# Slides

**Carlos Pereira dos Santos**, CEAFEL, University of Lisbon

**Title: **Scoring mean value theorem Download

**Abstract:**Regarding short Conway’s group, a value of a game can be thought of as an indeterminate cloud, covering the confusion interval. In such a situation it is natural to ask –

*Is there a fair settlement about the value of a game?*The answer is well-known; it is the mean value

*m(G)*(a number), satisfying

*n.m(G)-t*≤

*n.G*≤

*n.m(G)+t*(for some fixed perturbation

*t*).

In this work, we discuss the occurrence of a mean value in the guaranteed scoring universe which is, in some sense, two-dimensional due the different nature of moves and points.

**Israel Rocha**, Dalhousie University

**Title: **Eternal Picaria Download

**Abstract:****Picaria is a traditional board game, played by the Zuni tribe of the**

**American Southwest and other parts of the world, such as a rural Southwest**

**region in Sweden. It is related to the popular children's game of Tic-tac-toe, but the 2 players have only 3 stones each, and in the second phase of the game, pieces are slided, along speci_ed move edges, in attempts to create the three-in-a-row. We provide a rigorous solution, and prove that the game is a draw; moreover our solution gives insights to strategies that players can use.**

(joint work with Urban Larsson)

**Gabriel Renault**, Sopra Steria

**Title:**Solving misère illuNIMati through boomerang games Download

**IlluNIMati is a partizan variant of NIM in which tokens are triangles with a blue vertex, a green vertex and a red vertex. Only Left may play in a heap where the vertex facing North of the top triangle is blue. Only Right may play in a heap where the vertex facing North of the top triangle is red. They may both play in a heap where the vertex facing North of the top triangle is green. However, after removing any positive number of triangles of a heap, they may turn this heap, choosing which vertex faces North. We consider illuNIMati positions within a bigger set of games, called boomerang games, to solve the misère version.**

**Abstract:****, University of New England**

Craig Tennenhouse

Craig Tennenhouse

**Title:**A two-player pebbling game Download

**Abstract:**Blocking Pebbles is a partizan game based on graph pebbling directed acyclic graphs. We demonstrate some results on oriented trees, and find numbers, nimbers, and some infinitesimals on the out-star.

(joint work with Mike Fisher)

**R. J. Nowakowski**, Dalhousie University

**Title:**Cricket pitch perfection Download

**Abstract:**A cricket Pitch is a path with bumps (non-negative weights) and a roller that resides at the vertices. Left rolls (one or more edges) to the left and Right to the right reducing each bump by 1. The roller cannot go over a 0-edge, this part is `perfect'. Nowakowski and Ottaway partially solved the game. Nowakowski and (Angela) Siegel give a complete solution in terms of reductions and ordinal sums.

(joint work with Paul Ottaway and Angela Siegel)

**, Capilano University**

Paul Ottaway

Paul Ottaway

**Title:**The OSLO game of Hives Download

**Abstract:**Hives is a one-sided loopy (OSLO) game played on a graph with white and red vertices. On Right's turn, they may colour any white vertex red as long as the set of red vertices form an independent set. On Left's turn, they may swap the colour of any two adjacent vertices as long as the red vertices remain an independent set. Note that Left has the equivalent of a pass move by selecting a pair of adjacent white vertices while Right does not have a pass move. The game ends when the red vertices form a maximal independent set. In this talk, we will examine the game values and algebraic structure that exists in Hives, provide strategies for certain classes of graphs as well as decomposition theorems for general graphs.

**, West Chester University**

Michael Fisher

Michael Fisher

**Title:**Atomic weight calculus of Subversion Download

**Abstract:**In this work we analyze Subversion, a recently proposed all-small combinatorial ruleset. Often, the atomic weight calculus, allowing the recursive computation of the atomic weight of a game G with the atomic weights of its options, is not easy. However, Subversion is an interesting case such that the theoretical difficult cases are reduced to a finite number of cases and organized in a table. With the table, given a Subversion position (a,b), a≥b, using the Euclidean algorithm and the continued fraction representation of a/b, it is possible to compute aw(a,b). An algorithm for that is presented.

(joint work with Neil McKay, Richard J. Nowakowski, Paul Ottaway and Carlos Santos)

**Marc Heinrich**, LIRIS, University of Lyon

**Title:**Quotients for push-button games Download

**Abstract:**We study a new construction which aims at solving the following question: what happens if we allow players to change the rules of the game during the play? Given two rulesets R

_{0}and R

_{1}, we create a new ruleset R

_{0}⇒ R

_{1}corresponding to the following: the two players play according to rules R

_{0}until one of the players at its turn pushes an imaginary button, changing the rules to R

_{1}. We study how several classical impartial games like Nim, Wythoff, Euclid or Cram combine with this construction. By adapting indistinguishably quotients (technique of Plambeck, 2005), we look at how these new rules behave for sums of positions. We show that it is possible to give values to game positions which behave similarly to Grundy values.

(joint work with Eric Duchêne, Urban Larsson, Aline Parreau)

**, LIRIS, University of Lyon**

Valia Mitsou

Valia Mitsou

**Title:**The computational complexity of two card games with theoretical applications Download

**Abstract:**The main theme of this talk is card games that can be naturally reformulated as well-known graph-theoretic games. In particular, we will focus on two different card games: the game of SET and the game of UNO. The objective of the former is to form sets of cards that match in a certain sense, while in the latter the players need to discard their cards following a matching rule (for more details regarding the rules of the two games please visit the following websites: official website of SET, wikihow: how to play UNO). We show natural reformulations of SET as arc-kayles and of UNO as generalized geography and present some complexity results about them.

**Antoine Dailly**, LIRIS, University of Lyon

**Title:**Octal games on graphs Download

**Abstract:**Octal games are a well-defined family of two-player games played on heaps of counters, in which the two players take turns removing a certain number of counters from a heap, sometimes being allowed to split a heap in two nonempty heaps, until no counters can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path of n vertices is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph, on trees. We present a thorough study of the game on subdivided stars and bistars.

**, Université du Québec à Montréal**

Matthieu Dufour

Matthieu Dufour

**Title:**New results on circular NIM Download

**Abstract:**Matthieu Dufour and Silvia Heubach have been studying Circular Nim, a variation on the well-known game of NIM. This t:wo player game, described with two parameters and noted C(n,k), is played in the following way: n stacks of tokens are arranged in a circular way, and a move consists of choosing k consecutive stacks of this circular arrangement and then taking at least one token from one or more of the k stacks. In normal play, C(n,1), C(n,n-1) and C(n,n) are trivial. C(4,2) is easy, C(5,2), C(5,3) and C(6,3) are known since (Ehrenborg and, Steingrimsson, 1996) and C(6,4) and C(8,6) were solved by Dufour and Heubach (2012). We will present some more results, conjectures and comments on the misere play of that game.