The 2ⁿ X 2ⁿ Chess Games

n =Positive Integer

By A. Missoum

We will consider the geometric sequence of 2ⁿ x 2ⁿ Chess games, where n is an integer greater than zero. I will follow the line approach of the pieces (pawns) move and capture of the 8x8 chess. I will start the sequence with n=1.

1) For n=1, we have the 2 x 2 = 4 chess Game. The game is nothing else than a Merel game, where each player has two pawns, and the winner is one who first align diagonally, horizontally, or vertically two pawns, see board (1)

Board(1)

2) For n=2, we have the 2² x 2² =16 Chess game or the 4 x 4 chess game.

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The initial arrangement of pieces(pawns) is as in board(2). In this game each player has two pawns, one Firzan (Queen), which moves like the Firzan of the ancient Shatranj game( ie it moves and captures diagonally one square at a time), one king which moves and captures like the King of the 8x8 chess game. Two players , one with white pieces , resp the other with black pieces oppose each other. A promoted pawn can only be a firzan. This chess game is easy to play , but less important than the 8x8 chess game, because there is the lack of the queen, the castling, the prise en passant and other apects of the 8x8 chess games. However it belongs to the family of the 8x8 chess game.

3) For n=3, we have the 2³ X 2 ³ Chess game , or the 8x8 =64 chess game. This game is universally adopted by the orthodox chess federation and is the most popular game. The board is as follows:

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The initial arrangement of pieces(pawns) is as in board(3), where R=Rook, N=Knight, B=Bishop, K=King, and P=Pawn. The reader may found extensive literature about the 8x8 chess game.

The initial arrangement of the pieces(pawns) on the chessboard reflects that someone is playing against its mirror image. This is why the black king faces on the same file the white king.

4) For n=4, we have the 2⁴ x 2⁴ Chess game or the 16 x16 =256 chess game. The initial arrangement of pieces(pawns) is as in board(4). There are two players which face each other, one has white pieces, the other has black pieces . Each one has 2Rooks, 6Knights, 6 Bishops, 1 Queen, 1 King and 16 pawns , see board(4). The files are a, b, c,d, e, f, g, h, i, j, k , l m, n, o , p and the rows are from 1 to 16. The fundamental central squares are d8, e8, d9, e9.

The rules of pieces(pawns) moves and captures are as follows:

1) Pawns: At the start of the chess game, the Pawns have the option of moving, one, two or three squares at the time. Then after, they can only move one square or two squares at the time.

The prise en passant is like that of the 8x8 chess. The pawn captures like the pawn of the 8x8 chess When promoted, it can be any of the pieces Rook, Knight, Bishop, Queen , except for the King.

2) Knight : It has the option of moving 4 squares vertically and three squares horizontally and vice-versa,(ie the Knight can reach the frontier line in two moves and can protect the pawns near the central squares), or moving three squares vertically and two squares horizontally and vice-versa( here the pawn can reach the frontier line in three moves ) , or moving as the knight of the 8x8 chess .

3) The Rook, Bishop, and the Queen move and capture like the Rook, Bishop, and the Queen of the 8x8 chess

4) The King : It has the option of moving one square or two squares at the time It move diagonally, vertically or horizontally. The castling is done as follows:

Small Castling: The King on i1 moves to m1 and the Rook on p1 moves to l1

Great Castling: The King on i1 moves to h1 and the Rook on a1 moves to i1

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a b c d e f g h I J k l m n o p

Board(4)

The advantages of the 16 x 16 chess game is that it has the options of the Pieces(pawns )moves and captures of the 8x8 chess and new extended moves and captures of the King, the Knight and the Pawn. With the new extended moves and captures of the King, Knight and the pawn and the new castlings, the 16 x16 chess game cannot take longer time than the 8x8 chess game.

It can also be played as an 8x8 chess game , if we consider the board(4) a as a board of 64 squares, where each square represents four squares of the 16x16 chess. It can also be considered as a two handed 8x8 chess game , see board (5)). This two handed chess game can be played by four players one opoosing another, where , each one play on a chessboard of 8x8 squares However, one cannot simultaneously opposes the three other players. The game is played like the 8x8 chess game.

Two handed 8 x 8 chess game

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Board(5)

5) For n=5 we have a the 2⁵ x 2⁵ Chess game or the 32 x32 = 1024 chess game. The initial arrangement of pieces (pawns) is as in board(6). Two players oppose each other, one has white pieces, the other has black pieces. Each player has 2 Rooks, 14 Knights, 14 Bishops, 1 Queen,

1 King, and 32 pawns. The files are a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F.

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a b c d e f g h i j k l m n o p q r s t u v w x y z A B C D E F

Remark

Of course, the 32 x32 chessboard must be fairly large to have squares of the size of the squares of the standard 8x8 chessboard.

