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1 Shoebox, page 1 In his book Chess Variants & Games, A. V. Murali suggests playing chess on the exterior surface of a cube. This playing surface has intriguing properties: We can think of it as three interlocked cylinders. But finding a practical way to turn the concept into a playable game requires a bit of thought. A 4x4x4 cube is probably too small: The opposing armies are too close to one another, and if the pieces are surrounded on all sides by pawns in the opening setup, as they should be, there will be 12 pawns and only four pieces. On the other hand, a 5x5x5 cube is probably too large. There are 150 squares, and either too many pieces (25 if the top and bottom of the cube are filled with pieces and rings of 20 pawns circling them on the sides) or not enough (nine if the pieces are arrayed in the center squares of the top and bottom, with the rings of 16 pawns circling them also on the top and bottom). The solution to both problems one solution, anyway is to make the playing surface the exterior of a 6x4x3 rectangular solid. Such a solid is wider than it is thick, and longer than it is wide. It vaguely resembles a shoebox, which is why this variant is called Shoebox Chess, or simply Shoebox. The Playing Surface We can diagram the Shoebox playing surface (algebraic notation included) as shown in Figure 1. Figure 1: The Shoebox playing surface, unfolded as a flat diagram, with algebraic notation of the squares _ _ _ _ _ _ _ _ 2 _ _ _ _ We ll call the 4x3 rectangles the near and far ends (rows 1, 2, and 3 being the near end). The 3x6 rectangles we ll call the left and right sides, and the 6x4 rectangles the top and bottom (the top includes columns d through g; the bottom is at the far right in this diagram).

2 Shoebox, page 2 The 14x6 central area is functionally a cylinder: Pieces can travel off the left side of the diagram and return on the right side, or vice-versa. The other cylinders are a bit harder to see. The three-square-wide rectangles (the left side, far end, right side, and near end) appear visually in a more or less circular layout in this diagram, but in fact they form a cylinder. Likewise, the top, far end, bottom, and near end form a cylinder that is four squares wide. It s essential to understand that the near and far 4x3 rectangles are in fact contiguous to the central area on all four of their edges. To illustrate this, let s put a few random pieces on the ends and then mutate the diagram. The three diagrams in Figure 2 are all functionally identical; they all represent exactly the same configuration of pieces on the surface. Figure 2: Three functionally identical representations of the playing surface. The positions of the pieces shown are identical in all three diagrams. k _ _ _ _ _ q _ _ _ _ n _ K _ R _ _ _ _ _ _ B _ Figure 2b: _ _ _ n _ q _ _ _ _ k _ _ _ B _ _ _ _ _ _ R _ K _

3 Shoebox, page 3 Figure 2c: _ _ _ k _ _ _ _ q _ n _ _ _ _ _ _ _ _ K B _ _ _ _ R To visualize how the faces connect with one another, we can imagine folding up the playing surface so that it resembles a shoebox. Returning to Figure 1, visualize folding the near and far ends (the 4x3 rectangles) down from the top by 90 degrees. The 6x4 rectangle with columns d through g and rows 4 through 9 is the top. Fold the six-row rectangles down from the top, and join the right edge (the n column) to the left edge (the a column). At this point, g10-g11-g12 will be touching h9-i9-j9. That is, a rook positioned on e12 could travel to the right across f12 and g12 and then arrive on j9 and continue downward to j8, j7, and so on. This would not be a change of direction; the rook would be proceeding in a straight line from the top of the solid across an edge and onto the right side. In the same way, g1- g2-g3 are touching j4-i4-h4, d1-d2-d3 are touching a4-b4-c4, and d10-d11-d12 are touching c9-b9-a9. The rows shown at the top and bottom of the diagram touch the actual bottom surface (rows k through n) but in a slightly non-intuitive reversed way. That is, g12 is orthogonally adjacent to k9 (not to n9), f12 to m9, and so on. d1 is orthogonally adjacent to n4, e1 to m4, and so on. The question of how the arrangement of the faces of the shoebox affects the travel of knights and bishops we ll get to shortly. It turns out to be interesting. Basic Rules Of course, an infinite variety of pieces could be used in Shoebox. But the topology of the playing surface is challenging enough that a simple, straightforward set of pieces seems advisable. To the standard set of pieces (a king, queen, two rooks, two knights, two bishops, and ten pawns for each player), we will add a marshal and an archbishop (familiar from many variants, though not always given the same names), a pair of cannons (adapted from Chinese chess), and four enhanced pawns called lancers. Players move alternately, as in standard chess. All of the normal rules of chess, including en passant pawn capture (but not castling) apply. Capture is by displacement, the capturing piece moving onto the square of the captured piece, whereupon the latter is removed from play. A stalemate is a draw. A rule forbidding the opposing kings from facing one another is adapted from Chinese chess.

