Combinatorial Game Theory Colloquium V

Alda Carvalho, DCeT-Aberta University & CEMAPRE/ISEG Research-University of Lisbon, Portugal

Title:  Cyclic impartial games with carry-on moves (Part I)

Abstract: In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able of dealing with entailing moves.This talk presents a generalization that combines these two theories, making it suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response.

(Joint work with Carlos Pereira dos Santos, Koki Suetsugu, Richard J. Nowakowski, Tomoaki Abuku, and Urban Larsson)

Alfie Davies, Memorial University, Canada

Title: What if winning moves are banned?

Abstract: Learning the winning strategy for a game like Nim can make it less fun to play. But what if we play these games with the following alteration: we are not allowed to make a move which would have been winning before. Can we leverage our knowledge of the strategy for the original game to play this new paradigm well?

Ankita Dargad, Indian Institute of Technology Bombay, India

Title: Thermograph invariance for certain games

Abstract: In the context of disjunctive sums of combinatorial games, a natural question arises: Is it advantageous to play first in a specific component? And if so, can this advantage be quantified? Temperature theory seeks to answer this by measuring the incentive to move first. A key tool in this theory is the thermograph – a geometric representation that often simplifies the computation of a game’s temperature. For the Robin Hood game, an instance of the cumulative game: Wealth Nim (arXiv:2005.06326), the canonical forms become overwhelmingly large and lack discernible patterns. However, the thermographs for Robin Hood reveal a surprising regularity, exhibiting only a limited number of shapes as the heap sizes increase. This invariance was uncovered through an exploration of the monotonicity of the stops of the game’s options and the evolving shapes of its thermographs. This discovery leads to a compelling question: under what general conditions does this thermograph invariance hold?

(Joint work with Urban Larsson and Niranjan Balachandran)

Anjali Bhagat, Indian Institute of Technology Bombay, India

Title: Fork positions and 2-dimensional toppling dominoes

Abstract: Toppling Dominoes is known as a one-dimensional combinatorial game where the dominoes are arranged in a straight line. This project introduces fork positions where the dominoes are placed in a 2-dimensional plane. We define the rules for 2-dimensional fork positions in Toppling Dominoes. We explore how fork positions are different from 1-dimensional Toppling Dominoes. We prove that doubling a single domino to make the position two-dimensional favors the player whose domino was doubled. The game values become incomparable in the case of the neutral green domino. We also prove that when we make two 1-dimensional games into a fork position again, it favors the player whose domino was used to make the fork.

(Joint work with Urban Larsson)

Balaji Rohidas Kadam,  Indian Institute of Technology Madras, India

Title: Kotzig’s Nim under misère play

Abstract: We study Kotzig’s Nim game, a combinatorial game played on a directed cycle with labeled vertices, inspired by the geography game. In Kotzig’s Nim, two players take turns moving a token that starts at one node of the cycle, advancing it clockwise around a circular board of n nodes based on a predefined set of allowed step sizes M, a proper subset of natural numbers. Although Kotzig’s Nim rules are easy to understand and have been around for several decades, there are only a few known results. All existing results regarding Kotzig’s nim have concentrated on identifying P-positions in the normal play version – where the player who cannot make a move loses. We investigate P-positions in the misère play version, where the  player unable to move wins. We show that the game equivalence theorem applicable in normal play also applies to misère play. Our analysis mainly focuses on move sets M consisting of two elements. We also establish some general results about the winning player in the misère game Γ(M={a,a+1}; n).

(Joint work with Shaiju A. J.)

Bernhard von Stengel, London School of Economics, UK

Title: Combinatorial Game Theory Collection (IJGT)

Abstract: The International Journal of Game Theory (IJGT) is inviting submissions of significant papers in Combinatorial Game Theory. All articles are reviewed to the high standards of IJGT (one-sided blind peer review). Accepted papers are published online immediately after production, and are added to the CGT online Collection at https://link.springer.com/collections/jhjdeifbdf. Two editions of CGT papers have been and are about to be published in printed issues of IJGT, the first as Issue 2 of Volume 47 in 2018, the second as Issue 4 of Volume 53 in 2024. The CGT Collection has been aligned with the editions of the Combinatorial Game Theory Colloquia. The special collection does not represent formal proceedings. IJGT welcomes high quality submissions related to works presented at the conference. This brief talk serves to share some information and answer questions.

