This is a cute variant on chess that I bought in Japan as a souvenir for a friend. I wasn’t expecting it to be anything but a cute example of Japanese children’s game design, but it actually proved really interesting. The game layout and images of the pieces can be seen here: it’s obviously just a cute little chess game. The rules are similar to chess but with simpler moves and an additional way of winning. The board is a 3×4 matrix, with sky at the top and forest at the bottom. These regions constitute the players’ “areas”, which are similar to the back row of a chess board.

The pieces
Each side has only four pieces:

  • The lion, essentially the king in chess, that can move one square in any direction, making it as powerful as the queen on this board.
  • The Elephant, essentially a bishop, that can move one square diagonally
  • The Giraffe, essentially a rook, that can move one square horizontally or vertically
  • The Chick, essentially a pawn, that starts in the middle of the second row and can move forward one space. If it reaches the enemy area the chick becomes a chicken (which in play my friend called a “magic chicken” ) that can move sideways or forward diagonally, and backwards one step
  • The objective
    Winning is possible by catching your opponent’s lion or by advancing your lion into your opponent’s area. Catching the lion is called “catch” and winning by advance is a “try.”

    Replacable pieces
    The main change from the standard rules of chess is the ability to return captured pieces to the board. After you catch your enemy’s piece you put it next to your side of the board and can then place it on the board instead of moving an existing piece. You have to place them in the order you caught them, and you can put them in any empty square. It wasn’t clear from the explanation but the rules stated that the chick has to advance into the opponent’s area to become magical, so we figured that means you can’t enchant a chick by placing it in your opponent’s area.

    Differences from chess
    Replacable pieces on a board this size makes for an interesting variation on chess. You can see from the diagram that the chicks start off facing each other and able to take each other. This is of no benefit to the person who starts because both players end up with a chick in hand, but one player has his lion in the middle of the board. The lion is strong, not weak, so this is a good position to start.

    This is the other main difference from standard chess. Because no piece can take from range the lion is the strongest piece on the board, and moving it out early is good. Also, the ability to win by a try makes aggressive use of the lion a good tactic. In fact, over 10 or 12 games I got the impression that this game encourages aggressive play.

    Another difference from chess is the use of diversionary tactics, especially using captured pieces. For example, if you threaten the king with a newly-placed elephant from one side of the board, the king will have to take it. This gives your king a free run up the board on the other side. I don’t think these tactics are used as much in standard chess.

    Three special rules
    This game is a training game for child chess players (the website is on the women’s chess society homepage), and as such intended to introduce children to chess culture. So it introduces three special rules for all players:

  • Always say “please be good to me” (yoroshiku onegaishimasu) before you play and “thank you” at the end
  • Never let anyone help you: play under your own effort
  • Never say “again”: in mistakes are the foundation of learning, so try to accept your errors and play without taking moves back
  • Each game takes only 5 to 10 minutes, so it’s a pretty quick learning curve compared to chess and it’s cute and fun to play. I recommend giving it a go. It also has me wondering what other variations on chess might be possible. For example, if you doubled or tripled the board size could you play chess like a modern war-game, with great sprawling battles, and wargame-style tactics? I’ve not really seen variations of chess based on changing the board size and distribution of pieces, but it appears to offer opportunities to use the basic rules of chess to play a very different style of game. An interesting idea…

    Professor Quiggin at Crooked Timber has introduced a discussion of what President Obama will do after the next senate elections with the title “Zero Dimensional Chess,” which led some of the bigger wankers in the audience to wonder how 0-D Chess could occur, and from there to ponder the deeper details of the mathematics of chess. This got me thinking – I used to study mathematics, a long time ago, and I’m interested in some of its weirder applications – so I went for a brief internet wander, and while pondering the nature of how one would lay out a chess game mathematically, I thought of an amusing quantum mechanics analogy. So, here’s a brief look at what I found, plus my silly analogy.