4 Shoebox, page 4 If both players alternately make three null moves in a row, so that after three moves by white and three by black the board position remains unchanged, the game is a draw. (See the discussion of the rook s move, below, for an example of a null move.) The opening setup is shown in Figure 3. Figure 3: The opening layout in Shoebox. Black pieces are shown as lower-case letters, white pieces as capital letters. 12 r b a n 11 c k q c 10 n m b r 9 p p p l p p l p p p l p p l P P P L P P L P P P L P P L 4 3 R B A N 2 C K Q C 1 N M B R The Movement of the Pieces Because diagonal movement on the Shoebox has a peculiar property, we ll start by describing the move of the bishop. The peculiar property is that you can t move diagonally off of a corner. The orthogonals (rows and columns) on the Shoebox are essentially endless, but the diagonals aren t. To put it another way, the Shoebox has no edges, but it does have corners. Bishop The bishop moves diagonally, exactly like a standard chess bishop. However, there is no notion on the Shoebox of black and white squares. A moment s reflection will show why. Squares d10, d9, and c9 are all orthogonally adjacent to one another. If one of them is white and another is black, what color is the third one? A consequence of this is that even though the bishop always moves diagonally, it can (if no other piece gets in the way) sometimes end its move on one of two different squares that are orthogonally adjacent to one another. If the squares were colored black and white, a bishop would be able to start on a white square and end its move on a black square, or vice-versa.

5 Shoebox, page 5 Figure 4 shows just one of the four diagonal vectors that the bishop can use when starting from the square shown. Figure 4: A single diagonal vector of movement for the bishop on h7. The vector ends on g10, which is a corner. 12 _ * _ _ 11 * _ * _ 10 _ * _ x 9 * _ _ _ _ * _ _ _ _ _ _ _ * 9 8 * _ _ _ _ _ * _ _ _ _ _ _ * 8 7 _ * _ _ _ _ _ B _ _ _ _ * _ 7 6 _ _ * _ _ _ _ _ _ _ _ * _ _ 6 5 _ _ _ * _ _ _ _ _ _ * _ _ _ 5 4 _ _ _ _ * _ _ _ _ * _ _ _ _ 4 3 _ _ * _ 2 _ _ _ * Starting on h7, the bishop proceeds across the top (passing through g8 and f9), crosses the far end at e10 and d11, then touches the left side at a9 (moving downward but still leftward with respect to the flat diagram), continues down across the bottom from n8 to k5, touches the right side again at j4, crosses the near end via g2 and f3, cuts across the top at e4 and d5, heads across the left side again from c6 to a8, touches the bottom at n9, and finishes on the far end with e12, f11, and g10. Note that on a flat diagram, its last three squares appear to be a movement downward and to the right. Its travel ends at g10 (marked with an x) because it has now hit a corner head-on. Earlier it crossed two corners (a9 and n9) obliquely, but crossing a corner obliquely on a diagonal doesn t stop the movement. The diagonal does not continue from g10 to h9, because in spite of the appearance of the diagram, g10 and h9 are orthogonally adjacent, not diagonally adjacent. Note that the bishop can reach either a8 or a9, and either n8 or n9. Thus if the squares were colored in the usual way, it would be able to reach a white square when starting from a black square or vice-versa. Adding the other three diagonals to the moves of this bishop, we ll see that it s a powerful piece more powerful than the rook or knight. Starting from h7, as shown in Figure 5, it can reach 36 other squares (unless blocked by another piece; the bishop can t leap over pieces). It can reach a few of these squares using two different vectors. In the diagram, it can reach d3 either through g6-f5-e4 or through i6-j5-k4-f1-e2.

