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Construction of a Three-Dimensional Chess Board for Bishop Movement within the Forbidden Area with Vector Directives

Authors:
  • Chukwuemeka Odumegwu Ojukwu University, Uli

Abstract

Abstract- In this paper, the authors reviewed that the game of chess is the most highly played game in most part of the world. However, focusing on studies about bishop moves on a chessboard with some current and relevant state of knowledge regarding the two-dimensional and three-dimensional movement of a bishop/rook on a board with forbidden space. We also stated some basic results on vectors that are helpful in analyzing and studying the movements of a bishop on a three-dimensional chessboard. In addition, we showed that the three-dimensional boards with ๐ฅ non-attacking bishop can generate a three-dimensional bishop polynomial with a generating function ๐•ญ ๐ฑ,๐๐Ÿ,๐๐Ÿ . Furthermore, some problems on bishop moves were solved by applying bishop generating functions.
https://doi.org/10.31871/WJIR.9.1.27 World Journal of Innovative Research (WJIR)
ISSN:2454-8236, Volume-9, Issue-1, July 2020 Pages 101-105
101 www.wjir.org
๏€  Abstract- In this paper, the authors reviewed that the game of
chess is the most highly played game in most part of the world.
However, focusing on studies about bishop moves on a
chessboard with some current and relevant state of knowledge
regarding the two-dimensional and three-dimensional
movement of a bishop/rook on a board with forbidden space.
We also stated some basic results on vectors that are helpful in
analyzing and studying the movements of a bishop on a
three-dimensional chessboard. In addition, we showed that the
three-dimensional boards with ๎œ” non-attacking bishop can
generate a three-dimensional bishop polynomial with a
generating function ๎ฆœ๎ตซ๎œ ,๎›ฐ๎ซš,๎›ฐ๎ซ›๎ตฏ. Furthermore, some problems
on bishop moves were solved by applying bishop generating
functions.
Index Termsโ€”Chess movements; Vectors; three-dimensional
structures; Permutation;r-arrangement; combinatorial
structures; Disjoined chess board in three-dimension; Vector
generating function.
I. INTRODUCTION
The game of chess is highly played in most part of the world
and is considered as one of the most famous game that
originated from Northwestern India in 6th century from an
ancient Indian game called Chaturanga (Ashtapada). It is
played on an 8 ร— 8 board with a total of 64 square. Even
though, the Chaturanga game is not considered as a standard
chess game compared to todayโ€™s chess game that is played on
a similar board with black and white squares having two
players. However, in the 10th century chess game made a
wonderful break through by spreading to Persia, and further
extending to the Islamic Arabian Empire to Europe and to the
Asian continent in places where colonization and civilization
had moved to (Laisin, Chukwuma, Okeke, 2020). Although,
the playing of chess game was highly discouraged during this
period by rollers and Christian leaders, who considered the
game as an educator of warring techniques. However, with
complete license the game survived penalization from leaders
of the Christian churches for an uncountable number of times
during the end of the 15th Century (Laisin, Chukwuma,
Okeke, 2020). Thus, in the year 1880, the chess game
metamorphosed into different shape with completely
different appearances which has developed to todayโ€™s chess
game. However, the period from 1880 โ€“ 1950 was considered
M. Laisin, Department of Mathematics, Chukwuemeka Odumegwu
Ojukwu University, Anambra State, Nigeria
O. C. Okoli, Department of Mathematics, Chukwuemeka Odumegwu
Ojukwu University, Anambra State, Nigeria
C. A. Okaa-Onwuogu, Department of Mathematics, Chukwuemeka
Odumegwu Ojukwu University, Anambra State, Nigeria
E. I. Chukwuma. Department of Mathematics, Chukwuemeka Odumegwu
Ojukwu University, Anambra State, Nigeria
as the romantic era for chess game with many national chess
players round the globe. The players of this game depend on
tactics, sacrifices techniques, technical and dynamic playing
skills.