    Chess as Graph Theory

    Apparently there is a well-established notion in mathenatics of a Board, which is a finite subset of a lattice of a given size. I think a lattice will be subject to the standard rules of differential geometry in finite spaces – which I was “taught” in 4th year of University by a crazy Noumean chap, but didn’t understand a word of – but it is also subject to all the rules of graph theory, some definitions of which are laid out here. Moves can be described in terms of “tours” or “cycles”, with a tour of length c referred to as a “constant length tour,” and similar definitions for tours of all the squares in the board (commonly defined as Hamiltonian Tours or Cycles). Moves with a fixed length in two dimensions (such as the Knight) are called “Leapers.” So, for example, the night is a (1,2) Leaper.

    The mathematics of chess has been used to solve various forms of “Rook Problem,” which is the number of ways of placing k Rooks on a board such that no Rook can take any other Rook, for which closed solutions[1] can be found. But the fundamental problem appears to be the solution of problems called “series-movers” in which the aim is to take all of your opponent’s pieces. Unfortunately, the reference I found that introduces series movers (Kotesovec, 2009) is written in Czech, but for the abstract, so is kind of hard to read. The goal is to find solutions to such problems that are mathematically simple and to represent existing chess problems in terms of them. I’m not sure how “taking” a piece is expressed mathematically. According to Kotesovec, many of these chess problems have proven optimal solutions, but of course we know that ultimately chess problems are solved by path searching (checking all future moves), which implies that there is no optimal solution for most real-life chess situations.

    Interestingly, some of the work done on these problems has been done by Donald Knuth, who I think is the chap who invented LateX.

    Harvard University used to run a course on Chess and mathematics, which shows a lot of the terminology used and suggests that it relies on little more than specific applications of standard graph theory. The page seems to have the result that the number of solutions to the problem of “Mate in N” is a Fibonacci number, which is kind of surprising.

    It seems like the mathematics of chess is well understood and comes down to defining certain types of allowed paths on finite graphs, and using the usual range of graph theoretic methods to solve for optimal paths (the shortest number of moves) and to find algorithms for path finding.

    Chess as Quantum Mechanics

    Chess consists of a two dimensional space occupied by different kinds of particles (the pieces) that move according to strict physical laws, and are annihilated by anti-particles. There are also some strict rules about the creation of new particles, and a very strict relationship between the amount of movement that can occur and time. It struck me that if you define time in terms of turns, that is as if 1 unit of time were one turn, then you really have a two-dimensional finite geometry, with energy-constrained movement of particles according to strict laws, and a set of rules for whether they can exist in the same space at all (particles of the same type) or annihilate (particles of opposite type). Some particles (the King) exert action-at-a-distance (gravity/anti-gravity) and some may have a wormhole property (knights) and some can predict the future without breaking the energy constraints (pawns taking en passant, and maybe Rooks when castling). If you extend the dimension of the board to include time as a third dimension, then I think you can model chess as a general field theory[2] on a three dimensional space of finite size, with limited gravity, strict energy constraints[3] and complex rules about the creation, destruction and movement of particles. We even have a maximum speed of movement (the maximum number of squares a Queen can move). Of course, all your calculations would have to be done using differential geometry, and you’d probably have to invent some form of dark matter (invisible knights clustered 1000 per square), but it would be an interesting problem for someone with three brains and three lifetimes to waste on utter pointlessness.

    Zero-dimensional Chess

    Presumably 0D chess occurs on the “null” board ([0]x[0]) so is trivial and uninteresting, but can be defined rigorously, in that all moves are trivial. But I assume that a [0]x[0] board does not have even one point, so no pieces can be placed, preventing the game from occurring.

    fn1: For the non-mathematical reader, a closed solution is a solution which can be described by a formula rather than an approximation or an algorithm for getting the answer from a computer. Most interesting mathematical problems don’t have a closed solution. This is a very good reason not to do maths, if you value your sanity.

    fn2: If I have my terminology right, QFT is the merging of general relativity (which covers gravity) and quantum mechanics (which covers energy and matter).

    fn3: By energy constraints here I mean that only one move can occur per unit of time, so only one annihilation per unit of time. I think that mathematically this would act as a boundary constraint on solving Hamiltonian problems; and the finite nature of the space makes me wonder if the many-body-problem could be solved in a chess version of quantum field theory.