Combinatorial Game Theory Colloquium IV
(Lisbon, 2325 January, 2023)
Combinatorial Game Theory (CGT) is a branch of mathematics that studies sequential games with perfect information. Combinatorial games include wellknown rulesets like Amazons, Clobber, Domineering, Hackenbush, Konane, Nim, Octal Games, Wythoff’s Nim. After John Conway’s On Numbers and Games (1976), Elwyn Berlekamp, John Conway, and Richard Guy published “the book” Winning Ways (1982). In that work, one can find a unified mathematical theory able to analyze a large class of rulesets. The books Lessons in Play (2007), by Michael Albert, David Wolfe, and Richard Nowakowski, and Combinatorial Game Theory (2013), by Aaron Siegel, are also mandatory reading.
Combinatorial Game Theory Colloquia are held every two years, in Portugal. Associação Ludus will organize in S. Miguel, Azores, the fourth edition of the CGTC, January, 2325, 2023, with support of Câmara Municipal de Lagoa, Centro de Análise Funcional, Estruturas Lineares e Aplicações, Centro de Matemática Aplicada à Previsão e Decisão Económica, Centro de Matemática e Aplicações (NovaMath, FCT NOVA), Expolab – Centro de Ciência Viva, Faculdade de Ciências e Tecnologia – Universidade dos Açores, Governo dos Açores, Núcleo Interdisciplinar da Criança e do Adolescente, Sociedade Afonso Chaves, and Sociedade Portuguesa de Matemática.
The meeting will take place at Expolab – Centro de Ciência Viva. See a map here.
REGISTRATION
For informations about submissions and registrations, just mail us: cgtc@cgtc.eu
CALL FOR PAPERS
The International Journal of Game Theory (IJGT) invites submissions of significant papers in Combinatorial Game Theory. Combinatorial games are traditionally twoplayer perfectinformation games with compact rules such as Nim or Chess, with more general games now being studied. Their theory was pioneered by Elwyn R. Berlekamp, John H. Conway and Richard K. Guy in Winning Ways (1982), republished in 2009 by A. K. Peters.
All articles will be refereed to the high standards of IJGT. Accepted papers are published online first in a Collection of papers that groups them in a single place. At the appropriate time, they will be published in print in a special issue of the journal. The first special issue on Combinatorial Games appeared as issue 2 in volume 47 of IJGT in 2018.
In Portugal, under the auspices of Associação Ludus, Carlos Pereira dos Santos has been organizing the Combinatorial Game Theory Colloquia taking place every two years, starting in January 2015, 2017, 2019, 2023. Authors of significant original results related to (or/and presented at) the conference are also encouraged to submit them by November 1, 2023 to the International Journal of Game Theory, in order to, if accepted, become part of the Collection. It is important to note that it is not necessary to have participated in the conference to do so.
IJGT Collection on Combinatorial Games Guest Editors:
Urban Larsson, Indian Institute of Technology Bombay, Mumbai, India
Carlos P. Santos, Center for Mathematics and Applications, Nova University of Lisbon, Portugal
Bernhard von Stengel, London School of Economics and Political Science, London, UK
Scientific Committee
Aaron Siegel, San Francisco, California
Alda Carvalho, ISEL & CEMAPRE
Bernhard von Stengel, London School of Economics and Political Science
Carlos Pereira dos Santos, LA & CEAFELUniversity of Lisbon
Eric Duchene, IUT Lyon 1, LIRIS lab.
