Word Kayles and Dawsonword

In Wordnim and Grundyword in the Aug 1993 Word Ways, and Edith Plays Word Treblecross in the Aug 1999 issue, Jeremiah Farrell, David Wright and Christopher Mihelich show how various nim-like games can be cast in the form of word games. This article casts two more such games in a similar mould

Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

Compound Node-Kayles on Paths

In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound, Node-Kayles has been solved for a long time under normal play, while it is still unsolved under mis\`ere play. We thus focus on the ten remaining versions, leaving only one of them unsolved.Comment: Theoretical Computer Science (2009) to appea

Periodicità nei giochi combinatori: il caso di un gioco non ottale.

L'obiettivo di questa tesi è stato quello di analizzare un particolare gioco combinatorio propostomi dal Prof. Alessandro Berarducci, al fine di trovare esplicitamente una strategia vincente per uno dei due giocatori. Si tratta di un gioco di natura topologica non precedentemente studiato, affine come presentazione, ma non come struttura, al più famoso Sprouts. Nel primo capitolo vengono presentati i risultati essenziali nella teoria dei giochi combinatori simmetrici, tra cui il teorema di Sprague-Grundy e il teorema di periodicità per i giochi ottali. Nel secondo capitolo viene invece presentato il gioco insieme alla relativa analisi, per la quale si è rivelato necessario dimostrare un'estensione del teorema di periodicità per i giochi ottali

Taking and Breaking Games

V této práci analyzujeme několik otevřených problémů v oblasti nestranných i stranných her typu Taking and Breaking. Pro nestranné odčítací hry dokážeme existenci hry s aperiodickou nim-sekvencí a periodickou sekvencí výhra-prohra. Analyzujeme ekvivalenční třídy těchto her a nalézáme řešení jedné z těchto tříd. Také představujeme novou hru typu Taking and Breaking, kterou z velké části vyřešíme. V oblasti stranných her provedeme analýzu několika odčítacích her a her typu Pure Breaking. Pro tyto hry také představíme obecnou techniku testování aritmetické periodicity. Pro automatické řešení nestranných her typu Taking and Breaking navrhujeme několik algoritmů. Práci uzavíráme důkazem PSPACE-těžkosti nestranné zobecněné odčítací hry a EXPTIME-těžkosti této hry ve stranné variantě.In this thesis, we examine several open problems in taking and breaking games under the impartial and partizan setting. We prove the existence of an impartial subtraction game with aperiodic nim-sequence and periodic outcome sequence. We also analyze the equivalence classes of subtraction games and provide a solution to one of these classes. We introduce a new taking and breaking game and partially solve it. Then we solve several partizan subtraction games and partizan pure breaking games and describe a general technique for testing arithmetic periodicity of these games. Moreover, we design some game solving algorithms for impartial taking and breaking games. We prove PSPACE-hardness for a generalized subtraction game under the impartial setting and EXPTIME-hardness under the partizan setting