A book called Chess Variants & Games, by A. V. Murali, looked interesting, so I bought it. I haven’t read much of it yet. It strikes me as quite scattered — one of those books written by a bright guy with a ton of off-the-wall ideas, no editor, and perhaps not a great deal of writing experience. Even so, some of his ideas are intriguing.

Playing chess on the exterior surface of a cube, for instance. That’s today’s mind trip. Playing on the exterior surface is entirely different from 3-dimensional chess played in the interior of a cube, and seems like an excellent basis for a variant. Playing on the exterior surface is, in effect, equivalent to playing on three cylinders that interlock with one another. A 5x5x5 cube has 150 exterior squares, a large but not unreasonably large playing surface.

I think I’ve managed to improve it, though. If you’re curious, take a look at my rules for Shoebox (added on May 13, 2014).

Murali seems not to have spent much time pondering the optimum piece density or movement vectors on such a playing surface. His bishops don’t use a proper diagonal when sliding over an edge to an adjacent face, but his pawns do. He suggests giving each player eight pieces and 16 pawns, but this is probably not enough, as less than 1/3 of the squares will be occupied at the start of the game (compared to 1/2 of the squares in conventional chess). The ratio of pieces to pawns is not very good either.

He gives no suggestions for a system of algebraic coordinates with which to notate piece positions, which makes it tricky even to talk about how the pieces might be set up. I’ve worked out a coordinate system, but it’s actually easier (for me, anyway) to visualize how the pieces move on the exterior of a cube than to discuss it using coordinates. For now, let’s just use generic terminology and diagrams rather than algebraic coordinates.

If each player starts with a set of pieces and pawns congregated on one face, and if the pawns surround the pieces on all sides (as they should), you’ll have 9 pieces and 16 pawns per side. That’s not enough pieces for a board with 150 squares. To balance things a bit better, imagine extending the layout of forces in two directions (but not the other two). If we assume white’s army is stationed on the top face, we’ll put one row of pawns on the upper row of the front face and another row of pawns on the upper row of the rear face. Fill the gap in the top face with extra pieces (their precise nature not yet defined). Now we have 20 pawns and 15 pieces for each player, a 4/3 pawn-to-piece ratio, and the ratio of occupied squares to the total number of squares is now 7/15, which is closer to 1/2.

This playing surface has some interesting characteristics. A single bishop can eventually reach any square on the board; there’s no concept of black vs. white diagonals. A knight placed on an edge square near (but not on) a vertex square can reach nine other squares, some of which are adjacent to one another. And because we’ll need to come up with a couple of new types of pieces, we might consider a piece that teleports through the interior of the cube, emerging on the analogous square on the opposite face.

To illustrate the geometry, here’s how a bishop would travel along the diagonals:

cubic diagram bishopNotice that the two rows of squares it can reach on the top face are not related to one another diagonally; they’re separated by knight’s moves. And here’s what happens when a knight moves across a vertex:

cubic diagram knight

If we conceive of the knight’s move as starting with one square to the left and then two squares up, it travels across the vertical edge and then across the upper left edge and arrives at the square above the vertex. Conversely, if it starts by traveling up two squares and then turning to the left, it travels across the upper right edge and then across the upper left edge to arrive at the square down and to the left of the vertex.

How a knightrider would move on this board … I’m going to have to think about that. The concept of “continuing in the same direction” gets a little hazy when you cross two edges in a single leap.

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