João Pedro Neto, University of Lisbon
Jorge Nuno Silva, University of Lisbon
Richard Nowakowski, Dalhousie University
Thane Plambeck, Counterwave, Inc
Urban Larsson, Industrial Engineering and Management, Technion
Organizing Committee
Alda Carvalho, ISELIPL & CEMAPRE/REMUniversity of Lisbon
Ana Paula Garrão, NICAUAc & University of Azores
Carlos Pereira dos Santos, Center for Mathematics and Applications (NovaMath), FCT NOVA
Jorge Nuno Silva, CIUHCTUniversity of Lisbon
Margarida Raposo, NICAUAc & University of Azores
Ricardo Cunha Teixeira, NICAUAc & University of Azores
Tiago Hirth, Ludus Assoiation
Working Sessions, Afternoons (15:0018:00)
23/1 – Monday  24/1 – Tuesday  25/1 – Wednesday  
9:009:20  Opening ceremony  
9:209:30 
Aaron Siegel, Horizons in Combinatorial Game Theory


9:309:50  Milos Stojakovic, University of Novi Sad, Strong avoiding positional games 
Dana Ernst, Northern Arizona University, Impartial geodetic convexity achievement and avoidance games on graphs  
9:5010:10  Alda Carvalho, ISELIPL & CEMAPRE/REMUL, «All is number»? Not so easy, Mr. Pythagoras 
Danijela Popović, Mathematical Institute of SASA, A new approach to equivalence of games 
Neil McKay, University of New Brunswick, Numbers and ordinal sums 
10:1010:30  Paul Ellis, Rutgers University, The arithmeticperiodicity of Cut for C = {1, 2c} 
Michael Fisher, West Chester University, Olympic games: three impartial games with infinite octal codes 
Antoine Dailly, Laboratory of Informatics, Modelling and Optimization of the Systems, Subtraction games on graphs 
10:3010:50  Koki Suetsugu, National Institute of Informatics, Some new universal partizan rulesets 
Richard J. Nowakowski, Dalhousie University, The game of Flipping Coins 
Craig Tennenhouse, University of New England, Vexing vexillological logic 
10:5011:10  Carlos Pereira dos Santos, CMA (NovaMath), FCT NOVA, Chess and Combinatorial Game Theory  Tomoaki Abuku, National Institute of Informatics, A Multiple Hook Removing Game whose starting position is a rectangular Young diagram with unimodal numbering 
Urban Larsson, Indian Institute of Technology Bombay, Feasible outcomes of Bidding Combinatorial Games 
10:1011:40  Coffeebreak  Coffeebreak  Coffeebreak 
11:4012:00  Bernhard von Stengel, London School of Economics, Zerosum games and linear programming duality 
Kyle Burke, Florida Southern College, ForcedCapture Hnefatafl  Nandor Sieben, Northern Arizona University, Impartial hypergraph games 
12:0012:20  Bojan Bašić, University of Novi Sad, A twist on the classical prisonersinaline hatguessing game 
Kanae Yoshiwatari, Nagoya University, Complexity of Colored Arc Kayles  Svenja Huntemann, Concordia University of Edmonton, Temperature of Partizan ArcKayles Trees 
12:2012:40  Hironori Kiya, Kyushu University, Normalplay with deadendwinning convention  Eric Duchêne, LIRIS, Lyon 1 University, Partizan subtraction games  Florian Galliot, University of Grenoble Alpes, The MakerBreaker game on hypergraphs of rank 3: structural results and a polynomialtime algorithm 
12:4013:00  Silvia Heubach, California State University, On the Structure of the Ppositions of Slow Exact kNim 
Aline Parreau, CNRS, Lyon 1 University, MakerBreaker Domination Game 
Prem Kant, Indian Institute of Technology Bombay, Bidding combinatorial games 
13:0013:20  Keito Tanemura, Kwansei Gakuin University, Chocolate games with a pass and an application of symbolic regression to these games 
Hikaru Manabe, Keimei Gakuen Elementary Junior & Senior High School, Fourdimensional chocolate games and chocolate games with a pass move 
Nacim Oijid, University of Lyon, Bipartite instances of Influence 
13:2015:00  Break for lunch  Break for lunch  Break for lunch 
15:0018:00  Working sessions  Working sessions  Working sessions 
19:0023:00  Conference Dinner 
Aaron Siegel, United States of America
Title: Horizons in Combinatorial Game Theory
Abstract: This talk will be a brief, highlevel overview of three topics in combinatorial game theory: temperature theory, misere play of partizan games, and calculation of octal games. These topics (among many others) have attracted considerable attention over many years, are deeply interesting from a mathematical and historical perspective, and, I will argue, are fertile ground for further advances. I will highlight some research directions in each topic that appear especially promising.
Alda Carvalho, ISELIPL&CEMAPRE/REMUniversity of Lisbon, Portugal
Title: «All is number»? Not so easy, Mr. Pythagoras
Abstract: We present a method to evaluate if a ruleset only has positions whose game values are numbers.
