Fibonacci and Lucas numbers are numbers which obey the respective mathematical
recurrence relations: F_{n}=F_{n-1} + F_{n-2},
with F_{0}=1, F_{1}=1 and L_{n}= L_{n-1}
+ L_{n-2}, with L_{1}=1, L_{2}=3, n=1,2,3...

**Example**: the first 10 Fibonacci numbers are: 1,1,2,3,5,8,13,21,
34,55. That is; each Fibonacci number F_{n} is the sum of its two
successive preceding numbers F_{n-1}, and F_{n-2}. These
numbers have many interesting properties and applications and many elements
in nature obey them. Example: The rabbits and bees reproduction, the pineapple
design, etc... In this article, I will design and develop new chess games
based on Fibonacci and Lucas numbers.

The diagrams of fig(1), fig(2), fig(3), fig(4), fig(5), fig(6), fig(7), fig(8), and fig(9) represent nine distinct chess games with pieces(pawns) initial arrangement, and where the sides obey the Fibonacci , or Lucas numbers, or both. Fig(1) is a 14x14 Lucas-Fibonacci chess game. Fig(2) is a 10x10 squares Fibonacci chess game. Fig(3) is a 12x12 Lucas-Fibonacci chess game. Fig(4) is a 8x10 Golden rectangle Fibonacci chess game. Fig(5) is a 11x11 Cyclic rectangles Fibonacci chess game. Fig(6) is a 12x12 Lucas-Fibonacci chess game. Fig(7) is a 10x10 two handed Fibonacci chess game. Fig(8) is a 11x11 cyclic rectangle Fibonacci chess game. Fig(9) is a 10x10 Lucas-Fibonacci chess game. Now, except for the chess game of fig(7), the rules of play for the other chess games are almost similar to those of the usual 8x8 squares chess with the addition that the Pawn can either move forwards, or horizontally one square, and it captures diagonally. Each chess game has its number of Knights, Rooks, Bishops, Queens and Pawns. Because of the nature of the chessboard designs (fractal items designs), the process of identifying the files is as follows.

For example: consider the 12x12 Lucas-Fibonacci chess game of fig(3).
The chessboard squares coordinates are (x,y) with x=a,b,c,d,e,f,g,h,i,j,k,l,
and y=1,2,3,4,5,6,7,8,9,10,11,12. The quadrangles I, II, III, IV, V, VI,
VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX, XXI,
XXII, XXIII have the respective coordinates a1, d1, h1, k1, k4, d4, a4,
a8, d8, h8, k8, k12, h12, d12, a12, a6, j6, j7, a7. The remaining quadrangles
can be easily identified. The sides of the chessboard are Fibonacci, or
Lucas numbers or both. Example, for fig(3), the chessboard is a square
ABCD with sides equal to F_{n-1} + F_{n-2} + F_{n-1},
and DW=XC=CU=VB=BS=ZA=AL=F_{n-1}, and WX=UV=ZS=NL= F_{n-2},
and QR=MN=L_{n-2}. Where F_{n} are Fibonacci numbers and
L_{n} are Lucas numbers. The castlings are like those of the 8x8
squares chess.

Now, concerning the chess games of fig(2) and fig(7) there are some
special aspects to consider. Thus for the 10x10 Fibonacci chess game of
fig(2), all pieces (except the pawn) must cross the central square IJMK
(which comprises four sub squares) in order to reach one or the other side
of the opponent fields. The pawns move from square to square. In other
words, the zones I, II, III, IV constitutes barriers. Concerning the 10x10
two Handed chess game of fig(7), the game is as follows: white pieces have
the fields DKEU and UAJF, and black pieces have the fields HKCI, and GIBJ.
White pieces on QKEU move towards the opponent field GIBJ and vice-versa.
White pieces on the field KCIH move towards the opponent fields AUFJ(see
arrows on fig(7)). The white pieces(pawns) on DKEU cannot cross the segment
KH, and the pieces on AUFJ cannot cross the segment FJ . Similarly, black
pieces on KCIH cannot cross the segment KH, and those on the field GIBJ
cannot cross the segment FJ. In addition, the pawns on the column 10,9,
and 8(for white) can either move forwards or horizontally one square at
a time in order to move forwards through the column 7 towards the opponent
field. Similarly, the pawns on the columns a, b,c and d can move horizontally
or vertically in order to be on the column e and to move towards the opponent
field. This also applies for black pieces on the columns 1,2,3 and 4, and
j,i,h and g. The castlings are like those of the 8x8 squares chess. In
order to reach the opponent fields, all pieces(pawns) must cross the square
EFGH which comprises four sub squares, called the central squares. For
the chess game of fig(9), white pawns on b2 must move to the square c3
and then he can continue moving forwards along the column c towards the
opponent field. This also applies for the pawn on the square i2. Similarly,
black pawn on the square b9 must move to a8 and then he can continue moving
along this column. This also applies for the pawn on i9 who must move to
h8 and then continue along this column. Now all the nine chess games have
fractal designs, since their items can be repeated similarly (see the book
*Fractal everywhere* by B. MANDELBROT). The chess game of fig(4) can
be called 10x8 FIBONACCI GOLDEN SPIRAL CHESS GAME.

Using Fibonacci and Lucas numbers, I have designed and developed new chess games which have fractal nature. All these above mentioned chess games can be simply called FRACTAL SEQUENCE OF FIBONACCI -Lucas CHESS GAMES

Written by A. Missoum.

WWW page created: April 16, 1998.