible moves of a knight
starting on Ca3 on a hypercylindrical board:

_ _ _ _ _
|_|_|_|_|_|
|x|_|_|_|_|
|_|x|_|_|x|
|x|_|_|_|_|
|_|_|_|_|_|
_ _ _ _ _
|x|_|_|_|_|
|_|_|_|_|_|
|_|_|x|x|_|
|_|_|_|_|_|
|x|_|_|_|_|
_ _ _ _ _
|_|x|_|_|x|
|_|_|x|x|_|
|N|_|_|_|_|
|_|_|x|x|_|
|_|x|_|_|x|
_ _ _ _ _
|x|_|_|_|_|
|_|_|_|_|_|
|_|_|x|x|_|
|_|_|_|_|_|
|x|_|_|_|_|
_ _ _ _ _
|_|_|_|_|_|
|x|_|_|_|_|
|_|x|_|_|x|
|x|_|_|_|_|
|_|_|_|_|_|

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.