The rules of pieces(pawns) move and capture are as follows:

Pawn: At the start of the game, they have the option of moving one, two, three or four squares or 6 squares at a time. So, by moving 6 squares at a time, the pawn can reach the frontier line in 2 moves. After, they can only move one, two, or three squares or 4 squares at a time. They capture like the pawn of the 8x8 , or the 16x16 chess games . The prise en passant is the same as that of the pawn of the 8x8 or the 16x16 chess games. The pawns can be promoted as a Rook, a knight, a Bishop, or a Queen, except for the king.

Knight : It has the option of moving 7 squares vertically, and 4 squares horizontally and vice-versa( that is in two moves it can reach the frontier line and protect the pawns near the central squares), or 4 squares vertically, and 3 squares horizontally and vice-versa(like the extended move of the knight of the 16x16 chess), or moving like the knight of the 8 x 8 chess game.

The Rook, Bishop, and the Queen of the 32 x 32 chess move and capture like those of the 16x16 chess game or the 8x8 chess game.

King : It has the option of moving one, two, three or four squares at the time. It moves and captures vertically, horizontally, or diagonally. The castling is done as follows:

Small Castling: The king on the square q1 moves to y1 , and the Rook on the square F1 moves to x1.

Great Castling: The King on q1 moves to n1 and the Rook on a1 moves to o1.

With these afformentioned rules, the 32 x32 chess game comprises the Pieces (pawns) moves and capture of the 8x8 and the 16 x16 chess and extended moves which make it playable It do not take longer time than the 8x8 or the 16x16 chess games. The 32 x32 chess game can be also played as an 8x8 chess game, where the 32x32 chesboard can be transformed into an 8x8 chessboard where each square will represent 8 squares of the 32x32 chessboard. The 32 x32 chess game can be also played transformed into a 16x16 chess game , where each square will represent 4 squares of the 32x32 squares. It can also be considered as a two handed 16 x16 chess game(four players opposes each others, see board(6)) , or a 4 handed 8x8 chess game (eight players oppose each others) see board(7).

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BOARD(6)

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We have described the 4 handed 8x8 chess game which can be played between two players , 4 players, or eight players (however, I still did not experiment this sort of chess game.)

6) for N=6 we have the 2⁶ x 2⁶ chess games or the 64 x 64= 4096 chess game. The page cannot fill all the 64x64 chessboard squares. We will simply omit to gives the 64x64 ches board. However, from the previous 8x8, 16x16, 32x32 chess games, we can infer that each player has 2 rooks, 30 knight, 30 bishops, 64 pawns, 1 queen and 1 king. At the start of the chess game, the pawn has the option of moving 1, 2, 3, 4 6 or 8, 10, 12 or 14 squares at a time. After the start, it can move a maximum of 8 squares at a time. By moving 14 squares at the time it can reach the frontier line in two moves. The prise en pasant is like that of the 8x8 chess game. The Rook, Bishop and the Queen move and capture as those of the 8x8 chess game.

The knight can either move like the knights of the 8x8, 16x16, 32x32 chess games, or uses the following extended moves: 14 squares vertically and 8 squares horizontallyand vice-versa. So in two moves, the knight can reach the frontier line and protect its pawns near the frontier line.

The king can either move like the 8x8, 16x16, 32x32 chess games or use the extended moves:

eight squares at a time. If the files are abbreviated by: a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z, 1,2,3,4,5,6,7,8,9,10,11 , then for the small castling, the king moves from G1 to O1, the rook moves from 11 to N1.

For the great castling the king move from G1 to F1 and the rook move from a1 to G1.

So by continuing the sequence 2ⁿ x 2ⁿ Chess games, where n is a positive integer, we can infer that that the required numbers of Knights, respectively Bishops for this sequence of chess games is

X = An-1 + 2^n-1 With An-1 = An-2 + 2^n-2 Where A1,= 0, A2=2 and An-1 is the number of required Knights, respectively Bishops for the 2^n-1 x 2^n-1 Chess games.

So, for n= 4 , ie the 16 x16 chess game, A3= A2 + 2^4-2 = 2 + 4 = 6 Knights, respc 6 Bishops.

For n=5 , ie the 32 x32 chess game, A4 = A3 + 2^5-2 = 6 + 8 = 14 Knights, respc 14 Bihops.

