Impartial games emulating one-dimensional cellular automata and undecidability

We study two-player \emph{take-away} games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its behavior are algorithmically undecidable. We show that similar questions are undecidable for our \emph{rule 110} game.Comment: 22 pages, 11 figure

Computational Utilities for the Game of Simplicial Nim

Simplicial nim games, a class of impartial games, have very interesting mathematical properties. Winning strategies on a simplicial nim game can be determined by the set of positions in the game whose Sprague-Grundy values are zero (also zero positions). In this work, I provide two major contributions to the study of simplicial nim games. First, I provide a modern and efficient implementation of the Sprague-Grundy function for an arbitrary simplicial complex, and discuss its performance and scope of viability. Secondly, I provide a method to find a simple mathematical expression to model that function if it exists. I show the effectiveness of this method on determining mathematical expressions that classify the set of zero positions onseveral simplicial nim games

Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe

In this dissertation we explore variations on Tic-Tac-Toe. We consider positional games played using a new type of move called a hop. A hop involves two parts: move and replace. In a hop the positions occupied by both players will change: one will move a piece to a new position and one will gain a piece in play. We play hop-positional games on the traditional Tic-Tac-Toe board, on the finite planes AG(2, q) and PG(2, q) as well as on a new class of boards which we call nested boards. A nested board is created by replacing the points of one board with copies of a second board. We also consider the traditional positional game played on nested boards where players alternately occupy open positions. We prove that the second player has a drawing strategy playing the hop-positional game on AG(2, q) for q ≥ 5 as well as on PG(2, q) for q ≥ 3. Moreover we provide an explicit strategy for the second player involving weight functions. For four classes of nested boards we provide a strategy and thresholds for the second player to force a draw playing a traditional positional game as well as the new hop-positional game. For example we show that the second player has a drawing strategy playing on the nested board [AG(2, q1 ) : PG(2, q2 )] for all q2 ≥ 7. Other bounds are also considered for this and other classes of nested boards

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

The Vulcan game of Kal-toh: Finding or making triconnected planar subgraphs

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard

Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim

Given an impartial combinatorial game G, we create a class of related games (CIS-G) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such modifications amount to taking cofinite induced subgraphs (CIS) of the original game graph. Some recent numerical/heuristic work has suggested that the underlying structure and behavior of such "CIS-games" can shed new light on, and bears interesting relationships with, the original games from which they are derived. In this paper we present an analytical treatment of the cofinite induced subgraphs associated with the game of (three-heap) Nim. This constitutes one of the simplest nontrivial cases of a CIS game. Our main finding is that although the structure of the winning strategies in games of CIS-Nim can differ greatly from that of Nim, CIS-Nim games inherit a type of period-two scale invariance from the original game of Nim.Comment: 26 pages, 5 figure

Peg Solitaire on Trees with Diameter Four

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible

Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim

Given an impartial combinatorial game G, we create a class of related games (CISG) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such modifications amount to taking cofinite induced subgraphs (CIS) of the original game graph. Some recent numerical/heuristic work has suggested that the underlying structure and behavior of such “CIS-games” can shed new light on, and bears interesting relationships with, the original games from which they are derived. In this paper we present an analytical treatment of the cofinite induced subgraphs associated with the game of (three-heap) Nim. This constitutes one of the simplest nontrivial cases of a CIS game. Our main finding is that although the structure of the winning strategies in games of CIS-Nim can differ greatly from that of Nim, CIS-Nim games inherit a type of period-two scale invariance from the original game of Nim