6 Shoebox, page 6 Figure 5: On an unobstructed surface, the bishop on h7 can move to any of the 36 squares indicated. 12 _ * _ _ 11 * _ * _ 10 _ * _ * 9 * _ _ _ _ * _ _ _ * * _ _ * 9 8 * _ _ _ _ _ * _ * _ _ * _ * 8 7 _ * _ _ _ _ _ B _ _ _ _ * _ 7 6 _ _ * _ _ _ * _ * _ _ * _ * 6 5 * _ _ * _ * _ _ _ * * _ _ _ 5 4 _ * _ _ * _ _ _ _ * * _ _ _ 4 3 * _ * _ 2 _ * _ * 1 _ _ * _ Rook Compared to the move of the bishop, the move of the rook is simple indeed. It moves any number of squares along unobstructed orthogonals, as shown in Figure 6. Figure 6: The movement of the rook. 12 _ * _ _ 11 _ * _ _ 10 _ * _ _ 9 _ _ _ _ * _ _ _ _ _ _ _ * _ 9 8 * * * * R * * * * * * * * * 8 7 _ _ _ _ * _ _ _ _ _ _ _ * _ 7 6 _ _ _ _ * _ _ _ _ _ _ _ * _ 6 5 _ _ _ _ * _ _ _ _ _ _ _ * _ 5 4 _ _ _ _ * _ _ _ _ _ _ _ * _ 4 3 _ * _ _ 2 _ * _ _ 1 _ * _ _ Starting from e8, as shown, the rook can reach m8 along any of four different vectors. In addition, if its travel is unobstructed, the rook can return to its starting square using either of two vectors. Because of

7 Shoebox, page 7 this, the rook (and any other piece that moves like a rook) can potentially make a null move, leaving the position the same after its move as before. This isn t likely to be a useful capability in most situations, but it s a legal move. Such a move could conceivably prevent a stalemate, if the rook is pinned and no other piece can make a legal move. It will be observed that on an unobstructed board, the rook can reach 29 squares, making it somewhat less powerful than the bishop. Queen The queen combines the moves of the rook and bishop. This is almost too trivial to be worth diagramming, but for the record, Figure 7 shows the 60 distinct squares the queen can reach on an open board, starting from h7. Several of these can be reached using more than one vector. Figure 7: The movement of the queen. 12 _ * _ _ 11 * _ * _ 10 * * * * 9 * _ * _ _ * _ * _ * * _ _ * 9 8 * _ * _ _ _ * * * _ _ * _ * 8 7 * * * * * * * Q * * * * * * 7 6 _ _ * _ _ _ * * * _ _ * _ * 6 5 * _ * * _ * _ * _ * * _ _ _ 5 4 _ * * _ * _ _ * _ * * _ _ _ 4 3 * * * * 2 _ * _ * 1 _ _ * _ Knight The movement of a standard chess knight on the Shoebox has an odd feature: If it starts its move near a corner, it can reach two orthogonally adjacent squares (not possible for a knight on a flat surface). In addition, because the Shoebox is larger than a conventional chessboard, the Shoebox knight has an enhanced move. But we ll look first at Figure 8, which shows how a standard knight s move works on the Shoebox, both because it illustrates the concept more clearly and because the marshal and archbishop make use of exactly this type of knight-move.