(Joint work with Melissa Huggan, Richard J. Nowakowski, and Carlos Pereira dos Santos)
Aline Parreau, CNRS, Lyon 1 University, France
Title: MakerBreaker Domination Game
Abstract: The MakerBreaker Domination Game is played on a graph G with two players Dominator and Staller. At his turn, Dominator, selects a vertex in order to dominate the graph while at his turn Staller forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. We study the problem of deciding who has a winning strategy for a given graph G. We prove that this problem is PSPACEcomplete, even for bipartite graphs and split graphs but is polynomial for cographs, trees and block graphs. We in particular focus on pairing strategies for Dominator and prove that having a pairing strategy is the only way to win in cographs, trees, block graphs and interval graphs.
(Joint work with Guillaume Bagan, Eric Duchêne, Valentin Gledel, Tuomo Lehtilä, and Gabriel Renault)
Antoine Dailly, Laboratory of Informatics, Modelling and Optimization of the Systems, France
Title: Subtraction games on graphs
Abstract: Subtraction Games are takingbreaking games defined by a set S of integers, in which a player may remove k counters if k is in S. In (Beaudou et al., 2018) and (Dailly et al., 2019), we extended the definition of subtraction games to play them on graphs: a player may remove a connected subgraph of order k if k is in S, and if the resulting graph is still connected. A similar extension was also defined for octal games. This talk will be an overview of the results obtained in subtraction and octal games on graphs, ranging from structral (ultimate periodicity of a kind of Grundy sequence) and general complexity (PSPACEcompleteness of finite subtraction games with 1 not in S) to polynomialtime algorithms on specific games and graph families (S={1,…,N} on stars, and stronger results for small values of N). I will also highlight several open problems.
Bernhard von Stengel, London School of Economics, United Kingdom
Title: Zerosum games and linear programming duality
Abstract: The minimax theorem for zerosum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig (1951) is massively incomplete, as we argue in this article. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof.
Bojan Bašić, University of Novi Sad, Serbia
Title: A twist on the classical prisonersinaline hatguessing game
Abstract: The game in which a warden arranges n prisoners in a line and then each of them guesses (one by one) whether the hat on his head is white or black, aiming to maximize the total number of correct guesses, is a folklore thing. Those who hear it for the first time (though today it is not easy to meet such a person) are usually surprised when they learn that (spoiler alert!) all the prisoners with the exception of only one can secure correct guesses. And furthermore, nothing changes if the hats come in more than two colors: in that case, too, one prisoner can singlehandedly transmit enough information so that all the other prisoners be able to deduce their hat colors. They simply establish a bijection between the available hat colors and a complete residue system modulo the number of colors, and then the first prisoners declares the sum of hats of all the other prisoners (under the same modulus). This is one of many puzzles in circulation featuring prisoners and a warden. And although they are formulated in an informal fashion and often presented as a matter of recreational mathematics, many of them hide serious science under their facade. There are tight connections between these puzzles and game theory, information theory, coding theory etc. In this talk we introduce a variant of this game, where information that prisoners can relay to other prisoners is much more restricted. In particular, each prisoner is asked whether he wants to take a guess on his hat color, to which question he answers aloud (everybody hears that); if the answer is affirmative, he takes a guess but the other prisoners do not hear what his guess is, they only get to know whether the guess was correct. Therefore, basically, each prisoner is able to transmit only a single binary bit («yes»/«no» answer), and although the other prisoners do get a little more information (the outcome of the guess), the prisoner who takes a guess cannot directly control this further affair. Clearly, if there are 4 possible colors, the first two prisoners can encode the sum of colors of all the remaining prisoners, and thus all but the first two prisoners can assure correct guesses. What is, however, perhaps somewhat surprising, and which will be the first case seen in the talk, is that, if there are 5 hat colors, the prisoners still can arrange a strategy that will guarantee correct guesses for all of them but the first two (and 5 is the largest such number). We shall then present some results on the general question what the maximal possible number of colors is such that, given a positive integer m, if n prisoners are playing the game with the indicated number of colors, they can devise a strategy such that all but m prisoners are guaranteed to guess correctly.