Furthermore, the game of chess is tagged as one of the most
played World games with many fans but its origin is not easy
to trace and it is completely controversial. Chess game has
many kinds of legends, plain guesses, and stories together
with a dispute over where the game originated from and when
it started. Most people accepted that it was not an individual
that discovered the establishment of the game of chess
because the game is too complicated with all its concepts and
rules for an individual mind to have discovered. Then, the
chess game was in a steady flux until Steinitz Wilhelm was
called as the first famous World official chess champion in
1886. However, a wise man from Indian under the roll of a
famous tyrannical King called King Shahram (Laisin,
Chukwuma, Okeke, 2020) was considered as one of the first
man who discovered the game of chess in India. In addition,
the wise man decided to explain to King Shahram how
important the game is if everyone in his kingdom is having a
good knowledge of how this game is played. Thus, the chess
game he formulated and described to King Shahram includes
the following; the King, his Queen, the knights, bishops,
rooks and pawns that is played on an 8 ร— 8 board by two
players. With this, the king was surprised to see that everyone
in his kingdom was very important and well represented in
the chess game. In-addition, King Shahram discovered that
the game shows a warring kingdom where everyone is
important and also playing their part to protect the King with
his kingdom. The king instructed everyone in his kingdom to
study and play the game (chess). King Shahram with
gratitude thanked the wise man and also instructed and
played the game with his generals, the game of chess started
from there.
However, the first World official Championship for chess
game occurred in the year 1886 in which Steinitz Wilhelm
turns out to be the overall best official champion of the world.
Since 1948, the world champion has been regulated by the
federation international des echecs (FIDE), the games
international governing body. Even though, there are
varieties of hypothesis about the history of chess and there is
no specific person who invented the popular game owning to
its complexity and thus there are many interesting legends
pertaining to its origin the game still stands as the most highly
played game in the world. In-addition, during 20th century,
the game of chess was revived with chess engine and
database invention such that the game can further be played
Construction of a Three-Dimensional Chess Board
for Bishop Movement within the Forbidden Area
with Vector Directives
M. Laisin, O. C. Okoli, E. I. Chukwuma, C. A. Okaa-Onwuogu.
Construction of a Three-Dimensional Chess Board for Bishop Movement within the Forbidden Area with Vector
Directives
102 www.wjir.org
with computers. Thus, in 1997, Kasparov played and lost the
World chess championship for a six times match to the IBMโ€™s
computer Deep Blue. Thus, the website and online chess
game was developed which later became so popular between
the years 2007 โ€“ 2018 with many players round the world. In
2007, the website of chess game (chess.com) was introduced
together with Lichess in 2010 and chess 24โ€™s website in 2014
(Laisin, Okeke, Chukwuma, 2020).
In addition, the game of chess is one of the oldest and most
popular board games, that is played by two-players on a
checkered board with 64 squares arranged in an 8 ร— 8 grid
with alternating colors (usually white and black) as shown in
fig1 below, with the current world champion as Magnus
Carlsen of Norway.
Fig. 1
Thus, each player begins with six different types of pieces: 8
pawns, 2 rooks, 2 bishops, 2 knights, a queen and the most
important one the king all in the same color (Laisin,
Chukwuma, Okeke). The objective of the game is to
checkmate the opponentโ€™s king by placing it under an
inescapable threat of capture. Each piece moves differently
with the most powerful one been the queen and the least
powerful the pawn.
The Bishop is a tall slender piece with pointed tip that has a
strange cut made into it and it sits next to the knight piece. It
has a value which is less than that of a rook. The bishop can
move as many unoccupied squares as possible diagonally as
far as there is no piece obstructing its path. Bishops capture
opposing pieces by landing on the square occupied by an
opponent piece. A bishop potential is maximized by placing
it on an open, long diagonal such that it will not be obstructed
by friendly pawn or an opponentโ€™s piece. A quick
development of the bishop can be achieved by a special move
called fianchetto. How a bishop gets along with pawns
determines if it is a good or bad bishop. If your bishop and
most of your pawn are on the same color squares then it is a
bad bishop because it has fewer squares available to it. Each
player starts out with two bishop pieces, each one residing on
its own color of square. In addition, a bishop moves
diagonally and captures a piece if that piece rests on a square
in the same diagonal (LAISIN, 2018; Laisin, and Uwandu,
2019).
However, the polynomial for nonattacking bishop has a very
good part to play in the theory of permutations with forbidden
positions (Laisin, Okoli, &Okaa-onwuogu, 2019;
Laisin&Uwandu, 2019; Laisin, 2018; LAISIN, 2018;
Skoch, 2015; Jay &Haglund, 2000; Herckman, 2006; Chung,
& Graham,1995) have shown that polynomial of either the
bishop/rook on a given board can be generated recursively by
applying cell decomposition techniques of Riordan ( Abigail,
2004; Riordan, 1980; Riordan, 1958).