8 Shoebox, page 8 Figure 8: A standard knight-move on the Shoebox _ _ * * 9 _ _ _ _ * _ _ * * _ _ _ _ _ 9 8 _ _ _ _ _ _ N _ _ _ _ _ _ _ 8 7 _ _ _ _ * _ _ _ * _ _ _ _ _ 7 6 _ _ _ _ _ * _ * _ _ _ _ _ _ _ _ _ _ 2 _ _ _ _ To understand this diagram, it may be best to think of the knight s move not as one square orthogonally followed by one square on the outward diagonal but as either one square orthogonally followed by a right-angle turn and then two more squares, or as two squares orthogonally followed by a right-angle turn and then one more square. As usual, the intervening squares don t have to be empty; the knight can leap. In the diagram above, if we think of the knight on g8 as moving upward through g9 and g10 followed by a right-angle turn to the right, it will arrive on h9 (because g10 and h9 are orthogonally adjacent, across an edge). On the other hand, if we think of the knight as starting its move with a single square to the right, from g8 to h8, followed by two squares upward, it would pass through h9 and arrive at g10. The other seven squares are arrived at in a more standard way, even though four of them involve crossing an edge. Travel from g8 to h9 or g10, however, is a path across two edges, which is why this vector ends up being, in a sense, doubled, allowing the knight to reach 9 separate squares rather than only 8. If we position the knight on an end square, its travel gets still weirder. As shown in Figure 9, it can now reach ten squares, because it can travel across two edges in a couple of different ways.

9 Shoebox, page 9 Figure 9: The possible moves of a knight starting on an end square _ _ _ _ * _ * _ _ _ _ _ _ _ _ _ _ _ 5 4 * _ * * * _ _ _ _ _ _ _ * * 4 3 _ _ * _ 2 N _ _ _ 1 _ _ * _ On such a large board, a knight that could reach, at most, ten squares wouldn t be as useful tactically as we might like. Accordingly, the Shoebox knight can use either a 2/1 or a 3/1 vector. If there are no double edge crossings in its vector, the Shoebox knight can reach 16 squares, as shown in Figure 10. It s still considerably less powerful than a bishop, so the standard idea of trading a bishop for a knight would be inadvisable in Shoebox. Figure 10: The Shoebox knight can use either a 2/1 or a 3/1 vector, leaping over any intervening pieces. 11 * _ * _ 10 * _ * _ 9 _ * * _ _ _ * * _ _ _ _ _ _ 9 8 _ _ _ _ N _ _ _ _ _ _ _ _ _ 8 7 _ * * _ _ _ * * _ _ _ _ _ _ 7 6 _ _ _ * _ * _ _ _ _ _ _ _ _ 6 5 _ _ _ * _ * _ _ _ _ _ _ _ _ _ _ _ _ 2 _ _ _ _

10 Shoebox, page 10 If the knight starts from a square at or near a corner, it can reach as many as 18 squares. In Figure 11, for instance, it may appear that the knight has traveled diagonally from g9 to f10, but in fact the vector that leads it to f10 is g9-h9-g10-f10, a standard knight s move. Figure 11: Depending on where its move starts, the Shoebox knight can sometimes reach more than 16 squares. 12 _ _ * * 11 _ _ * * 10 * * * _ 9 _ _ _ _ _ _ N _ * * _ _ _ _ 9 8 _ _ _ * * _ _ * * * _ _ _ _ 8 7 _ _ _ _ _ * _ * _ _ _ _ _ _ 7 6 _ _ _ _ _ * _ * _ _ _ _ _ _ _ _ _ _ 2 _ _ _ _ King The Shoebox king moves and captures exactly like a standard chess king. As with the movement of the bishop, however, there is no diagonal square outward from a corner square. Accordingly, a king on d9 can reach only seven other squares (c9, c8, d8, e8, e9, e10, and d10) rather than eight. Because the Shoebox has no edges, checkmating the king will often be more difficult than in standard chess. To ease the difficulty, a rule from Chinese chess has been adapted to Shoebox. When either of the kings has left its end of the board and is located on a square in the top, bottom, or left or right side, the two kings cannot face one another on an unobstructed vertical or horizontal. The king cannot move onto a square where it would face the enemy king. If there is exactly one piece between the two kings, that piece is pinned. It can only be moved if its move is to the same or another square where it will be interposed between the two kings. (See the discussion of the null move in the section on the rook.) As long as both kings are in their respective ends of the Shoebox, the no-facing rule does not apply. The tactical advantage of giving the Shoebox a movable edge to keep the enemy king from moving freely can be seen in the position in Figure 12.