(Joint work with Vlado Uljarević)
Carlos Pereira dos Santos, Center for Mathematics and Applications (NovaMath), FCT NOVA, Portugal
Title: Chess and Combinatorial Game Theory
Abstract: Combinatorial Game Theory was born in twentieth century. The seminal works On Numbers and Games and Winning Ways are masterpieces of mathematical creativity. Some games are numbers, some games are not. John Horton Conway said in some interviews that the coinvention of Combinatorial Game Theory was one of his mathematical achievements he was most proud of. In some conferences he explained how happy he was when discovered the value 1/2 in the context of games. Here, a not artificial Chess problem related to the discovery of the dyadic numbers is presented. Its construction was inspired by a “mysterious” wellknown game played by the world chess champions Jose Raul Capablanca and Emanuel Lasker.
Craig Tennenhouse, University of New England, United States of America
Title: Vexing vexillological logic
Abstract: We define a new impartial combinatorial game, Flag Coloring, based on flood filling. We then generalize to a graph game and demonstrate that the generalized game is PSPACEcomplete for two colors or more via a reduction from the game Avoid True, determine the outcome classes of games based on realworld flags, and discuss remaining open problems.
(Joint work with Kyle Burke)
Dana Ernst, Northern Arizona University, United States of America
Title: Impartial geodetic convexity achievement and avoidance games on graphs
Abstract: A set P of vertices of a graph G is convex if it contains all vertices along shortest paths between vertices in P. The convex hull of P is the smallest convex set containing P. We say that a subset of vertices P generates the graph G if the convex hull of P is the entire vertex set. We study two impartial games Generate and Do Not Generate in which two players alternately take turns selecting previouslyunselected vertices of a finite graph G. The first player who builds a generating set for the graph from the jointlyselected elements wins the achievement game GEN(G). The first player who cannot select a vertex without building a generating set loses the avoidance game DNG(G). Similar games have been considered by several authors, including Harary et al. In this talk, we determine the nimnumber for several graph families, including trees, cycle graphs, complete graphs, complete bipartite graphs, and hypercube graphs.
(Joint work with Bret Benesh, Marie Meyer, Sarah Salmon, and Nandor Sieben)
Danijela Popović, Mathematical Institute of SASA, Serbia
Title: A new approach to equivalence of games
Abstract: The wellknown SpragueGrundy theory states that every impartial combinatorial game played under the socalled normal play convention is equivalent to a single Nim heap. However, this theory does not tell anything about the structure of game graphs of the concerned games, and does not work under the misère play convention. We suggest a new notion of equivalence of games, named emulational equivalence. It is stronger than the SpragueGrundy equivalence and weaker than the isomorphism of game graphs, and it can be applied regardless of whether the game is played under normal or under misère convention. Additionally, we introduce a new game on graphs named Hackenforb, which turns out to have a great emulational potential, namely, for various impartial games we were able to construct Hackenforb instances emulationally equivalent to them.
(Joint work with Bojan Bašić and Nikola Milosavljević)
Eric Duchêne, LIRIS, Lyon 1 University, France
Title: Partizan subtraction games
Abstract: Partizan subtraction games are combinatorial games where two players, Left and Right, alternately remove a number n of tokens from a heap of tokens, with n∈SL (resp. n∈SR) when it is Left’s (resp. Right’s) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current work, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, namely weak dominance, fairness and ultimate impartiality. We consider the problem of computing this behavior with respect to SL and SR, which is connected to the wellknown Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets SL and SR have size at most 2.
(Joint work with Marc Heinrich, Richard J. Nowakowski, and Aline Parreau)
Florian Galliot, University of Grenoble Alpes, France
Title: The MakerBreaker game on hypergraphs of rank 3: structural results and a polynomialtime algorithm
Abstract: In the MakerBreaker positional game, Maker and Breaker take turns by picking vertices from a hypergraph H, and Maker wins if and only if he claims all the vertices of some edge of H. We introduce a general notion of danger at a vertex x, which is a subhypergraph representing an urgent threat that Breaker must hit with his next pick if Maker picks x. Applying this concept in hypergraphs of rank 3, we get a structural characterization of the winner with perfect play as well as optimal strategies for both players based on danger intersections. More specifically: we construct a family F of dangers such that a hypergraph H of rank 3 is a Breaker win if and only if the Fdangers at x in H intersect for all x. By construction of F, this will mean that H is a Maker win if and only if Maker can guarantee the appearance, within the first three rounds of play, of a specific elementary subhypergraph (on which Maker easily wins) consisting of a linear path or cycle. This last result has a consequence on the algorithmic complexity of deciding which player has a winning strategy on a given hypergraph: this problem, which has been shown by Rahman and Watson to be PSPACEcomplete on 6uniform hypergraphs, is in polynomial time on hypergraphs of rank 3. This validates a conjecture by Rahman and Watson. Another corollary of our result is that, if Maker has a winning strategy on a hypergraph of rank 3, then he can ensure to claim an edge in a number of rounds that is logarithmic in the number of vertices.