Furthermore, Laisin, Okoli, &Okaa-onwuogu, 2019;
Laisin&Uwandu, 2019; LAISIN, 2018; Laisin&Ndubuisi,
2017; Jay Goldman, and James Haglund, 2000 studied,
examined and investigated movement of bishop/rook on a
board with forbidden area to develop techniques for
polynomials using generating functions. Thus, to determine
the solutions for fundamental problems by examining the
existence, enumeration and structure of the bishop/rook
generating function on an ๎Šร—๎‰ board. Informally, the
bishop moves for a nonattacking bishop can be classi๏ฌed into
three categories: search, generation, and enumeration
(LAISIN, 2018; Bona, 2007).
The polynomials generated by nonattacking bishop provide a
way of enumeration for permutation with forbidden positions
that was developed by Kaplansky, and Riordan, 1946.
LAISIN, 2018; Nickolas and Feryal, 2009 generalized these
properties and theorems for two-dimensional bishop
polynomials. However, more advanced dimensions were
partially done for the three-dimensional cases (Laisin, Okeke,
Chukwuma, 2020; Laisin, Chukwuma, Okeke, 2020; Zindle,
2007).
Now, we shall be focusing on the three-dimensional boards
for non-attacking bishop to generate a three-dimensional
bishop polynomial within the forbidden area (Michaels,
2013; Shanaz, 1999). In addition, we shall apply the bishop
generating functions for a twoโ€“ dimensional and the
three-dimensional cases on disjoined sub-boards.
A. Basic definitions
A ring R is a set with two laws of composition +and ร— called
addition and multiplication, which satisfy these axions;
a. With the composition +, ๎œด is an abelian group,
with identity denoted by 0. This abelian group
is denoted by ๎œด+
b. Multiplication is associative and has an identity
denoted by 1.
c. Distributive law for all ๎œฝ,๎œพ,๎œฟ ๎— ๎œด,๎–œ
๏ˆบ๎œฝ+๎œพ๏ˆป๎œฟ=๎œฝ๎œฟ +๎œพ๎œฟ & ๎œฟ๏ˆบ๎œฝ+๎œพ๏ˆป=๎œฟ๎œฝ +
๎œฟ๎œพ(Artin, 1991)
A chess board B of a ring is a chess board which is closed
under the operations of addition subtraction, and
multiplication and which contains the first placement
(๎œพ0๏ˆบ๎œค๏ˆป= 1). A bishop polynomial with forbidden positions
is denoted as ๎ฅ€(๎”,๎œค), given by
๎ฅ€๏ˆบ๎”,๎œค๏ˆป=๎ท๎œพ๎…(๎œค)๎”๎…
๎‡
๎…=1
,
where ๎ฅ€(๎”,๎œค) has coefficients ๎œพ๎…(๎œค) representing the
number of ways of bishopโ€™s placements on B. Furthermore,
on m ร—n board B, we have ๎œพ0๏ˆบ๎œค๏ˆป= 1 and the coefficients
are determined by
๎ฅ€๏ˆบ๎”,๎œค๏ˆป=๎ท๏‰€๎‰
๎‡๏‰๏‰€๎Š
๎‡๏‰
min ๎ผŒ(๎‰,๎Š)
๎‡=0
๎‡!๎”๎‡
=๎ท๎Š!๎‰!
๎‡!๏ˆบ๎Š ๎ต† ๎‡๏ˆป! (๎‰ ๎ต†๎‡)!
min ๎ผŒ(๎‰,๎Š)
๎‡=0
๎”๎‡.
