# Sequence Of M X N Chess Games

In previous articles published on the Chess Variants Web pages of H. Bodlaender, I designed chessboards with initial arrangements of pieces and formulated rules for the sequence of N x N chess and 2N x 2N chess games, where N may be even or odd. In this article, following the same approach, I will design and formulate chess rules for the sequence of M x N chess games, where M and N are positive integers. To do so, I will divide the sequence of M x N chess into the following groups:

## Group 1

```M=1, N=j   j=1,2,3,...,r   (k, r positive integers)
N=i, N=1   i=1,2,3,...,k
```

The corresponding boards are the 1x1, 1x2, 1x3, 1x4,..., 1xr, and 2x1, 3x1, 4x1, 5x1, 6x1,..., kx1 boards; that is, boards one row of r squares, or respectively, one column of k squares. We may simply disregard this group as chessboards.

## Group 2

```M=2, N=j   j=2,3,4,...,r   (k,r positive integers)
N=i, N=2   i=2,3,4,...,k
```

The corresponding boards are the 2x2, 2x3, 2x4, ..., 2xr, and 3x2, 4x2, 5x2,..,kx2 boards. That is, boards of two rows of r squares, and two columns of k squares. This group will also be disregarded as chessboards.

## Group 3

```M=3, N=j   j=3,4,5,...,r   (k,r positive integers)
M=i, N=3   i=3,4,5,...,k
```

The corresponding boards are the 3x3, 3x4, 3x5, up to 3x6, and the 4x3, 5x3, 6x3 boards. They can be playable as Merels games but not as chess games because of the lack of sufficient rows/columns.

## Group 4

```M=4, N=j   j=4,5,6,...,r   (k,r positive integers)
M=i, N=4   i=4,5,6,...,k
```

The corresponding boards are illustrated on the diagrams for "Group 4". The 4x4, 4x5, 4x6, 4x7 and the 4x8 boards can be playable as Checkers or Merels games. but not as consistent chess games. On the other hand, the 5x4, 6x4, 7x4, up to the 8x4 games can be practically playable as chess games (see initial arrangements of pieces on the diagrams for "Group 4"). The Queen moves and captures diagonally one square at a time, while the King moves like the King in orthodox chess. In this group, there is no need for Knights, Bishops or Rooks. The pawns move one square and capture like the pawns in orthodox chess.

## Group 5

```M=5, N=j   j=5,6,7,...,r   (k, r positive integers)
M=i, N=5   i=5,6,7,...,k
```

The corresponding boards with initial arrangements of pieces are illustrated on diagrams for "Group 5". The games 5x5, 5x6, 5x7 up to 5x8, and the 6x5, 7x5, 8x5, 9x5, 10x5. 11x5 up to 12x5 games can be played as chess games, while the remaining are not. Now for the 5x5, 5x6, 5x7 and 5x8 chess games, the Queen moves and captures one square diagonally and the King moves and captures like the King in orthodox chess. For the 8x5, 9x5, 10x5, 11x5 and 12x5 the Queen moves like the Queen in orthodox chess. From the 8x5 to the 12x5 board, the Rooks can be introduced. The Queen move can be limited to up to three squares.

## Group 6

```M=6, N=j   j=6,7,8,...,k   (k,r positive integers)
M=i, N=6   i=6,7,8,...,r
```

The corresponding boards are illustrated on the diagrams for "Group 6". The 6x6, 6x7, 6x8, up to 6x9 boards can be playable as chess games, but the Queen's horizontal, vertical and diagonal move and capture are limited up to three squares at a time. The Rook and King move like those in orthodox chess. For the 7x6, 8x6, 9x6, 10x6 and 11x6 boards, the Queen, Knight and Rook move and capture like those in orthodox chess. For the 12x6, 13x6 and maybe the 14x6 boards, we can rule that the Pawn moves three squares at a time at the start of the game. Then after, it can move only one square at a time. The King can either move one square or two squares at a time. The Knight can move three squares up and one square horizontal and vice-versa or like the Knight in orthodox chess. Rook moves and captures like the Rook in orthodox chess. Beyond 14x6 the games are not practically playable as chess games.

## Group 7

```M=7, N=j   j=7,8,9,...,r   (k,r positive integers)
M=i, N=7   i=7,8,9,...,k
```

The corresponding boards relative to this group are illustrated on diagrams for "Group 7". The 7x7, 7x8 and 7x9 can be practically playable as chess games. Beyond 7x9, they are not really playable. Because there are more columns and less rows necessary for the consistency of play. The Queen's diagonal, horizontal and vertical moves and captures are limited to up to three squares at the time. There is no need of Rooks, Knights and Bishops. On the other hand, for the 8x8, 9x7, 10x7, 11x7 games, we can adopt the rules of orthodox chess and introduce the Rooks, Knight on both sides of the Queen, and the King. From 13x7, 14x7, 15x7, 16x7, 17x7 games, the Pawn can either move two or three squares at a time (optional) at the start of the game. The Knight moves and captures three squares up and one square horizontal and vice-versa, or like the Knight in orthodox chess. All the other pieces move and capture like those of orthodox chess. Castlings are necessary and there are performed like those of orthodox chess, and are symmetrical. The Pawns prise en passant is valid. Beyond 17x7 the games are not practically playable as chess games, because of the insufficient number of columns.