(Joint work with Sylvain Gravier and Isabelle Sivignon)
Hikaru Manabe, Keimei Gakuen Elementary Junior & Senior High School, Japan
Title: Fourdimensional chocolate games and chocolate games with a pass move
Abstract: The authors have studied threedimensional chocolate bar games that are variants of game of Nim, and they will generalize them to the case of fourdimensional chocolate bar games. They presented the necessary and sufficient condition whereby a threedimensional chocolate bar may have a Grundy number (p1) xor (q1) xor (r1), where p, q, and r are the length, height, and width of the bar in 2021. In this talk, they will present a fourdimensional version of the theorem and use it for the threepile nim with a pass move. Here, we modify the game’s standard rules to allow a onetime pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once either player has used a pass, it is no longer available. It is wellknown that in classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity. Using the fourthdimensional coordinate for the pass move, we can treat a threepile nim with a pass move as a fourdimensional chocolate bar game. This approach opens a new perspective on the complexity of the traditional threepile with a pass.
(Joint work with Aditi Singh, Yuki Tokuni, and Ryohei Miyadera)
Hironori Kiya, Kyushu University, Japan
Title: Normalplay with deadendwinning convention
Abstract: We consider a new winning convention of partizan games. The loser of a game under this convention is the player who cannot play anymore except for the case that the position is dead for the player, that is, the player can no longer make moves; In this case, they win. A famous game under this convention is Shichinarabe, which is similar to card Dominoes, sevens, Fan Tan, and Showdown. We characterize the game such that in each turn the number of options is at most one.
(Joint work with Koki Suetsugu)
Kanae Yoshiwatari, Nagoya University, Japan
Title: Complexity of Colored Arc Kayles
Abstract: Cram, Domineering, and Arc Kayles are wellstudied combinatorial games that can be interpreted as edgeselectingtype games on graphs. In this talk, we introduce a generalization, called Colored Arc Kayles (which includes these games), discussing its complexity.
(Joint work with Tesshu Hanaka, Hironori Kiya, and Hirotaka Ono)
Keito Tanemura, Kwansei Gakuin University, Japan
Title: Chocolate games with a pass and an application of symbolic regression to these games
Abstract: The authors present their research on several combinatorial games with a pass move, and the application of symbolic regression to these games. These games are chocolate games, Moore’s Nim, and Restricted Nim. Here, the game’s standard rules are modified to allow a onetime pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once either player has used a pass, it is no longer available. It is wellknown that in classical threepile Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity, but in chocolate games, Moore’s Nim and Restricted Nim the pass move was found to have a minimal impact. There is a simple formula for the previous player’s position for these games. In chocolate games and Restricted Nim, there are also simple formulas for the cases in that Grundy numbers are 1,2,3, …. When the authors studied the previous player’s positions and formulas for Grundy numbers, they used a symbolic regression library that the authors made. Each formula is different for each Grundy number, and discovering the formula is a timeconsuming task for human beings. Therefore, symbolic regression is a valuable tool in the research of combinatorial games. The authors present the basic structure of their software in this article.
(Joint work with Yuji Sasaki, Hikaru Manabe, Yuki Tokuni, and Ryohei Miyadera )
Koki Suetsugu, National Institute of Informatics, Japan
Title: Some new universal partizan rulesets
Abstract: A universal partizan ruleset is a ruleset in which every finite value can appear as a position. Generalized Konane is the only ruleset which has been proven as a universal partizan ruleset. In this talk, we introduce three new partizan univesal rulesets. One is proved by a constructive method. The others are proved by reduction. Reduction is often used for proofs of complexity (like PSPACEComplete or NPComplete). In this talk, we show that it can also be used for proving that a ruleset is a universal partizan ruleset.
