FORMS AND VALUES OF NUMBER-LIKE AND NIMBER-LIKE GAMES by Neil Anderson McKay Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Dalhousie University Halifax, Nova Scotia August 2016 ⃝c Copyright by Neil Anderson McKay, 2016 To people and other living things of the past, present, and future whose lives benefit us all in ways we may not know. ii Table of Contents List of Tables ................................... vii List of Figures .................................. viii Abstract ...................................... ix List of Abbreviations and Symbols Used .................. x Acknowledgements ............................... xii Chapter 1 Introduction .......................... 1 1.1 Introduction . .1 1.1.1 Background . .1 1.1.2 Purpose . .2 1.1.3 Contents . .4 1.2 Starting from empty . .6 1.3 Outcomes . .8 1.4 Equality and other relations . 13 1.5 Simplification and Canonical Form . 15 1.6 Hackenbush . 18 1.7 Form and heredity . 19 1.8 Incentives . 20 1.9 Numbers . 21 1.10 Nimbers . 24 1.11 Dicots . 26 1.12 Stops . 28 1.13 Infinitesimals . 29 1.14 Reduced Canonical Form . 31 iii Chapter 2 Ordinal sum ........................... 33 2.1 Introduction . 33 2.1.1 Form and Outcomes . 34 2.2 Ordinal Sum Basics . 35 2.2.1 Norton’s Lemma . 36 2.2.2 Atomic Weight . 37 2.2.3 The Atomic Weight Calculus . 39 2.2.4 The Colon Principle . 40 2.3 Domination and Reversibility in Ordinal Sums . 40 2.3.1 Associativity of ordinal sum . 44 2.4 Dicots and ordinal sum . 45 2.4.1 Star-like positions . 46 2.4.2 Atomic weights of ∗-based positions . 47 2.5 Recognizing Ordinal Sums . 48 2.5.1 Literal ordinal sums . 48 2.5.2 Positions equal to ordinal sums . 50 2.5.3 Terminology . 52 Chapter 3 Transitive games ........................ 53 3.1 Introduction . 53 3.2 Transitivity and ordinal sum . 55 3.3 Stops of transitive positions . 56 3.4 Infinitesimals and transitivity . 57 3.4.1 Reduced values of transitive positions . 57 3.5 Hereditarily transitive values . 61 3.6 Making transitive positions . 62 3.7 Primary Options . 65 3.8 Transitive domination . 65 3.9 Simplifying hereditarily transitive positions . 67 3.10 Stratified positions . 72 3.11 HT-values . 74 3.12 Partial order of hereditarily transitive positions . 75 iv Chapter 4 Stalks ............................... 79 4.1 Introduction . 79 4.2 Families of Stalks . 80 4.3 A Simpler Stalk . 84 4.4 Two Stalks . 87 4.4.1 Stalk Incentives . 88 4.5 Three Stalks . 90 4.6 Outcomes of Sums of Nimber-based Numbers . 97 4.7 Recognizing Sums of Stalks . 102 4.7.1 Incentives of Sums of Stalks . 103 4.8 Values of Sums of ∗-based Numbers . 104 4.8.1 Values of Uptimals . 104 4.9 Sums of ∗-based Numbers . 107 4.9.1 Incentives of ∗-based numbers . 107 4.9.2 Comparing killing and non-killing moves . 110 4.10 Group structure . 112 4.10.1 On the Order of ∗-based Positions Including Stalks . 112 4.10.2 On the Group Structure of Stalks . 114 4.11 On the relation of ∗-based positions to ∗-based integers . 115 Chapter 5 Misère-play and ∗-based Dyadic Rationals ........ 120 5.1 Introduction to Misère-play . 120 5.1.1 Ordinal Sums Under Misère-play . 124 5.2 On ∗-based Dyadic Rationals under Misère-play . 126 5.3 Outcomes of Sums . 128 5.4 Distinguishability . 131 5.5 Discussion . 132 Chapter 6 Partizan Euclid ......................... 133 6.1 Game Tree Structure . 135 6.2 Reducing the Signature . 139 v 6.2.1 Algorithm . 143 6.3 Outcome Observations . 144 6.4 Open Questions . 145 Chapter 7 Conclusion ............................ 148 7.1 Open Problems . 148 7.1.1 Ordinal Sum . 149 7.1.2 Hereditarily Transitive . 149 7.1.3 Hackenbush . 149 7.1.4 Infinitesimals . 150 7.1.5 Partizan Euclid . 150 Bibliography ................................... 152 Appendix A .................................... 155 vi List of Tables 1.1 Some very short game trees and their names . .9 1.2 Players . .9 1.3 hackenbush positions with at most one edge . 18 2.1 Outcome of the ordinal sum of G and H ............. 35 3.1 The number of HT-values born by day n ≤ 5 .......... 75 4.1 Families of stalks ......................... 81 4.2 Outcomes of ∗n-based integers . 99 5.1 Misère-play outcome of ordinal sums with non-end bases . 