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

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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 𝐥 non-attacking bishop can generate a three-dimensional bishop polynomial with a generating function 𝕭 𝐱,𝐁𝟏,𝐁𝟐 . 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 non-attacking bishop can
generate a three-dimensional bishop polynomial with a
generating function ,,. Furthermore, some problems
on bishop moves were solved by applying bishop generating
functions.
Index TermsChess 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 × 8 board with a total of 64 square. Even
though, the Chaturanga game is not considered as a standard
chess game compared to today’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
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 today’s chess
game. However, the period from 1880 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 × 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 IBM’s
computer Deep Blue. Thus, the website and online chess
game was developed which later became so popular between
the years 2007 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 24’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 × 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 opponent’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 opponent’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 × board. Informally, the
bishop moves for a nonattacking bishop can be classiﬁed 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 two 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 × called
addition and multiplication, which satisfy these axions;
a. With the composition +, is an abelian group,
with identity denoted by 0. This abelian group
is denoted by +
b. Multiplication is associative and has an identity
denoted by 1.
c. Distributive law for all ,,  ,
+= + & += +
(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
(0= 1). A bishop polynomial with forbidden positions
is denoted as (,), given by
,=()
=1
,
where (,) has coefficients () representing the
number of ways of bishop’s placements on B. Furthermore,
on m ×n board B, we have 0= 1 and the coefficients
are determined by
,=
min (,)
=0
!
=!!
!  ! ( )!
min (,)
=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 × board and its diagonal
denoted by and let;
1,2,,=1,2,,1
12
2
 [1,2,,]
Then, the is the power series in a single variable y defined
by
==(,,,)
(LAISIN, 2018)
C. Standard Basis
Suppose = is the space of diagonal vectors and let
the diagonal vector be denote with 0= 1 in the 
position and zeros elsewhere. Then, the m vectors from a
basis for . That is every vector =1,2,, has
the unique expression;
 =11+ 22++
as the linear combination of = (1,2,,)
(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;
 =()2+ ()22°1
2
=()2+ ()2+ 2(180° °)1
2
By sine rule we have;
°
 =°
 =°

 =°
°=°
°
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 90°,,.
Fig. 2.1 (Laisin, Okeke, Chukwuma; 2020).
A. Theorem
Suppose M is the rook movement and the distance from a
fixed point (1,2,3) to any point (1,2,3). Then, the
 is a unit vector in the direction of the rook movement
 = (Laisin, Okeke, Chukwuma; 2020).
B. Theorem: Angle between two vectors
Suppose 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
(  )through an angle of =450for movement on the
board with forbidden positions is;
,, =1
=0
, 
Proof
The proof of theorem 3.1 follows immediately from Lemma
2.1 in arranging n bishops among m positions ( 
)through a direction of movement in an angle of 450 with
forbidden positions is as follows;
Case 1>
,, =, 1
1,1
+2
2 ,2
3
3,3+ . . . (1)
(),0
=1
=0
,
Case 2 =
,, =, 1
1,1
+2
2,2
3
3,3+ . . . 1
()0,0
=1
,
=0
( LAISIN, 2018; Abigail, 2004)
D. Theorem(n-disjoint sub-boards with
movements through an angle of )
Suppose, isan ×board ofdarkenedsquares with bishops
that move through a direction of an angle of =450then,
, 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
,=,(),
=0
=0
= 1,2, ( LAISIN, 2018)
III. RESULTS
THEOREM 3.1
Suppose B is a three-dimensional disjoined bishop boards
with forbidden squares and let non-attacking bishop
movements generate a bishop function, then, the generating
function is;
,1,2=,1
+ ,+1 × ,
= 1,2, ,  1
where the number of disjoint boards is denoted as .
Proof
Let 1 be a two-dimensional × board, with
non-attackingbishop, then, we have the following;
,1=1=
1
=0
1 + 1
+21+ . . .+11
1
Now, considering 1 as an 8 × 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;
0,1 , 1,1,2,1,3,1,4,1,
5,1, 6,1,,13,1, then we have;
1=
13
=0
1 + 1
+21+ . . .+113
1
=0,1+1,1+ 2,12+3,13
+4,14+5,15
+ 6,16+ 7,17
+ 8,18+ 9,19
+ 10,110 + 11,111
+ 12,112 +13,113
= ,1
Thus, this is the maximum number of bishop placements on
an 8 × 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;
0,11,1,2,1,3,1,4,1,
5,1, 6,1,,1,1 respectively.