## Group 8

```M=8, N=j   j=8,9,10,...,k   (k,r positive integers)
M=i, N=8   i=8,9,10,...,r
```

The corresponding boards are illustrated on diagrams for "Group 8". The 8x9, 8x10, 8x11 can be simply playable like orthodox chess except that there are more Knights, Bishops and Queens than in orthodox chess. Castlings are respectively performed like the 9x9, 10x10, 11x11 chess (see my article Sequence of NxN chess games). Beyond 8x11, the games are not practically playable as chess games. The 9x8, 10x8, 11x8 and 12x8 boards can be played like orthodox chess, while for the 13x8, 14x8, 15x8, 16x8 boards, the Knight can either move three squares up and one square horizontal and vice-versa or like the Knight of orthodox chess. At the start of the Game, the Pawn can move three squares (optional) at a time, then after, one square at a time. The King moves two squares or one square at a time. Castling are done like orthodox chess. Beyond 17x8 the games are not practically playable as chess games, because of the great inequality between the number of Rows and the columns.

## Group 9

```M=9, N=j   j=9,10,11,...,k   (k,r positive integers)
M=i, N=9   i=9,10,11,...,r
```

The corresponding boards are illustrated on diagrams for "Group 9". The 9x9, 9x10, 9x11 can be played like the 9x9 chess with either the Missoum option (limited Queens move of up to three squares at a time, see my article Sequence of NxN chess games), or as the Queen in orthodox chess. Castlings are done,like the 10x10, and 11x11 chess (see my previously mentioned article). Beyond the 9x11 board, the games are not practically playable as chess games. The 10x9, 11x9, 12x9 up to 13x9 can be playable as the 9x9 chess. On the other hands, for the 13x9, 14x9, 15x9, 16x9, 17x9, 18x9 games, we can adopt that the knight moves and captures three squares up and one square horizontal and vice-versa. The pawn can move three squares at a time (optional) at the start of the game. Castlings are symmetrical. Beyond the 18x9 the games are not practically playable as chess games.

By continuing designing and formulating rules for the remaining MxN chess games, M greater than 9 and N greater than 9 (see diagrams at the end of this article) we can conclude as follows:

We observe from the aformentioned chess games and their corresponding chessboards with initial arrangement of pieces pawns, the required number of pieces necessary for the sequence of MxN chess games can be formulated as follows:

• #(PAWNS) = J For J = 4,5,6,....k, i=4,5,6,..., r, M=i, N=j
0 Otherwise
• #(KNIGHTS) = 2u For I greater than 6 and j greater than 5, M=i, N=j
0 Otherwise
• #(BISHOPS) = 2v For I greater than 6 and j greater than 9, M=i, N=j
0 Otherwise
• #(ROOKS) = 2 For M greater than 6, and N greater than 5, M=i, N=j
0 Otherwise
• #(QUEENS) = 1 Or 2 For i greater than 5 M=i, N=j
0 Otherwise

With the number of columns (N) of the chessboards increasing, The castlings will obey the same rules as the castling of the sequence of NxN chess games (see my article Sequence of NxN chess games).

Moreover, for M=i, N=j, i, j greater than 6, the sequence of MxN chess games for which i is greater than j and j is greater than 12 and less than 20, we adopt the rule that the Knight can either move and capture 3 squares up and 1 square horizontal and vice-versa or like the knight of orthodox chess. The Queen, Bishop, Rook move and capture like those of orthodox chess.

But the King can either move one square or two squares at a time. At the start of the game, the Pawn can move 3 squares, two or one square (optional). Then after, he can only move one square at a time. The prise en passant is still valid.

Remark: For M=i, N=9, i greater or equal to 9 and less than 20, the Queen may either move like the Missoum option (limited move of up to three squares) or like orthodox chess. We can conclude that The sequence of MxN chess games are more interesting than the NxN and the 2N x 2N Chess games, because it comprises them (example: for M=8, N=8, we have orthodox chess). Moreover, for each fixed number of columns N we can have many related chess games (example: for N=8 we have the 8x8, 9x8, 10x8, 11 x8, 12x8, 13x8, 14x8, 15x8, 16x8, 17x 8, 18x8, 19x8, etc.). Also to each fixed number of rows (M), will correspond many distinct chess games with distinct chess rules. While for the NxN chess or the 2N x 2N Chess, we are limited to only one chess game for each fixed number of columns N. In other words, the MxN chessboards are expandable in rows and columns. Also, in terms of chess openings, we can infer that they are more extensive than those of the NxN or the 2N x 2N chess games.

Remark: Imagine yourself a chess master playing simultaneously against many chess players, each one in front of a k x 8 chess board, k=8,9,10,11,12,13,14,15,16,17,18,19,20; that is playing simultaneously against one in front of an 8x8 chess board, another in front off a 9x8 chessboard, etc,.. Up to a last one in front of a 20x8 chessboard. So, for each specific chessboard, you have to concentrate on its specific chess openings. Suppose your are facing fifty players, each one playing the 8x8 chess, 60 others each playing the 9x8 chess, 100 others each one playing the 10x8 chess, 60 others each one playing the the 11x8 chess, etc... Then you will find yourself playing against a matrix of chess players, and the simultaneous chess play could be a headache for the master. We would like to know the performance of a Grandmaster in such a simultaneous chess playing.!!!

Written by A. Missoum. Edited by David Howe.
WWW page created: August 19, 1997.