Kyle Burke, Florida Southern College, United States of America
Title: ForcedCapture Hnefatafl
Abstract: Hnefatafl is an ancient game that has enjoyed recent popularity, though the exact original rules are uncertain. We describe a variant, ForcedCapture Hnefatafl, where players are required to make a capturing move when available. In this talk, we will show that it is PSPACEhard to determine which player has a winning strategy.
(Joint work with Craig Tennenhouse)
Michael Fisher, West Chester University, United States of America
Title: Olympic games: three impartial games with infinite octal codes
Abstract: In this talk we will look at three subtraction games, each describable using an infinite octal code. The Ppositions will be enumerated. Additionally, we will see why knowing the complete Grundy function is likely out of reach.
Milos Stojakovic, University of Novi Sad, Serbia
Title: Strong avoiding positional games
Abstract: Given an increasing graph property F, the strong AvoiderAvoider F game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses F first loses the game. If the property F is «containing a fixed graph H», we refer to the game as the H game. We show that Blue has a winning strategy in two strong AvoiderAvoider games, P4 game and CC>3 game, where CC>3 is the property of having at least one connected component on more than three vertices. These are some of the first nontrivial AvoiderAvoider games for which the outcome is determined. We also study a variant, the strong CAvoiderCAvoider games, with the additional requirement that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in several trong CAvoiderCAvoider games.
(Joint work with Jelena Stratijev)
Nacim Oijid, University of Lyon, France
Title: Bipartite instances of Influence
Abstract: The game Influence is a scoring combinatorial game that has been introduced in 2021 by Duchene et al. It is a good representative of Milnor’s universe of scoring games, i.e. games where it is never interesting for a player to miss his turn. New general results are first given for this universe, by transposing the notions of mean and temperature derived from nonscoring combinatorial games. Such results are then applied to Influence to refine the case of unions of paths started in the previous paper. The computational complexity of the score of the game is also solved and proved to be PSPACEcomplete. We finally focus on some specific cases of Influence when the graph is bipartite, by giving explicit strategies and bounds on the optimal score on structures like grids, hypercubes or tori.
Nandor Sieben, Northern Arizona University, United States of America
Title: Impartial hypergraph games
Abstract: We study two building games and two removing games played on a finite hypergraph. In each game two players take turns selecting vertices of the hypergraph until the set of jointly selected vertices satisfies a condition related to the edges of the hypergraph. The winner is the last player able to move. The building achievement game ends as soon as the set of selected vertices contains an edge. In the building avoidance game the players are not allowed to select a set that contains an edge. The removing achievement game ends as soon as the complement of the set of selected vertices no longer contains an edge. In the removing avoidance game the players are not allowed to select a set whose complement does not contain an edge. We develop some generic tools for finding the nimvalue of these games and show that the nimvalue can be an arbitrary nonnegative integer. The outcome of many of these games were previously determined for several special cases in algebraic and combinatorial settings. We provide several examples and show how our tools can be used to refine these results by finding nimvalues.
Neil McKay, University of New Brunswick, Canada
Title: Numbers and ordinal sums
Abstract: There are many rulesets in which all values are numbers and yet the values are difficult to compute. BlueRed Hackebush stalk values are easy to compute as the positions are ordinal sums of numbers in canonical form. Recently some rulesets have had positions described using ordinal sums but as the summands are not in canonical form the values are hard to understand and compute. We explore forms of numbers for which values are easy to compute and a ruleset, Teetering Towers, where we can effectively compute values.
Paul Ellis, Rutgers University, United States of America
Title: The arithmeticperiodicity of Cut for C = {1, 2c}
Abstract: Cut is a class of partition games played on a finite number of finite piles of tokens. Each version of Cut is specified by a cutset C ⊆ N. A legal move consists of selecting one of the piles and partitioning it into d + 1 nonempty piles, where d ∈ C. No tokens are removed from the game. It turns out that the nimset for any C = {1, 2c} with c ≥ 2 is arithmeticperiodic, which answers an open question of Dailly et al. (Discrete Applied Mathematics, 285, 2020). The key step is to show that there is a correspondence between the nimsets of Cut for C = {1, 6} and the nimsets of Cut for C = {1, 2c}, c ≥ 4. The result easily extends to the case of C = {1, 2c_{1} , 2c_{2}, 2c_{3}, …}, where c_{1}, c_{2}, c_{3},… ≥ 2.