11 Shoebox, page 11 Figure 12: Checkmate by the black king and rook against the bare white king _ _ _ _ _ _ _ _ _ _ _ r _ _ _ _ _ _ 5 4 _ _ _ _ _ _ _ _ k _ _ _ _ _ 4 3 _ K _ _ 2 _ _ _ _ The white king on e3 in Figure 12 is checkmated by the black rook on h5. It can t move downward to d2, e2, or f2, because those squares face the black king (e2-f2-g2-i4 being on an orthogonal). Likewise, the white king can t move up to d4, e4, or f4. The squares d3 and f3 are, of course, covered by the black rook. Marshal and Archbishop The marshal combines the moves of a rook and a 2/1 standard chess knight. The archbishop combines the moves of a bishop and a 2/1 knight. Neither piece is as powerful as the queen. The archbishop is more powerful than the marshal, just as the bishop is more powerful than the rook. Cannon The rook being less powerful in Shoebox, another pair of pieces with a rook-type move seems a good way to fill out the piece complement. The cannon makes non-capturing moves like a rook. In addition, it can leap over a single piece (either friendly or enemy) and continue its vector of movement on the far side of the leaped piece. The cannon can capture enemy pieces only in a move that includes a leap. For this reason it will be more useful in the earlier part of the game, when there are more pieces available to leap over, and less powerful (though still dangerous) in the endgame. See below for a cannon variant called the cannonpult. Because the Shoebox surface is cylindrical, a cannon can sometimes capture a piece that you might not notice it threatens. Two possible cannon attacks are shown in Figure 13.

12 Shoebox, page 12 Figure 13: Assuming there are no other pieces on the board, the white cannon on i6 can capture the black knight on k6, and the black cannon on e8 can capture either the white archbishop on e7 or the white lancer on e9, because these captures can be made following leaps _ _ _ _ 9 _ _ _ _ L _ _ _ _ _ _ _ _ _ 9 8 _ _ _ _ c _ _ _ _ _ _ _ _ _ 8 7 _ _ _ _ A _ _ _ _ _ _ _ _ _ 7 6 _ _ _ _ _ _ _ _ C _ n _ _ _ _ _ _ _ 2 _ _ _ _ Assuming there are no other pieces on the board, the black cannon on e8 in Figure 13 can capture either the white archbishop on e7 (by leaping over the white lancer on e9 and then traveling upward across the far end, down the bottom, across the near end, and up the front) or the white lancer (by leaping the archbishop and then continuing around the surface). The white cannon on i6 can capture the black knight on k6 by leaping over the knight, traveling all the way around the Shoebox, and capturing the knight on its second pass. A possible variant: The cannons could be replaced with catapults, which move like rooks and throw a single piece positioned behind them (with no other pieces intervening between the catapult and the piece to be thrown) forward past the square where the catapult s move ends (possibly even passing over another single piece). In a given move, the catapult can either throw another piece or move or capture like a rook; it can t both throw another piece and capture an enemy piece in the same move. Catapulting another piece is an interesting and dangerous capability, especially as there s no reason not to allow the catapult to throw enemy pieces and kings. The thrown piece can land on another piece and capture it. In this manner, one enemy piece could be used to capture another enemy piece. In rare circumstances the catapult might be used to throw an enemy piece onto a friendly piece, capturing the friendly piece. This could be useful, for example, if throwing the enemy piece creates a discovered check and there is no other desirable place for the enemy piece to end up; or if the enemy king is thrown onto a friendly piece and ends up in check on that square. On a cylindrical board, a catapult can even capture itself by throwing an enemy piece entirely around the Shoebox so that it ends up on the same square the catapult ends on probably not a useful capability, but logically allowable. A single piece, the cannonpult, could be both a cannon and a catapult at the same time. If the catapult and cannon are combined into a single cannonpult, this piece would obviously capture only as a