125 6.1 Outcome of a position with exactly one option for each player . 135 6.2 Irreducible signatures with corresponding positions and outcomes 142 6.3 aw(E(q, p)), for q = 11,..., 20, p = 10,..., 15........... 146 6.4 Some atomic weights from partizan euclid .......... 147 vii List of Figures 1.1 The partial order of outcomes . 11 1.2 Relations ρ and possible outcomes of G given G ρ 0....... 15 1.3 A stalks position . 18 1.4 The dyadic rationals . 22 4.1 Values and outcomes of stalks with at most one edge . 79 4.2 Non-green monochrome stalks are integers . 81 4.3 blue-red stalks are dyadic rationals . 81 4.4 green stalks are nimbers . 82 4.5 A sprigs position . 82 4.6 A flowers position . 83 6.1 Some of the game tree of (11, 8) ................. 134 6.2 Paired outcome of signatures. 140 viii Abstract We consider combinatorial games (positions) played by two players who move alter- nately. In a disjunctive sum of positions a player may play in any one summand. Plays in a particular summand may not alternate between players. Play ends in a finite number of moves when a player cannot move in any summand. The winneris determined by the last player to move. We primarily consider the case where the last player to move wins (the normal-play convention). In a sum of nimbers both players have the same available moves (options). In a sum of numbers both players would rather it not be their turn. The canonical form theory partitions positions into equivalence classes which form a group with disjunc- tive sum. The unique (canonical) representative of an equivalence class is called a value. Nimber-valued and number-valued positions are closed under disjunctive sum. It has long been known how to identify positions that are nimbers or numbers. There are other methods to analyze positions that are not nimber-like and number-like, such as reduced canonical forms and atomic weight. Nimbers and numbers are both hereditarily transitive (where no player would benefit from moving twice in a row) and both are hackenbush positions. Dicotic positions (where both players have an option or neither does) are like nimbers and numbers. The dicotic hereditarily transitive positions are described using ordinal sum. We show how to recognize positions whose values are from the ruleset hackenbush stalks (whose summands are described by ordinal sums). We then consider outcomes of hackenbush stalks under the misère-play convention. We end by considering the ruleset partizan euclid, a partizan dicotic ruleset, that is like nimbers and numbers in other ways. ix List of Abbreviations and Symbols Used b(˜ G) formal birthday of G. b(G) birthday of G. GL the Left options of G. GR the Right options of G. G the negative of G, used in misère-play. G → H the move from G to H. G : H the ordinal sum of G and H. L the set of Left win positions. N the set of first player win positions. P the set of second player win positions. R the set of Right win positions. L the outcome (Left, Left). N the outcome (Left, Right). P the outcome (Right, Left). R the outcome (Right, Right). rcf(G) the reduced canonical form of G. G˜ the set of short positions. G the set of short values. LS(G) the Left stop of G. RS(G) the Right stop of G. TL(G) Left-transitive closure of G. TR(G) Right-transitive closure of G. T(G) transitive closure of G. HT(G) hereditary transitive closure of G. D the universe of dicotic positions. 0 {· | ·} “zero”, “the empty position”. 1 {0 | ·} “one”. n {n − 1 | ·} “n”. x ∗ {0 | 0} “star”. ∗n {0, ∗,..., ∗n−1 | 0, ∗,..., ∗n−1} “star-n”. F “far star”. ¨x {0 | {0 | −x}} “tiny-x”. ↑ {0 | ∗} “up”. ↑∗ {0, ∗ | 0} “up star”. ↓ {∗ | 0} “down”. ⇑ {0 | ↑∗} “double-up”. {0 | ⇑∗} “triple-up”. xi Acknowledgements I do not have the space to list the names of all the people I thank. I sincerely thank everyone who has helped me; if you read this wondering if I mean you, I almost certainly do. I thank many people multiple times as they play multiple important roles. For some people I do not have the space to state all the ways they have helped me. First, I thank the Department throughout my time at Dalhousie. In particular I.