(Joint work with Carlos Pereira dos Santos and Urban Larsson)

Bojan Bašić,  University of Novi Sad, Serbia

Title: Yet another (not so well-known) funny afternoon in a jail courtyard, and its (even less well-known) variation with quite unexpected consequences

Abstract: There are many conundrums in circulation depicted as a game between prisoners and a warden. It is often the case that, in spite of their seemingly naive (cheerful?) presentation, there are deep mathematical theories that lurk beneath them. For some of them, however, it looks as though they remain just (light) brain-teasers and nothing more. As a warmer-upper, we present something from this latter class (which is, in contrast to Ebert’s n-prisoners puzzle, the hats-in-a-line game, etc., significantly lesser known). We then ask the same question for a slight variation of the same problem. Think very conscientiously before trying to guess the answer, it is very likely that you will be wrong!

Carlos Pereira dos Santos, Center for Mathematics and Applications (NOVA Math), FCT NOVA, Portugal

Title:  Cyclic impartial games with carry-on moves (Part II)

Abstract: Look at Alda Carvalho’s abstract.

(Joint work with Alda Carvalho, Koki Suetsugu, Richard J. Nowakowski, Tomoaki Abuku, and Urban Larsson)

Craig Tennenhouse, University of New England, USA

Title: Temperature of Partizan Arc Kayles

Abstract: Motivated by the longstanding conjecture that the temperature of Domineering is bounded by two we examine Partizan Arc Kayles as a generalization, and show that this game has unbounded temperature on trees.

(Joint work with Svenja Huntemann and Neil McKay)

Craig Tennenhouse, University of New England, USA and Kyle Burke, Florida Southern College, USA

Title: A CGT Book for early undergraduates

Abstract: In this talk, we present a textbook written in January 2021 (Craig Tennenhouse and Kyle Burke; https://kyleburke.info/CGTBook.php). The main goal of the book is to provide a free introductory CGT text suitable for undergraduate students interested in the subject. The book can also be used as a discrete math text, as there is no requirement for prior knowledge of discrete math.

Danijela Popović, Mathematical Institute of SASA, Serbia

Title: Hackenforb vs. Invariant games

Abstract: Hackenforb is a game on graphs in which players are given a graph G and a set of forbidden connected graphs F. A move consists in removing one edge and additionally all the remaining connected components that are isomorphic to some forbidden graph. In previous work it was shown that Hackenforb is a powerful game that can mimic various other games. In this talk we will consider if Hackenforb can mimic Invariant games which were defined by Duchêne and Rigo (2010).

(Joint work with Bojan Bašić)

François Carret, École Normale Supérieure de Lyon, France

Title: Split sums of Nim games with a large number of piles

Abstract: In a split sum of two games, a player can play on either game but, once the left game is finished, the entire game finishes. It works as a generalisation of a game with a PASS option, where a player on their turn can pass, as long as it is not the final move. We have found that in split sums of Nim games with a significant number of piles, the split sum closely approximates the classic disjunctive sum with respect to the P-positions.

Harman Agrawal, Indian Institute of Technology Bombay, India

Title: QuadroCount: A Combinatorial Game

Abstract: We define QuadroCount, a two-player grid-based partizan game. A given position is a configuration with n pieces for each player Left and Right. The individual area of each pair of pieces is computed by treating them as corner stones, as shown in the figures, which have areas 8 and 4, respectively.

The black squares represent Left’s pieces and the yellow squares represent Right’s pieces. The OverLapping Area or olá is the sum of all individual areas, computed as

where (x_i,y_i) denotes the coordinates of the i-th piece. Every move must decrease the olá, and a player who cannot do so loses (normal play). We propose sequences of terminal configurations that range over N and identify all Nash equilibrium-type local minima for n = 2. We also establish game values and conjecture that a player can have at most a one-move advantage over the other player.

