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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 Terms—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 × 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

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 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+ . . .+11

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+

+ 13,11

,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+

+ 13,11

=(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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