https://doi.org/10.31871/WJIR.9.1.27 World Journal of Innovative Research (WJIR)
ISSN:2454-8236, Volume-9, Issue-1, July 2020 Pages 101-105
103 www.wjir.org
(LAISIN, 2018)
B. Definition
Suppose that B be is an ๎‰ร—๎‰ board and its diagonal
denoted by ๎ฅ๎Ÿ  and let;
๎œจ๏ˆบ๎•1,๎•2,๎‡ฅ,๎•๎‡๏ˆป=๎ท๎‚๏ˆบ๎‰1,๎‰2,๎‡ฅ,๎‰๎‡๏ˆป๎•1
๎‰1๎•2
๎‰2๎‡ฅ๎•๎‡
๎‰๎‡
๎— ๎œญ[๏ˆพ๏ˆบ๎•1,๎•2,๎‡ฅ,๎•๎‰๏ˆป๏ˆฟ]
Then, the ๎ฅ๎Ÿ  is the power series in a single variable y defined
by
๎ฅ๎Ÿ =๎ฅ๎Ÿ ๏ˆบ๎•๏ˆป=๎ท๎‚(๎‰,๎‰,๎‡ฅ,๎‰)๎•๎‰
๎‰
(LAISIN, 2018)
C. Standard Basis
Suppose ๎ฅ๎Ÿ =๎œจ๎‰ is the space of diagonal vectors and let
the diagonal vector be denote ๎๎… with ๎œพ0๏ˆบ๎œค๏ˆป= 1 in the ๎…๎๎„
position and zeros elsewhere. Then, the m vectors ๎๎… from a
basis for ๎œจ๎‰ . That is every vector ๎œบ=๏ˆบ๎”1,๎”2,๎‡ฅ,๎”๎‡๏ˆป has
the unique expression;
๎œบ๎œง =๎”1๎1+ ๎”2๎2+๎šฎ+ ๎”๎‰๎๎‰
as the linear combination of ๎œง= (๎1,๎2,๎‡ฅ,๎๎‰)
(LAISIN, 2018).
D. Vector quantities
These are those quantities that have both magnitude and
direction.
E. Resultant vector
The resultant vector is that single vector which would have
the same effect in magnitude and direction as the original
vectors acting together.
F. Parallelogram law
The parallelogram law of vectors states that if two vectors are
represented in magnitude and direction by the adjacent sides
of a parallelogram drawn from the point of intersection of the
vectors represents the resultant vector in magnitude and
direction.
Fig 2
By cosine rule we have;
๎กผ๎กพ =๏ˆพ(๎œฒ๎œณ)2+ (๎œณ๎œด)2๎ต†2๏ˆบ๎œฒ๎œณ๏ˆป๏ˆบ๎œณ๎œด๏ˆป๎œฟ๎‹๎๎Ÿšยฐ๏ˆฟ1
2
=๏ˆพ(๎œฒ๎œณ)2+ (๎œณ๎œด)2+ 2๏ˆบ๎œฒ๎œณ๏ˆป๏ˆบ๎œณ๎œด๏ˆป๎œฟ๎‹๎(180ยฐ๎ต† ๎Ÿ ยฐ)๏ˆฟ1
2
By sine rule we have;
๎๎…๎Š๎Ÿ ยฐ
๎œต๎œด =๎๎…๎Š๎Ÿฎยฐ
๎œฒ๎œต =๎๎…๎Š๎Ÿšยฐ
๎œฒ๎œด
๎กผ๎กพ =๎œฒ๎œต๎๎…๎Š๎Ÿšยฐ
๎๎…๎Š๎Ÿฎยฐ=๎œต๎œด๎๎…๎Š๎Ÿšยฐ
๎๎…๎Š๎Ÿ ยฐ
II. THEOREM
If the movement on fig. 2.1 is a rook movement, then the
angle between the vertical and the horizontal rook movement
must be 90ยฐ๎ต†๏ˆพ๎Ÿ™,๎Ÿš,๎Ÿ ๏ˆฟ.
Fig. 2.1 (Laisin, Okeke, Chukwuma; 2020).
A. Theorem
Suppose M is the rook movement and the distance from a
fixed point ๎œจ(๎’1,๎’2,๎’3) to any point ๎œถ(๎1,๎2,๎3). Then, the
๎พŽ๎œฏ is a unit vector in the direction of the rook movement
๎กฒ๎ข€ = ๎กน (Laisin, Okeke, Chukwuma; 2020).
B. Theorem: Angle between two vectors
Suppose ๎Ÿ› is the angle between two vectors. Then, the sum of
products of the corresponding direction of the rook
movement is the cosines from the two generated vectors by
the restricted area. Hence, the rook movement is a
three-dimensional structure (Laisin, Chukwuma, Okeke;
2020).
