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An Investigation of Partizan Misere Games

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An Investigation of Partizan Misere Games AN INVESTIGATION OF PARTIZAN MISERE` GAMES by M.R. Allen Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Dalhousie University Halifax, Nova Scotia July 2009 arXiv:1008.4109v1 [math.CO] 24 Aug 2010 c Copyright by M.R. Allen, 2018 DALHOUSIE UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled “AN INVESTIGATION OF PARTIZAN MISERE` GAMES” by M.R. Allen in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dated: July 17, 2009 External Examiner: Research Supervisor: Examining Committee: ii DALHOUSIE UNIVERSITY Date: July 17, 2009 Author: M.R. Allen Title: AN INVESTIGATION OF PARTIZAN MISERE` GAMES Department or School: Department of Mathematics and Statistics Degree: PhD Convocation: October Year: 2009 Permission is herewith granted to Dalhousie University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author’s written permission. The author attests that permission has been obtained for the use of any copyrighted material appearing in the thesis (other than brief excerpts requiring only proper acknowledgement in scholarly writing) and that all such use is clearly acknowledged. iii nnnnnnnnnn (she’ll know what it means) iv Table of Contents List of Tables ix List of Figures xii Abstract xiv Acknowledgements xv Chapter 1 Introduction 1 1.1 Introduction.................................... 1 1.2 IntroducingCombinatorialGameTheory . .. 3 1.3 AllAboutMis`erePlay .............................. 11 1.3.1 Problems with Mis`ere Play . 11 1.3.2 Genus ................................... 14 1.3.3 Indistinguishability and Mis`ere Monoids . 19 1.4 ThePartialOrderonPositions ......................... 27 1.5 ThesisLayout................................... 30 Chapter 2 Examples of Mis`ere Monoids for Certain Partizan Positions 32 2.1 Introduction.................................... 32 2.2 The Mis`ere Monoid of cℓ (1, 1)..........................¯ 32 v 2.3 The Mis`ere Monoid of cℓ (σ, σ).......................... 38 2.4 The mis`ere monoid of cℓ (ρ) ........................... 41 2.5 The Mis`ere Monoid of cℓ (ρ, ρ).......................... 50 2.6 The Mis`ere Monoid of cℓ (τ)........................... 77 2.7 Conclusion..................................... 81 Chapter 3 The Cardinality of Partizan Mis`ere Monoids 82 3.1 Introduction.................................... 82 3.2 Closures of Binary Positions of Birthday Two or Less and their Conjugates Under Indistinguishability . 82 3.3 The Closure of L(ξ)................................ 84 3.4 The Closure of L(τ n)............................... 88 3.5 Conclusion..................................... 97 Chapter 4 Stars 98 4.1 Introduction.................................... 98 4.2 All-Small...................................... 99 4.3 Extending the ∗ + ∗ ≡ 0ResultstoNon-All-SmallGames . 102 4.4 Conclusion..................................... 104 Chapter 5 Zeroes 105 5.1 Introduction.................................... 105 vi 5.2 ξ + ξ in ab3 positions .............................. 107 5.3 Tweedledum-Tweedledee on ab3 ......................... 114 5.4 Conclusion..................................... 119 Chapter 6 A Brief Categorical Interlude 120 6.1 Introduction.................................... 120 6.2 AVarietyofSums ................................ 122 6.3 AdjointsandDisjunctiveSum . .. .. 128 6.4 Restricting ourselves to ab3 ........................... 130 6.5 Conclusion..................................... 131 Chapter 7 Isomorphic Monoids 135 7.1 Introduction.................................... 135 7.2 To the Left: L(ξ) ................................. 138 7.3 Isomorphic to Mcℓ(∗) ............................... 139 7.4 Conclusion..................................... 182 Chapter 8 Two Examples of Partizan Heap-based Mis`ere Monoids 184 8.1 Introduction.................................... 184 8.2 The Partizan Subtraction Game L(1, 2), R(1) ................. 186 8.3 The Partizan Subtraction Game L(1), R(2) .................. 194 8.4 Conclusion..................................... 199 vii Chapter 9 Conclusion 201 Appendix A Frequently Used Positions 203 A.1 0.......................................... 203 A.2 ∗ .......................................... 203 A.3 1.......................................... 204 A.4 1.......................................... 204 A.5 σ .......................................... 204 A.6 σ .......................................... 205 A.7 ρ .......................................... 205 A.8 ρ .......................................... 