Combinatorial Game Distributions of Steiner Systems

The $P$-position sets of some combinatorial games have special combinatorial structures. For example, the $P$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, denoted by $D_{\text{sh}}$. However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose $P$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $D_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, ..., 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $P$-position set is its block set. From $D_{\text{sh}}$, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $D$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $D$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

A Cop and Drunken Robber Game on n-DimensionalInfinite-Grid Graphs

A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is s+o(s).

Peg solitaire on graphs with large maximum degree

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

CINE, ÉTICA Y ALTERIDAD. METÁFORAS DEL ENCUENTRO CON EL OTRO

Resumen: El título del artículo me va a permitir un juego combinatorio tomando sus términos por parejas. De tal manera que lo que viene a continuación constará de tres partes. Una primera sobre la relación entre el cine y la ética; una última, en torno al cine y la alteridad. Y, entre ambas, un breve exordio que, siguiendo el pensamiento de Emmanuel Lévinas, reflexiona sobre el vínculo entre la ética y la alteridad. Para ello me valdré del análisis de tres películas: El lector (Stephen Daldry, 2008), Rosetta (Jean-Pierre y Luc Dardenne, 1999) y Lone Star (John Sayles, 1996).Abstract: The article title will allow me a combinatorial game matching its terms in pairs. Thereby, what comes next will consist of three parts. The first one will address the relationship between cinema and ethics, while the last one will approach cinema and the otherness. In between both, there will be a brief exordium that, following the thought of Emmanuel Lévinas, reflects on the link between ethics and the otherness. All of this will be based on the analysis of three films: The Reader (Stephen Daldry, 2008), Rosetta (Jean-Pierre and Luc Dardenne, 1999) and Lone Star (John Sayles, 1996).

Automated Conjecturing II: Chomp and Reasoned Game Play

We demonstrate the use of a program that generates conjectures about positions of the combinatorial game Chomp—explanations of why certain moves are bad. These could be used in the design of a Chomp-playing program that gives reasons for its moves. We prove one of these Chomp conjectures—demonstrating that our conjecturing program can produce genuine Chomp knowledge. The conjectures are generated by a general purpose conjecturing program that was previously and successfully used to generate mathematical conjectures. Our program is initialized with Chomp invariants and example game boards—the conjectures take the form of invariant-relation statements interpreted to be true for all board positions of a certain kind. The conjectures describe a theory of Chomp positions. The program uses limited, natural input and suggests how theories generated on-the-fly might be used in a variety of situations where decisions—based on reasons—are required.

Mityzacja i mitologizacja. O książce Doroty Siwor Tropy mitu i rytuału

Mythicising and Mythologising: On Dorota Siwor’s book, Traces of Myth and Ritual The author discusses the issues touched upon in the book penned by Dorota Siwor, Traces of Myth and Ritual. On Polish Contemporary Prose – Not Only the Latest such as myth, mythologising and mythicising the presented world in the analysed books. She presents them in the context of research theories concerning myth, also, analysing the interpretative conceptions proposed by Siwor. She emphasises the quality of the publication, author’s erudition and the freedom in the selection of threads and motifs, which create a multidimensional space for combinatorial game suggesting potential Fullness, Jung’s coniunctio oppositorum – the union of form and matter, the imagined and real-world, culture and nature. She also indicates the fields on which polemics and criticism of the selection of the theories applied are possible.