C. Theorem
The number of ways to arrange n bishops among m positions
(๎‰ ๎ต’ ๎Š)through an angle of ๎Ÿ =450for movement on the
board with forbidden positions is;
๎ฅ€๏ˆบ๎•,๎œค๏ˆป๎œฒ๏ˆบ๎‰, ๎Š๏ˆป=๎ท๏ˆบ๎ต†1๏ˆป๎‡๎œพ๎‡
๎Ÿ 
๎Š
๎‡=0
๎œฒ๏ˆบ๎‰๎ต†๎‡, ๎Š๎ต†๎‡๏ˆป
Proof
The proof of theorem 3.1 follows immediately from Lemma
2.1 in arranging n bishops among m positions (๎‰ ๎ต’
๎Š)through a direction of movement in an angle of 450 with
forbidden positions is as follows;
Case 1๎‰>๎Š
๎ฅ€๏ˆบ๎•,๎œค๏ˆป๎œฒ๏ˆบ๎‰, ๎Š๏ˆป=๎œฒ๏ˆบ๎‰, ๎Š๏ˆป๎ต†๎œพ1
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎‰๎ต†1,๎Š๎ต†1๏ˆป
+๎œพ2
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎‰๎ต†2 ,๎Š๎ต†2๏ˆป
๎ต†๎œพ3
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎‰๎ต†3,๎Š๎ต†3๏ˆป+ . . . (๎ต†1)๎‰๎œพ๎‰
๎Ÿ (๎œค)๎œฒ๏ˆบ๎‰๎ต†๎Š,0๏ˆป
=๎ท๏ˆบ๎ต†1๏ˆป๎‡๎œพ๎‡
๎Ÿ 
๎‰
๎‡=0
๎œฒ๏ˆบ๎‰๎ต†๎‡,๎Š๎ต†๎‡๏ˆป
Case 2 ๎‰=๎Š
๎ฅ€๏ˆบ๎•,๎œค๏ˆป๎œฒ๏ˆบ๎Š, ๎Š๏ˆป=๎œฒ๏ˆบ๎Š, ๎Š๏ˆป๎ต†๎œพ1
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎Š๎ต†1,๎Š๎ต†1๏ˆป
+๎œพ2
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎Š๎ต†2,๎Š๎ต†2๏ˆป
๎ต†๎œพ3
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎Š๎ต†3,๎Š๎ต†3๏ˆป+ . . . ๏ˆบ๎ต†1๏ˆป๎Š๎œพ๎Š
๎Ÿ (๎œค)๎œฒ๏ˆบ0,0๏ˆป
=๎ท๏ˆบ๎ต†1๏ˆป๎‡๎œพ๎‡
๎Ÿ ๏ˆบ๎œค๏ˆป๎œฒ๏ˆบ๎Š๎ต†๎‡,๎Š๎ต†๎‡๏ˆป
๎Š
๎‡=0
( LAISIN, 2018; Abigail, 2004) ๎—–
D. Theorem(n-disjoint sub-boards with
movements through an angle of ๎ซ๎ซž๎ซ™)
Suppose, ๎œค isan ๎Šร—๎Šboard ofdarkenedsquares with bishops
that move through a direction of an angle of ๎Ÿ =450then,
๎ฅ€๏ˆบ๎”,๎œค๏ˆป for thedisjointsub-boards is;
Construction of a Three-Dimensional Chess Board for Bishop Movement within the Forbidden Area with Vector
Directives
104 www.wjir.org
๎ฅ€๏ˆบ๎”,๎œค๏ˆป=๎ท๎ท‘๎ฃฒ๎œค๎†,๎œญ(๎”)๎…๎œพ๎…๎Ÿ ๎ตซ๎œค๎†๎ตฏ, ๎†
๎Š
๎‡=0
๎Š
๎…=0
= 1,2, ๎‡ฅ๎Š ( LAISIN, 2018)
III. RESULTS
THEOREM 3.1
Suppose B is a three-dimensional disjoined bishop boards
with forbidden squares and let ๎ˆ non-attacking bishop
movements generate a bishop function, then, the generating
function is;
๎ฅ€๎ตซ๎”,๎œค1,๎œค2๎ตฏ=๎ฅ€๏ˆบ๎”,๎œค1๏ˆป๎”๎…
+ ๎ฅ€๎ตซ๎”,๎œค๎†+1๎ตฏ๎Ÿฎ๎๎…๎ƒ๎Š๏ˆบ๎ขœ๎ซš ร— ๎ขœ๎ซ›๎Žฎ ๎ขœ๎ซœ๏ˆป๎”๎…,
๎—Š ๎†= 1,2, ๎šฎ,๎ˆ ๎ต† 1
where the number of disjoint boards is denoted as ๎Ÿฎ.