For n=6, ie the 64x 64 chess game, A5 = A4 + 2^6-2 = 14 + 16 = 30 Knights, respec 30 Bishops.

For n=7, ie the 128 x 128 chess game, A6 = A5 + 2^7-2 = 30 + 32 = 62 Knights, resp 62 Bishops.

For n=8, ie the 256 x 256 chess game, A7 = A6 + 2^8-2 = 62 + 64 = 128 Knights, resp 128 Bishops

etc....

Now, if we consider the required small and great castlings for the sequence 2ⁿ x 2ⁿ we can infer that the small castling requires that the King moves 2^n-2 squares, and the Rook moves 2^n-2 squares

Thus for the 8 x8 chess , the King moves 2 squares, and the Rook moves 2 squares.

For the 16 x16 chess the King moves 4 squares, and Rook moves 4 squares.

For the 32 x32 chess , the King moves 8 squares, and the Rook moves 8 squares.

For the 64 x64 chess, the King moves 16 squares, and the Rook moves 16 squares

etc.....

For the great castling, the King moves 2⁰ square , ie one square, and the Rook moves (2ⁿ /2 ) + 1 squares . If we consider the handed chess game, then the sequence 2ⁿ x 2ⁿ chess games can also be played as 2 handed 2^n-1 x 2^n-1 chess games, or 4 handed 2^n-2 x 2^n-2 chess games, or 8- handed 2^{n- 3}x 2^n-3chess games, etc.... Or 2^n-1 handed 4 x 4 chess games.

Written by A. Missoum. WWW conversion by Hans Bodlaender.

WWW page created: March 17, 1997. Last modified: April 16, 1998.

The html-file was obtained from a document in Wordperfect, which was first converted to Word, and then from Word to html. Both conversions may have introduced errors, apologies for this. Also, early versions of Microsoft Explorer seem to display the tables in a wrong way - Netscape Explorer seems not to have this problem.

Comments

Geometric sequence of Chess Games. Chess variants as large as you want.

George Duke wrote on Fri, Aug 6, 2010 11:51 AM EDT:

How many Bishops? Black's generalization of Karakus is a kind of power series, exponential, because he can as well use 3x3 next after or skipping 2x2, which expands each space just from 1 to 4 pieces/spaces. [Reference is to Quadd Shogi Chess and to Giant Chess.] In 3x3 there are nine Queens where there was one, and so on. With all squares three-by-threes, an 8x8 board becomes 24^2, 576 squares. They are like Frolov's moving palace nine-space, out of one or as if one; but in pure Black each of the nine has the same residual piece-type without variation. That is the methodology of Karakus and the Black twosome, incidentally specified by that Xiangqi-size restricted area having 3 its specific instance. Missoum's approach here differs still in growing boards. 'N=3' is the first nontrivial case, where we find ourselves today; or as Missoum writes, ''The reader may have found extensive literature about the 8x8 chess game.'' Some systems characterize by period-doubling towards Chaos abruptly, but Chaos(or Chess) can prove manageable mathematically. Let's say the play is rather on a Missoum board n=8. You want to know how many Bishops you have. There is the formula used in the first (2005) of these three comments, taken from Missoum's text, arriving at 126 Bishops with n=8. (The example is Missoum's late in the article.) Why? Knowing the number of Bishops and full piece disposition allow interim values to be calculated, settled on, and adjusted player and computer. As for Knights, closely following or duplicating Bishop in numbers, notice Missoum's method of extending their reach with longer leap-lengths available by board sizes emerged. Sustainable growth; where governments fail we succeed.

John Smith wrote on Fri, Dec 19, 2008 12:07 PM EST:Good ★★★★

Your 4-player 16x16 Chess is a good idea.

George Duke wrote on Tue, Feb 22, 2005 05:24 PM EST:Good ★★★★

'GHI,LargeCV': In sequence 2(to n) x 2(to n), integer n > 0, 8x8 is the
first non-trivial case. It is where we find ourselves today. 'The reader
may (have) found extensive literature about the 8x8 chess game',
featuring mirror-image symmetry.--A. Missoum. With n=4, 16x16: h8,i8,h9,i9
central squares and requisites are for Pawn 1,2,3 initially; Knight as
(4,3)Antelope plus (3,2)Zebra. Where n=5, N is (7,4)Ibex + N + Antelope. 
(n=6) would be quite sufficient. However, exercise covers n=8; for example,
number of Bishops is given by X = (An-1)+2(to n-1) with (An-1) =
(An-2)+2(to n-2), and so on, so that 62+64= 126 Bishops.

Number of ratings: 2, Average rating: Good, Number of comments: 3