13 Shoebox, page 13 cannon, not as a rook. Logically, a cannonpult should be able to both leap and throw in the same move. It can even leap over a piece and then throw the piece it has just leaped over. Pawns & Lancers Shoebox pawns are similar to standard chess pawns. They advance toward the enemy, and if they reach the enemy s home squares (the other end of the Shoebox) they promote. The pawn layout also includes four lancers, which are essentially a more powerful type of pawn, so we can profitably discuss the two pieces together. For the opening pawn/lancer layout, refer back to Figure 3. Pawns can advance either one or two squares on their first move, and thereafter can advance only one square. They capture by moving one square diagonally. Nothing new here. Like a pawn, the lancer can t back up. It can only advance straight or diagonally toward the enemy s home squares, or move sideways. It can always move either one or two squares, and its capturing move is the same as its non-capturing move. In addition, when the lancer moves by two squares, it can leap over an intervening piece. These possibilities are shown in Figure 14. Figure 14: The basic moves of pawn and lancer. The black pawn on c9 can make a non-capturing move to c8 or c7, and can capture to b8 or d8. The black pawn on h4, however, has only one possible capturing move (to g2), because its other diagonal would be off the corner of the right side. It can move orthogonally to g3. The lancer on k9 can move or capture to any of the squares marked with asterisks _ _ _ _ 9 _ _ p _ _ _ _ _ * * l * * _ 9 8 _ *. * _ _ _ _ _ * * * _ _ 8 7 _ _. _ _ p _ _ * _ * _ * _ 7 6 _ _ _ _ *. * _ _ _ _ _ _ _ _ _ _ _ _ _ _ p _ _ _ _ _ _ 4 3 _ _ _. 2 _ _ _ * As shown in Figure 15, lancers and pawns can mutually capture one another en passant whenever the piece to be captured has made a two-square move. As in standard chess, a capture en passant can be made only in the move immediately following the move of the piece to be captured. Because the lancer can leap the intervening square even if it s occupied, if it has leaped over one of its own friendly pieces the en passant capture of a lancer could conceivably capture two pieces (the lancer and the piece it just

14 Shoebox, page 14 now leaped over) in a single move. Figure 15: Capture en passant by pawn and lancer _ _ _ _ 9 _ _ p _ _ p _ _ * * l _ _ _ 9 8 _ _. L _. _ _ P _ _ * _ _ 8 7 p. P _. _ _ _ _ P _ * _ 7 6 N _ _ _ L _ _ _ _ L _ _ _ _ 6 5 L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 _ _ _ _ Let s take a close look at Figure 15. It shows four separate situations. To begin with, the black pawn on c9 can advance to either c8 or c7. If it advances two squares, to c7, it can be captured en passant by the white pawn on d7, or by the white lancer on e6 (which jumps over the white pawn to make the capture). The black pawn on f9 can advance to f7. If it does so, it can be captured en passant by the white lancer on d8 using the lancer s sideways move. The black lancer on k9 can move, among its other possibilities, two squares to either i9 or m7. If it moves to i9 it can be captured (to j9) by the white pawn on i8. If it moves to m7, it can be captured (to L8) by the white pawn on k7. Finally, the white lancer on a5 can advance to a7 by leaping over the white knight on a6. If it does so, the black pawn on b7 can capture both the lancer and the knight by moving to a6. An important reason for including lancers in Shoebox is this: In the opening setup (referring back again to Figure 3), all of the pawns are defended by lancers. The lancers, in turn, are defended by the cannons. Conveniently, a Shoebox pawn requires exactly the same number of moves to reach one of the enemy home squares for promotion as a standard chess pawn. A lancer, however, can reach the enemy home squares in as little as three moves. On reaching the enemy s home squares, a pawn or lancer can promote to any of the pieces (except a pawn, lancer, or king) defined by the rules of Shoebox. A possible variant: Allowing pawns to make a non-capturing move of a single square sideways would

15 make the pawns and lancers more similar to one another, and would increase the flexibility (or turbulence, if you prefer) of the opening. Shoebox, page 15