Proof
Let ๎œค1 be a two-dimensional ๎Šร—๎Š board, with
๎ˆnon-attackingbishop, then, we have the following;
๎ฅ€๏ˆบ๎”,๎œค1๏ˆป=๎ท๏ˆบ๎”๏ˆป๎…๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป=
๎ˆ๎ต†1
๎…=0
1 + ๎”๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป
+๎”2๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป+ . . .+๎”๎๎ต†1๎œพ๎ˆ๎ต†1
๎Ÿ ๏ˆบ๎œค1๏ˆป
Now, considering ๎œค1 as an 8 ร— 8 two-dimensional
chessboard, then, the maximum number of bishops moves on
the forbidden squares for non-attacking bishops is as follows;
fig. 3.1
Thus, we have the following bishop placements for a
two-dimensional chessboard;
๎œพ0๏ˆบ๎”,๎œค1๏ˆป , ๎œพ1๏ˆบ๎”,๎œค1๏ˆป,๎œพ2๏ˆบ๎”,๎œค1๏ˆป,๎œพ3๏ˆบ๎”,๎œค1๏ˆป,๎œพ4๏ˆบ๎”,๎œค1๏ˆป,
๎œพ5๏ˆบ๎”,๎œค1๏ˆป, ๎œพ6๏ˆบ๎”,๎œค1๏ˆป,๎šฎ,๎œพ13๏ˆบ๎”,๎œค1๏ˆป, then we have;
๎ท๏ˆบ๎”๏ˆป๎…๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป=
13
๎…=0
1 + ๎”๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป
+๎”2๎œพ๎…๎Ÿ ๏ˆบ๎œค1๏ˆป+ . . .+๎”๎๎ต†1๎œพ13
๎Ÿ ๏ˆบ๎œค1๏ˆป
=๎œพ0๏ˆบ๎”,๎œค1๏ˆป+๎œพ1๏ˆบ๎”,๎œค1๏ˆป๎”+ ๎œพ2๏ˆบ๎”,๎œค1๏ˆป๎”2+๎œพ3๏ˆบ๎”,๎œค1๏ˆป๎”3
+๎œพ4๏ˆบ๎”,๎œค1๏ˆป๎”4+๎œพ5๏ˆบ๎”,๎œค1๏ˆป๎”5
+ ๎œพ6๏ˆบ๎”,๎œค1๏ˆป๎”6+ ๎œพ7๏ˆบ๎”,๎œค1๏ˆป๎”7
+ ๎œพ8๏ˆบ๎”,๎œค1๏ˆป๎”8+ ๎œพ9๏ˆบ๎”,๎œค1๏ˆป๎”9
+ ๎œพ10๏ˆบ๎”,๎œค1๏ˆป๎”10 + ๎œพ11๏ˆบ๎”,๎œค1๏ˆป๎”11
+ ๎œพ12๏ˆบ๎”,๎œค1๏ˆป๎”12 +๎œพ13๏ˆบ๎”,๎œค1๏ˆป๎”13
= ๎ฅ€๏ˆบ๎”,๎œค1๏ˆป
Thus, this is the maximum number of bishop placements on
an 8 ร— 8 two-dimensional chessboard. Similarly, as the
board increases in size the total number of nonattacking
bishops will also increase as new diagonals are introduced.
Then, the following bishop placements can now follow;
๎œพ0๏ˆบ๎”,๎œค1๏ˆป๎œพ1๏ˆบ๎”,๎œค1๏ˆป,๎œพ2๏ˆบ๎”,๎œค1๏ˆป,๎œพ3๏ˆบ๎”,๎œค1๏ˆป,๎œพ4๏ˆบ๎”,๎œค1๏ˆป,
๎œพ5๏ˆบ๎”,๎œค1๏ˆป, ๎œพ6๏ˆบ๎”,๎œค1๏ˆป,๎šฎ,๎œพ๎ˆ๎ต†1๏ˆบ๎”,๎œค1๏ˆป respectively.
However, the two-dimensional bishop boards generate the
bishop function with the generating function.
