n-Player Chess

Have you ever wanted to play a single chess game with two other people? What about with three or four other people? Is it possible to design a chess game that supports any finite number of players? Here’s a precise description of this challenge, followed by my attempt at a solution.

The Challenge: Create an n-player chess board that (i) preserves all the traditional pieces and their degrees of freedom; (ii) is divided into cells with eight directions each (F B L R + diagonal); and (iii) minimally changes the strategy of traditional chess.

First example: here is an interesting 3-player board (link no longer exists). But it doesn’t meet the challenge, because it’s composed cells with 12 directions each. This means that the pieces also don’t move in the traditional ways (For example, the queen can move in 12 directions instead of 8).

Second example: Meignorant found a 3-player board that has 4-sided cells. This board almost preserves the traditional motion of the pieces — but not quite! In the middle six cells, there are 10 directions, which means that the pieces acquire additional degrees of freedom there.

Another drawback of both of these examples is that it’s not obvious how n players can be added, instead of only 3. Here’s a board design that makes this trivial.

A Solution. Start with a circle. Divide it into n equal portions. Then divide each portion into an 4×8 grid (4 cells along the radius, 8 along the circumference). For n=3 players, what you’ll get is a board that looks like this:
One new board rule: the only way that a piece can pass from one of the n regions to another is by going through the center. In the board drawn above, this means that the three white lines separating the 3 regions are impassible barriers. However, straight lines pass through the central point just as they appear to.

The first two conditions are easily satisfied. All the pieces can keep their original degrees of freedom, because every cell (including each middle cell) has 8 directions (F B L R diagonal). Indeed, each cell even lies in one column and one row, each of length 8 — just like in traditional chess.

But what about the strategy? The strategy changes about as little as I think is possible. For example, when n=2, the game just reduces to something very similar to traditional chess. (Can you figure out what’s different? This difference can actually be avoided, but not without giving the board an unusual topology.)

On the other hand, things definitely start to change in higher-player chess games, although I don’t know the extent of this until someone actually plays a game.

Note that from any given player’s side of the board, only 1/(n-1) of each of the other players’ sides of the board is accessible. (In the image above, 1/(3-1) = 1/2 of the other two regions are accessible from any given region). This might be insignificant when n is small. However, it means that for games with more than 10 players, there are always be regions of the board that you cannot access without first moving to another region.

This feature of many-player games may actually help to limit the chaos, by limiting how many people can attack a single player in a single round. It also makes the following question especially interesting.

A Final Question. For what values of n can a `Knight’s Tour‘ be completed in n-player chess?

A Knight’s Tour is possible iff it’s possible for a knight to complete a circuit in which it lands on every cell exactly once, and ends up back where it started. Many theorems have been proved about this for different board shapes. It has been proven, for example, that a Knight’s Tour is possible on a traditional chess board.

However, solving the problem for n-player chess seems to be quite a bit more complex. But let me leave it as an open question for now.

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