(Joint work with Thotsaporn Aek Thanitapinonda)
Prem Kant, Indian Institute of Technology Bombay, India
Title: Bidding combinatorial games
Abstract: We generalize the alternating play convention in normal play combinatorial games by means of Discrete Richman Auctions (Develin et al. 2010, Larsson et al. 2021, Lazarus et al. 1996). Under this framework, for infinitely many monoids of short games, we propose algorithmic play solutions to compare games. We then establish various general results such as group structures of integers and dyadic rationals, a simplicity theorem and existence of infinitesimals.
(Joint work with Urban Larsson, Ravi K. Rai, and Akshay V. Upasany)
Richard J. Nowakowski, Dalhousie University, Canada
Title: The game of Flipping Coins
Abstract: We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers (there is a link to Alda Carvalho’ talk), and these are found by first applying a reduction, then decomposing the position into an iterated ordinal sum. This is unusual since moves in the middle of the line do not eliminate the rest of the line. Moreover, when G is decomposed into lines H and K, then G = H:R, where R are the right options of K. This is in contrast to Hackenbush Strings where G = H:K^{}.
(Joint work with Anthony Bonato and Melissa Huggan)
Silvia Heubach, California State University, United States of America
Title: On the Structure of the Ppositions of Slow Exact kNim
Abstract: Slow Exact kNim is a variant of the wellknown game of Nim. The rules of this variant are that in each move, k of the n stacks are selected and then one token is removed from each of the k stacks. The last player to move wins. We prove results on the structure of the Ppositions for the infinite family of games where we play on all but one of the n stacks.
(Joint work with Matthieu Dufour)
Svenja Huntemann, Concordia University of Edmonton, Canada
Title: Temperature of Partizan ArcKayles Trees
Abstract: Partizan ArcKayles (PArcK) is a generalization of both ArcKayles and Domineering. It is played on any finite, simple graph in which the edges are coloured red or blue. The players take turns removing an edge of their colour, including the two incident vertices, until the active player has no possible moves. Motivated by the conjecture that the temperature of Domineering is at most 2, we are studying the temperature of PArcK, concentrating on trees to begin with.
(Joint work with Neil McKay)
Tomoaki Abuku, National Institute of Informatics, Japan
Title: A Multiple Hook Removing Game whose starting position is a rectangular Young diagram with unimodal numbering
Abstract: We introduce a new impartial game named Multiple Hook Removing Game (MHRG for short). We also determine the Gvalues of some game positions (including the starting positions) in MHRG(m,n), the MHRG whose starting position is the rectangular Young diagram of size m×n with the unimodal numbering. In addition, we prove that MHRG(m,n) is isomorphic, as games, to MHRG(m,n+1) (if m ≤ n and m+n is even), and give a relationship between MHRG(n,n+1) (and MHRG(n,n)) and HRG(S_{n}), the Hook Removing Game in terms of shifted Young diagrams.
(Joint work with Masato Tada)
Urban Larsson, Indian Institute of Technology Bombay, India
Title: Feasible outcomes of Bidding Combinatorial Games
Abstract: This is the first part of two talks on normalplay discrete Richmantype Bidding Combinatorial Games that generalize the alternating play convention. Previous work on such games did not generalize the normalplay convention. For any fixed total budget (a nonnegative integer), 1) we define the notion of a feasible outcome and prove that every game has a feasible outcome; 2) we construct a game for any given feasible outcome class. The second talk is by Prem Kant, Indian Institute of Technology Bombay.
