I Don't Play Chess: A Study of Chess Piece Generating Polynomials

120 Pages·2016·1.23 MB·English

The College of Wooster Libraries
Open Works
Senior Independent Study Theses
2015
I Don't Play Chess: A Study of Chess Piece
Generating Polynomials
Stephen R. Skoch
The College of Wooster, [email protected]
Follow this and additional works at:https://openworks.wooster.edu/independentstudy
Part of theDiscrete Mathematics and Combinatorics Commons
Recommended Citation
Skoch, Stephen R., "I Don't Play Chess: A Study of Chess Piece Generating Polynomials" (2015).Senior Independent Study Theses.
Paper 6559.
https://openworks.wooster.edu/independentstudy/6559
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© Copyright 2015 Stephen R. Skoch
on t lay hess
I D ’ P C :
tudy of hess iece
A S C P
enerating olynomials
G P
ndependent tudy hesis
I S T
PresentedinPartialFulfillmentofthe
RequirementsfortheDegreeBachelorofArtsin
theDepartmentofMathematicsandComputer
ScienceatTheCollegeofWooster
by
StephenSkoch
TheCollegeofWooster
2015
Advisedby:
Dr. MatthewMoynihan

Abstract
Thisindependentstudyexaminescountingproblemsofnon-attackingrook,
andnon-attackingbishopplacements. Weexamineboardsforrookand
bishopplacementwithrestrictedpositionsandvarieddimensions. Inthis
investigation,wediscussthegeneralformulaofageneratingfunctionfor
unrestricted,squarebishopboardsthatreliesontheStirlingnumbersofthe
secondkind. Wediscussthemaximumnumberofbishopswecanplaceona
rectangularboard,aswellasabriefinvestigationofnon-attackingrook
placementsonthree-dimensionalboards,drawingaconnectiontolatin
squares.
iii

Dedication
Tomyparents,
foralloftheirmanysacrificestogivemeaneducation.
Thiswasonlypossiblebecauseofyou.
v

Acknowledgements
Iwouldliketotakethistimetothankmyadvisor,Dr. Moynihan,forhis
patience,encouragement,andguidancethroughouttheprocessofwritingthis
independentstudy. Icannotexpressmygratitudeforallofyourhelpand
beliefinmyabilities. ThankyoutoDr. Pierce,whogavemesomuchguidance
andsupportthroughoutmyyearsatthecollege. Iwouldalsoliketothankthe
entireMathematicsdepartmentatTheCollegeofWoosterforbeingsohelpful
andapproachable. Inaddition,Iwouldliketothankallofthewonderful
friendsIhavemadeatWoosterwhohavealwaysbeenthereforme. Youhave
alltaughtmesomuchandkeptmegrounded. Iamsogratefulforeachand
everymemory,andIamlookingforwardtoseeingallofyouchangethe
world. Mostimportantly,Iwouldliketothankmyparentsforbeingaconstant
sourceoflove,support,andwisdomthroughouteveryaspectofmylife.
vii

Contents
Abstract iii
Dedication v
Acknowledgements vii
1 Introduction 1
2 RookBoardsofTwoDimensions 5
3 TheProblemoftheBishops 27
3.1 The Maximal Number of Bishops and Bishop Placements on
SquareBoards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 SquareBishopBoardswhennisEven . . . . . . . . . . . . 31
3.1.2 SquareBishopBoardswhennisOdd . . . . . . . . . . . . 34
3.2 TheGeneralBishopPolynomialforSquareBoards . . . . . . . . . 41
3.2.1 ThePolynomialB (x)whennisEven . . . . . . . . . . . . 45
n
3.2.2 ThePolynomialB (x)whennisOdd . . . . . . . . . . . . . 52
n
ix