๎ฅ€๏ˆบ๎”,๎œค1๏ˆป=๎ท๏ˆพ๎œพ0๏ˆบ๎”,๎œค1๏ˆป , ๎œพ1๏ˆบ๎”,๎œค1๏ˆป,
๎ˆ๎ต†1
๎…=0 ๎œพ2๏ˆบ๎”,๎œค1๏ˆป, ๎šฎ, ๎œพ๎ˆ๎ต†1๏ˆบ๎”,๎œค1๏ˆป๏ˆฟ๎”๎…
=๎œพ0๏ˆบ๎”,๎œค1๏ˆป+ ๎œพ1๏ˆบ๎”,๎œค1๏ˆป๎”+๎œพ2๏ˆบ๎”,๎œค1๏ˆป๎”2+ ๎šฎ
+ ๎œพ13๏ˆบ๎”,๎œค1๏ˆป๎”๎ˆ๎ต†1
๎ฅ€๏ˆบ๎”,๎œค1๏ˆป
=๎ท(1, 14,78,220,330,126,28, 1, 15,10, 1, 3,2, 1)๎”๎…
13
๎…=0
Thus,๎ฅ€๏ˆบ๎”,๎œค1๏ˆป is the maximum bishop generating function
for an ๎Šร—๎Š two-dimensional board. Hence, a projection of
the vector of (๎ˆ๎ข+๎ˆ๎ข +๎ˆ๎ข‘on the vector ๎ข+๎ข+๎ข‘ gives a
unit vector 1
๎Žพ3๏ˆบ๎ข+๎ข+๎ข‘๏ˆป with a bishop projection movement
as 3๎ˆ
๎Žพ3 ๎‘๎Š๎…๎๎.
Now, considering the bishop movement on a 3-dimensional
space, denoted as ๎ฅ€๏ˆบ๎”,๎œค1,๎œค2๏ˆป. We can now construct a
3-dimensional chess board that decomposes into disjoint
boards with vectors ๎ขœ๎ซš , ๎ขœ๎ซ›and๎ขœ๎ซœ respectively.
fig. 3.2
then, it follows the placement of bishops on the chess board,
thus we have;
๎ฅ€๎ตซ๎”,๎œค1,๎œค2๎ตฏ=๎ฅ€๏ˆบ๎”,๎œค1๏ˆป๎”๎…
+ ๎ฅ€๎ตซ๎”,๎œค๎†+1๎ตฏ๎Ÿฎ๎๎…๎ƒ๎Š๏ˆบ๎ขœ๎ซš ร— ๎ขœ๎ซ›๎Žฎ๎ขœ๎ซœ๏ˆป๎”๎…,
๎—Š ๎†= 1,2, ๎šฎ,๎ˆ ๎ต† 1
= [ ๏ˆพ๎œพ0๏ˆบ๎œค1๏ˆป+ ๎œพ1๏ˆบ๎œค1๏ˆป+๎šฎ+๎œพ๎ˆ๎ต†1๏ˆบ๎œค1๏ˆป๏ˆฟ๎”๎…
+ ๎Ÿฎ๏ˆพ๎œพ0๏ˆบ๎œค2๏ˆป+ ๎œพ1๏ˆบ๎œค2๏ˆป+๎œพ2๏ˆบ๎œค2๏ˆป+๎šฎ
+๎œพ๎ˆ๎ต†1๏ˆบ๎œค2๏ˆป๏ˆฟ๎”๎…
=๎ท๏ˆบ๎”๏ˆป๎…๎œพ๎…๎Ÿ 
๎ˆ๎ต†1
๎…=0
(๎ขž,๎œค1,๎œค2)
=๎ท๎ท‘๎œพ๎…๎Ÿ ๎ตซ๎œค๎†๎ตฏ
๎ˆ๎ต†1
๎œต=0
๎ˆ๎ต†1
๎…=0
๎ฃฒ๎œค๎†,๎œต๎”๎…
+๎ท๎ท‘๎ฃฒ๎œค๎†,๎œต๎œพ๎…๎Ÿ 
๎ˆ๎ต†1
๎œต=0
๎ˆ๎ต†1
๎…=0
(๎œค๎†+1)๎Ÿฎ๎๎…๎ƒ๎Š( ๎ขœ๎ซš
ร— ๎ขœ๎ซ›๎Žฎ ๎ขœ๎ซœ)๎”๎…
=๎ท๎ท‘๎ฃฒ๎œค๎†,๎œต๎œพ๎…๎Ÿ ๎ตซ๎œค๎†๎ตฏ
๎ˆ๎ต†1
๎=0
๎ˆ๎ต†1
๎…=0
๏ˆพ1 + ๎Ÿฎ๎๎…๎ƒ๎Š๏ˆบ๎ขœ๎ซš ร— ๎ขœ๎ซ›๎Žฎ ๎ขœ๎ซœ๏ˆป๏ˆฟ๎”๎…, ๎—Š ๎†
= 1,2, ๎šฎ๎Žฎ,๎ˆ ๎ต† 1;
where the number of disjoint boards is denoted as ๎Ÿฎ.