(Joint work with Prem Kant, Ravi K. Rai, and Akshay V. Upasany)
Aaron Siegel, United States of America
Alda Carvalho, ISELIPL&CEMAPRE/REMUniversity of Lisbon, Portugal
Alfie Davies, Memorial University, Canada
Aline Parreau, CNRS, Lyon 1 University, France
Ana Paula Garrão, University of Azores, Portugal
Antoine Dailly, Laboratory of Informatics, Modelling and Optimization of the Systems, France
Bernhard von Stengel, London School of Economics, United Kingdom
Bojan Bašić, University of Novi Sad, Serbia
Carlos Pereira dos Santos, Center for Mathematics and Applications (NovaMath), FCT NOVA, Portugal
Colin Wright, Solipsys Limited, United Kingdom
Craig Tennenhouse, University of New England, United States of America
Dana Ernst, Northern Arizona University, United States of America
Danijela Popović, Mathematical Institute of SASA, Serbia
David Wolfe, Verisk, Canada
Eric Duchêne, LIRIS, Lyon 1 University, France
Florian Galliot, University of Grenoble Alpes, France
Hikaru Manabe, Keimei Gakuen Elementary Junior & Senior High School, Japan
Hironori Kiya, Kyushu University, Japan
Hirotaka Ono, Nagoya University, Japan
Jeanette Shakalli, FUNDAPROMAT, Panama
Kanae Yoshiwatari, Nagoya University, Japan
Keito Tanemura, Kwansei Gakuin University, Japan
Koki Suetsugu, National Institute of Informatics, Japan
Kyle Burke, Florida Southern College, United States of America
Margarida Raposo, University of Azores, Portugal
Matthieu Dufour, University of Quebec, Canada
Michael Fisher, West Chester University, United States of America
Milica Maksimović, University of Novi Sad, Serbia
Milos Stojakovic, University of Novi Sad, Serbia
Nacim Oijid, University of Lyon, France
Nandor Sieben, Northern Arizona University, United States of America
Neil McKay, University of New Brunswick, Canada
Paul Ellis, Rutgers University, United States of America
Prem Kant, Indian Institute of Technology Bombay, India
Ricardo Teixeira, University of Azores, Portugal
Richard J. Nowakowski, Dalhousie University, Canada
Silvia Heubach, California State University, United States of America
Svenja Huntemann, Concordia University of Edmonton, Canada
Thotsaporn Aek Thanitapinonda, Mahidol University, United States of America
Tiago Hirth, University of Lisbon, Portugal
Tomoaki Abuku, National Institute of Informatics, Japan
Urban Larsson, Indian Institute of Technology Bombay, India
EVENT LOCATION AND TRANSPORTATION
The meeting will take place at Expolab – Centro de Ciência Viva. See a map here.
Please be aware that the meeting will be held at Lagoa. A daily shuttle will be in place everyday, to take people from Ponta Delgada and back.
Mornings – Departures from Hotel Canadiano (8:00 am and 8:30 am)
Afternoons – Departures from Expolab (6:00 pm and 6:30 pm)
For adventurers who like to walk, we recommend this trail (google maps).
Accommodations (some suggestions):
 Hotel Canadiano de Ponta Delgada
 Hotel Marina Atlântico
 NEAT Hotel Avenida
 Hotel Ponta Delgada
 MS Vila Nova
 Hotel Camões
 Thomas Place
Tourist Information
São Miguel has a lot of fantastic programs. A lot! We leave here three suggestions, according to our personal taste.
Suggestion 1: Visit Caldera of the Furnas Volcano + Terra Nostra Garden + Lunch at Tony’s
In Terra Nostra (google maps), it is mandatory to take a bath and visit the garden. You should take a bath, be prepared for that!
At Tony’s (google maps) you really have to eat a volcanic dish (Cozido das Furnas)!
All this takes place in Furnas. Everything can be done in the same day.
Suggestion 2: Take a bath in Dona Beija Thermal Pools (google maps). In fact, we could have mentioned this in the first suggestion, as it is also in Furnas.
But, there would be many activities in the same day. It is open at night and it must have a special status!
Suggestion 3: Visit Lagoon of the Seven Cities. You should go down and have a coffee at Green Love (google maps).
More info here
Conference Dinner and Restaurants
Conference Dinner
The Conference Dinner is scheduled for January 24th, Tuesday, at 7:00 PM. The event will take place at the VIP Executive Azores Hotel, whose address is Rotunda de São Gonçalo nº 131 – S. Pedro, 9500343, Ponta Delgada, Portugal (google maps).
The menu is available here .
The price is €35 per person. Payment can be made on the first day of the meeting.
It is important for us to inform the restaurant about the number of people. Confirm your presence until the 15th of January.
Lunches
During the conference, we recommend the restaurant Q’enosso, whose address is Rua Tenente Coronel Ângelo Albergaria Pacheco, 12 Lagoa, São Miguel, Portugal (google maps). The restaurant will organize a Buffet for lunch with a price of €15 per person with soup, fish dish, meat dish and vegetarian dish, drinks included. If you are interested, you should inform the organization so that the reservation is guaranteed.