23 research outputs found
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
Games on interval and permutation graph representations
We describe combinatorial games on graphs in which two players antagonistically build a representation of a subgraph of a given graph. We show that for a large class of these games, determining whether a given instance is a winning position for the next player is PSPACE-hard. In contrast, we give polynomial time algorithms for solving some versions of the games on trees
A codex of n-and p-positions in harary’s ‘caterpillar game’
Frank Harary proposed the following game: Given a caterpillar C, two players take turns removing edges of a path. The player who takes the last edge wins the game. In this paper, we completely characterize the N-and P-positions for all caterpillars with spine length zero, one, two and three. Furthermore, we analyze approximately 94% of the caterpillars with spine length greater than or equal to four. In those cases, they all turn out to be N-positions
Identifying adequate models in physico-mathematics: Descartes' analysis of the rainbow
The physico-mathematics that emerged at the beginning of the seventeenth century entailed the quantitative analysis of the physical nature with optics, meteorology and hydrostatics as its main subjects. Rather than considering physico-mathematics as the mathematization of natural philosophy, it can be characterized it as the physicalization of mathematics, in particular the subordinate mixed mathematics. Such transformation of mixed mathematics was a process in which physico-mathematics became liberated from Aristotelian constraints. This new approach to natural philosophy was strongly influenced by Jesuit writings and experimental practices. In this paper we will look at the strategies in which models were selected from the mixed sciences, engineering and technology adequate for an analysis of the specific phenomena under investigation. We will discuss Descartes’ analysis of the rainbow in the eight discourse of his Meteorology as an example of carefully selected models for physico-mathematical reasoning. We will further demonstrate that these models were readily available from Jesuit education and literature
Winning an Independence Achievement Game.
The game Generalized Kayles (or Independence Achievement) is played by two players A and B on an arbitrary graph G. The players alternate removing a vertex and its neighbors from G, the winner being the last player with a nonempty set from which to choose. In this thesis, we present winning strategies for some paths
The Computational Complexity of Some Games and Puzzles With Theoretical Applications
The subject of this thesis is the algorithmic properties of one- and two-player
games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles
and games in this context are of the following type: can we design efficient computer
programs that play optimally given any opponent (for a two-player game), or solve
any instance of the puzzle in question?
We examine four games and puzzles and show algorithmic as well as intractability
results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport
a wolf, a goat, and a cabbage across a river by using a boat that can carry only one
item at a time, making sure that no incompatible items are left alone together. We
study generalizations of this puzzle, showing a close connection with the Vertex
Cover problem that implies NP-hardness as well as inapproximability results.
Second, we study the SET game, a card game where the objective is to form
sets of cards that match in a certain sense using cards from a special deck. We
study single- and multi-round variations of this game and establish interesting con-
nections with other classical computational problems, such as Perfect Multi-
Dimensional Matching, Set Packing, Independent Edge Dominating Set,
and Arc Kayles. We prove algorithmic and hardness results in the classical and
the parameterized sense.
Third, we study the UNO game, a game of colored numbered cards where players
take turns discarding cards that match either in color or in number. We extend results
by Demaine et. al. (2010 and 2014) that connected one- and two-player generaliza-
tions of the game to Edge Hamiltonian Path and Generalized Geography,
proving that a solitaire version parameterized by the number of colors is fixed param-
eter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard.
Finally, we study the Scrabble game, a word game where players are trying to
form words in a crossword fashion by placing letter tiles on a grid board. We prove
that a generalized version of Scrabble is PSPACE-hard, answering a question posed
by Demaine and Hearn in 2008