๎—–
https://doi.org/10.31871/WJIR.9.1.27 World Journal of Innovative Research (WJIR)
ISSN:2454-8236, Volume-9, Issue-1, July 2020 Pages 101-105
105 www.wjir.org
IV. NUMERICAL APPLICATIONS
Example 4.1
A rice farm of eight square kilometers is to be worked by a
maximum number of controllers and each controller works
diagonally. If each diagonal plot can only be worked by one
controller. How many controllers can be given this
assignment and how in many ways?
Solution
Considering the rice farm of eight square kilometers, then, we
have a maximum of 64 square plots with forbidden squares of
non-attacking controllers is as follows;
Rice farm 4.1
Thus, we have 14 controller placements for plots as on the
rice farm 4.1;
๎œพ0๏ˆบ๎”,๎œค1๏ˆป , ๎œพ1๏ˆบ๎”,๎œค1๏ˆป,๎œพ2๏ˆบ๎”,๎œค1๏ˆป,๎œพ3๏ˆบ๎”,๎œค1๏ˆป,๎œพ4๏ˆบ๎”,๎œค1๏ˆป,
๎œพ5๏ˆบ๎”,๎œค1๏ˆป, ๎œพ6๏ˆบ๎”,๎œค1๏ˆป,๎šฎ,๎œพ13๏ˆบ๎”,๎œค1๏ˆป, where the controller is
denoted by ๎œพ๎…:๎…= 0, 1, 2, 3, ๎šฎ,13 then we have;
๎ฅ€๏ˆบ๎”,๎œค1๏ˆป=๎œพ0๏ˆบ๎”,๎œค1๏ˆป+ ๎œพ1๏ˆบ๎”,๎œค1๏ˆป๎”+๎œพ2๏ˆบ๎”,๎œค1๏ˆป๎”2+ ๎šฎ
+ ๎œพ13๏ˆบ๎”,๎œค1๏ˆป๎”๎ˆ๎ต†1
=๎ท(1, 14,78,220,330,126,28, 1, 15,10, 1, 3,2, 1)๎”๎…
13
๎…=0
= 1 + 14๎”+78๎”2+220๎”3+330๎”4+126๎”5+28๎”6
+๎”7+
15๎”8+10๎”9+๎”10 + 3๎”11 + 2๎”12 +๎”13
= 1 + 14๎œฒ๏ˆบ13,13๏ˆป+78(๎œฒ๏ˆบ12,12๏ˆป)2+220(๎œฒ๏ˆบ11,11๏ˆป)3
+330(๎œฒ๏ˆบ10,10๏ˆป)4+126(๎œฒ๏ˆบ9,9๏ˆป)5
+28(๎œฒ๏ˆบ8,8๏ˆป)6+ (๎œฒ๏ˆบ7,7๏ˆป)7+15(๎œฒ๏ˆบ6,6๏ˆป)8+10(๎œฒ๏ˆบ5,5๏ˆป)9
+ (๎œฒ๏ˆบ4,4๏ˆป)10 + 3(๎œฒ๏ˆบ3,3๏ˆป)11 +
2(๎œฒ(2,2))12 + (๎œฒ(1,1))13
=65,529,875 ๎“๎œฝ๎’๎
Thus, the rice farm can be given to 14 controllers and in
65,529,875 ways.
V. CONCLUSION
The polynomials generated by bishop movements on a
forbidden space are very interesting for both two and
three-dimensional cases. We were able to realize the
objectives of this paper by showing that, the generating
function for ๎ˆ non-attacking bishop movements generate a
bishop function. Finally, we applied this formula to solve the
rice farm problem.
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