17 research outputs found
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
Building Nim
The game of nim, with its simple rules, its elegant solution and its
historical importance is the quintessence of a combinatorial game, which is why
it led to so many generalizations and modifications. We present a modification
with a new spin: building nim. With given finite numbers of tokens and stacks,
this two-player game is played in two stages (thus belonging to the same family
of games as e.g. nine-men's morris): first building, where players alternate to
put one token on one of the, initially empty, stacks until all tokens have been
used. Then, the players play nim. Of course, because the solution for the game
of nim is known, the goal of the player who starts nim play is a placement of
the tokens so that the Nim-sum of the stack heights at the end of building is
different from 0. This game is trivial if the total number of tokens is odd as
the Nim-sum could never be 0, or if both the number of tokens and the number of
stacks are even, since a simple mimicking strategy results in a Nim-sum of 0
after each of the second player's moves. We present the solution for this game
for some non-trivial cases and state a general conjecture
Impartial geodetic building games on graphs
A subset of the vertex set of a graph is geodetically convex if it contains
every vertex on any shortest path between two elements of the set. The convex
hull of a set of vertices is the smallest convex set containing the set. We
study two games where two players take turns selecting previously-unselected
vertices of a graph until the convex hull of the jointly-selected vertices
becomes too large. The last player to move is the winner. The achievement game
ends when the convex hull contains every vertex. In the avoidance game, the
convex hull is not allowed to contain every vertex. We determine the nim-number
of these games for several graph families.Comment: 30 pages, 19 figures, 1 tabl
Chomp on generalized Kneser graphs and others
In chomp on graphs, two players alternatingly pick an edge or a vertex from a
graph. The player that cannot move any more loses. The questions one wants to
answer for a given graph are: Which player has a winning strategy? Can a
explicit strategy be devised? We answer these questions (and determine the
Nim-value) for the class of generalized Kneser graphs and for several families
of Johnson graphs. We also generalize some of these results to the clique
complexes of these graphs. Furthermore, we determine which player has a winning
strategy for some classes of threshold graphs.Comment: 17 pages, 4 figures, removed a wrong theorem about almost bipartite
graphs from a previous versio
Kommutative Algebra
The workshop brought together researchers on various areas of Commutative Algebra. New results in combinatorial commutative algebra, homological methods and invariants, characteristic p-methods, and in general commutative algebra and algebraic geometry were presented in the lectures of the workshop