205 A.9 τ .......................................... 206 A.10 τ n ......................................... 206 A.11 L(ξ) ........................................ 207 A.12 R(ξ) ........................................ 207 A.13 ξ .......................................... 207 A.14 ∗n ......................................... 208 A.15 η .......................................... 208 A.16 θ .......................................... 209 Bibliography 210 viii List of Tables Table 1.2.1 Determining the outcome class of a position ξ given information about the outcome classes of ξ’soptions. .................... 10 Table 1.3.1 Positions with varying outcomes in normal play and mis`ere play. 11 Table 1.3.2 Normal play outcomes under disjunctive sum where ? denotes that the outcomecouldbeinanyofthefouroutcomeclasses. 12 Table 1.3.3 Mis`ere play outcomes under disjunctive sum where ? denotes that the outcomecouldbeinanyofthefouroutcomeclasses. 12 Table 2.4.1 Outcomes of positions n ∗ +mρ where n ≡ 0or1(mod2). 42 Table 2.4.2 Positions of cℓ (ρ) up to Indistinguishability. 46 Table 2.4.3 Positions of cℓ (ρ) and the elements which distinguish them. 46 Table 2.4.4 Showing o−(ax) ≥ o−(p2x) and o−(px) ≥ o−(p2x) for x ∈{1, a, ap} in Mcℓ(ρ). ................................... 49 Table 2.4.5 Showing o−(apx) ≥ o−(ap2x) and o−(1x) ≥ o−(ap2x) for x ∈ {1,a,p} in Mcℓ(ρ)................................... 49 Table 2.4.6 Determining which elements are incomparable in Mcℓ(ρ)......... 50 Table 2.5.1 The outcome classes of positions in n∗+mρ+ℓρ where m ≡ ℓ+i (mod 4). 51 Table 2.5.2 The outcome classes of positions in cℓ (ρ, ρ) where n ≡ 0(mod2).. 52 Table 2.5.3 The outcome classes of positions in cℓ (ρ, ρ) where n ≡ 1(mod2).. 52 n n+4 Table 2.5.4 Showing q < q in Mcℓ(ρ,ρ). ...................... 67 ix n n+5 Table 2.5.5 Showing q < q in Mcℓ(ρ,ρ). ...................... 67 n n+6 Table 2.5.6 Showing q < q in Mcℓ(ρ,ρ). ...................... 68 n t Table 2.5.7 Showing q < q where t − n ≥ 7 in Mcℓ(ρ,ρ)................ 68 Table 2.5.8 Showing aqn < qt for t − n ≥ 6....................... 73 Table 2.5.9 Showing ap < qn for n ≥ 5. ........................ 73 Table 2.5.10 Showing ap2 < qn for n ≥ 4. ....................... 74 Table 2.5.11 Showing apn < qt for 3 ≤ n ≤ t. ..................... 74 Table 2.5.12 The incomparability of elements of the form (ax, y) in Mcℓ(ρ,ρ)..... 76 Table 2.5.13 The incomparability of elements of the form (ax, y) in Mcℓ(ρ,ρ) - Con- tinued. ................................... 76 Table 7.3.1 224 positions born on day 3 with monoids isomorphic to Mcℓ(∗)..... 179 Table 8.2.1 The outcomes for positions (x1) in the game L(1, 2), R(1).. 187 Table 8.2.2 The outcomes for positions (x1, x2) in the game L(1, 2), R(1). 187 2 2 M Table 8.2.3 The indistinguishability of b and ab in cℓ(h1,h2) in the game L(1, 2), R(1). .................................... 189 Table 8.2.4 The outcomes for positions (x1, x2, x3) in the game L(1, 2), R(1). 190 2 2 M Table 8.2.5 The indistinguishability of b , c, and ab in cℓ(h1,h2,h3) in the game L(1, 2), R(1)................................. 191 Table 8.3.1 The outcomes for positions (x1) in the game L(1), R(2)......... 195 Table 8.3.2 The outcomes for positions (x1, x2) with x1 ≤ 7 and x2 ≤ 6 in the game L(1, 2), R(1)................................. 195 x anbm M ♥ n m Table 8.3.3 The outcomes for positions in cℓ(h1,h2)with ≤ 7 and ≤ 6 in the game L(1, 2), R(1). .......................... 197 xi List of Figures Figure 1.2.1 The game trees of 0, 1, and ∗....................... 7 Figure 1.3.1 Commutativity of o− between positions in Υ and their equivalence classesunderΥ............................... 25 Figure 1.4.1 Outcome Class Partial Order. .. 27 Figure 2.2.1 The game trees of 1 and 1......................... 33 Figure 2.2.2 Outcome Class Partial Order. .. 37 Figure 2.3.1 The game tree of σ............................. 38 Figure 2.4.1 The game tree of ρ............................. 42 Figure 2.4.2 cℓ (ρ)’sPartiallyOrderedSet. .. .. 48 Figure 2.5.1 A snippet of the partially ordered set of elements of the form pm and n q in Mcℓ(ρ,ρ)................................ 70 Figure 2.5.2 A snippet of the partially ordered set of elements of the form apm and n aq in Mcℓ(ρ,ρ). .............................. 70 Figure 2.6.1 The game tree of τ............................. 78 Figure 3.3.1 The game trees of σ and L(σ)....................... 84 Figure 4.1.1 The game tree of L(σ)........................... 99 Figure 4.3.1 The game trees of η and L(η),respectively. 102 Figure 4.3.2 Right loses moving first in L(η)+ ∗ + ∗.................. 103 xii − Figure 5.1.1 o (∗2 + ∗2)= P.
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