However, the two-dimensional bishop boards generate the
bishop function with the generating function.
,1=0,1 , 1,1,
1
=0 2,1, , 1,1
=0,1+ 1,1+2,12+
+ 13,11
,1
=(1, 14,78,220,330,126,28, 1, 15,10, 1, 3,2, 1)
13
=0
Thus,,1 is the maximum bishop generating function
for an × two-dimensional board. Hence, a projection of
the vector of (+ +on the vector ++ gives a
unit vector 1
3++ with a bishop projection movement
as 3
3 .
Now, considering the bishop movement on a 3-dimensional
space, denoted as ,1,2. We can now construct a
3-dimensional chess board that decomposes into disjoint
boards with vectors , and respectively.
fig. 3.2
then, it follows the placement of bishops on the chess board,
thus we have;
,1,2=,1
+ ,+1 × ,
= 1,2, ,  1
= [ 01+ 11++11
+ 02+ 12+22+
+12
=
1
=0
(,1,2)
=
1
=0
1
=0
,
+,
1
=0
1
=0
(+1)(
× )
=,
1
=0
1
=0
1 +  × ,
= 1,2, ,  1;
where the number of disjoint boards is denoted as .
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;
0,1 , 1,1,2,1,3,1,4,1,
5,1, 6,1,,13,1, where the controller is
denoted by := 0, 1, 2, 3, ,13 then we have;
,1=0,1+ 1,1+2,12+
+ 13,11
=(1, 14,78,220,330,126,28, 1, 15,10, 1, 3,2, 1)
13
=0
= 1 + 14+782+2203+3304+1265+286
+7+
158+109+10 + 311 + 212 +13
= 1 + 1413,13+78(12,12)2+220(11,11)3
+330(10,10)4+126(9,9)5
+28(8,8)6+ (7,7)7+15(6,6)8+10(5,5)9
+ (4,4)10 + 3(3,3)11 +
2((2,2))12 + ((1,1))13
=65,529,875 
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 non-attacking bishop movements generate a
bishop function. Finally, we applied this formula to solve the
rice farm problem.
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The Bishop polynomial on a board rotated in an angle of 45^0 is considered a special case of the rook polynomial. Rook polynomials are a powerful tool in the theory of restricted permutations. It is known that the rook polynomial of any board can be computed recursively, using the cell decomposition technique of Riordan. This independent study examines counting problems of non-attacking bishop placements in the game of chess and it's movements in the direction of O = 45^0 to capture pieces in the same direction as the bishop with restricted positions. In this investigation, we developed the total number of ways to arrange n bishops among m positions ( m>=n) and also constructed the general formula of a generating function for bishop polynomial that decomposes into n disjoint sub-boards B1, B2, ..., Bn by using an man array board. Furthermore, we applied it to combinatorial problems which involve permutation with forbidden positions to construct bishop polynomials in a combinatorial way.
Article
Full-text available
Article
Full-text available
It is shown how the placement of non-attacking bishops on a chessboard C is related to the matching polynomial of a bipartite graph. Reduction algorithms for finding the bishop polynomial of C are given. We interpret combinatorially the coefficients of this polynomial and construct some interesting boards. Some applications of the bishop polynomials are given. Resumen Se muestra cómo la colocación de alfiles que no atacan en un tablero de ajedrez C se relaciona con el polinomio de apareamiento de un grafo bipartito. Se dan algoritmos de reducción para encontrar el polinomio del alfil de C. Se interpretan combinatoriamente los coeficientes de este polinomio y se construyen algunos tableros interesantes. Se dan algunas aplicaciones de los polinomios de alfiles.
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Introduction to Enumerative and Analytic Combinatorics fills the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The book first deals with basic counting principles, compositions and partitions, and generating functions. It then focuses on the structure of permutations, graph enumeration, and extremal combinatorics. Lastly, the text discusses supplemental topics, including error-correcting codes, properties of sequences, and magic squares. Strengthening the analytic flavor of the book, this Second Edition: • Features a new chapter on analytic combinatorics and new sections on advanced applications of generating functions • Demonstrates powerful techniques that do not require the residue theorem or complex integration • Adds new exercises to all chapters, significantly extending coverage of the given topics Introduction to Enumerative and Analytic Combinatorics, Second Edition makes combinatorics more accessible, increasing interest in this rapidly expanding field. Outstanding Academic Title of the Year, Choice magazine, American Library Association.