(Joint work with Urban Larsson)

Helena Verrill, University Warwick, UK

Title: Grundy sequence of the Cut Game C(1,2)

Abstract: I will describe a formula for the values G(n) for the Cut Game C(1,2), which is a heap game, where players alternately divide heaps of counters into 2 or 3 piles.

(Joint work with Alfie Davies)

Hideki Tsuiki, Kyoto University, Japan

Title: A cellular automaton for Blocking Nim

Abstract: In a k-blocking play of a game,  the previous player can block up to k-1 potential moves of the opponent at each turn, restricting the opponent’s options. When k=1, the game reverts to its original form, and for k=2, Holshouser and Reiter analyzed the 3-pile Blocking Nim. However, for k≥3, the complexity of analysis increases significantly. A useful tool to analyze a blocking game is the surplus number, defined as the number of winning moves minus k.  Cook, Larsson, and Neary introduced a cellular automaton for calculating surplus numbers of blocking Wythoff’s game. In this talk, we adapt their approach to n-pile Nim games, introducing a cellular automaton for calculating surplus numbers for blocking Nim, particularly in the 3-pile case.  We reveal intriguing patterns in the arrangement of surplus numbers generated by this cellular automaton, and examine some properties of this cellular automaton.

(Joint work with Hiromi Hasegawa and Katsunobu Imai)

Hikaru Manabe, Keimei Gakuen Elementary Junior & Senior High School, Japan

Title: Maximum Nim and the Josephus problem

Abstract: This study examines the relation between the Grundy numbers of Maximum Nim and the Josephus problem.  In the original Josephus problem, every k-th number is to be removed from (0,1,2,…, n-1) arranged in a circle for some natural numbers n and k. Note that in the second round of the removing process, we neglect numbers that are removed in the first round when we remove the k-th number, and in the third, fourth round,.., we do the same.  Let f(x) = ëx/kû , where ë û  is the floor function and k is a natural number larger than 1. We have the following:

Theorem. Let G_f (m) be the Grundy number of Maximum Nim with the rule function f. Then, G_f (nk-1-m) = n–i if and only if m is the i-th number to be removed in the Josephus problem when i < n and m is the last number that remains when i = n.

We generalize the Josephus problem. There are numbers (0,1,2,…,n-1) arranged in a circle, and we remove the k₁-th, k₂-th, …,k_n-th numbers. Note that when k₁=k₂=k₃=⋯=k_n, this is the same as the original Josephus problem. This generalization include many variants. For example, block Josephus problem in which we skip s numbers and then remove k numbers proceeding around the circle.

We also have a theorem with respect to this generalized Josephus problem and Maximum Nim. We define g as the following. If

we define g(x)=i. Then, we have the following result that is a generalization of the previous theorem.

Theorem.  When G_g (m) is the Grundy number of Maximum Nim with the rule function g, we have

if and only if m is the t-th number to be removed in the Josephus problem when t < n and m is the last number that remains when t = n.

Using the above two theorems, we can build a new algorithm for the Josephus problem.

(Joint work with Koki Suetsugu, Shoei Takahashi, and Ryohei Miyadera)

Hikaru Manabe, Keimei Gakuen Elementary Junior & Senior High School, Japan

Title: Amalgamation Nim with a restriction on amalgamation

Abstract: S.C. Locke and B. Handley presented the original Amalgamation Nim in which players can use one of the following two options:

(i) To choose one of the piles and remove any number of stones from the pile;

(ii) To amalgamate two piles of stones into one pile when the number of stones in two piles is not zero.

The player who removes the last stone or stones is the winner.

I propose a variant of Amalgamation Nim in which we can amalgamate two piles of stones into one pile when the number of stones in two piles equals to or exceeds 2. This variant of Amalgamation Nim has very elegant features. I present the formula that describes the set of P-positions for this game, and the set of P-positions is the union of a set of positions whose Nim-Sum is 0 and a set of positions whose Nim-Sum is 1. I have the following conjecture. In this game, the set of positions whose Grundy number is 2k+1 is the sum of a set of positions whose Nim-Sum is 0 and a set of positions whose Nim-Sum is 1.

Hiroki Inazu, Hiroshima University, Japan

Title: Ending Partizan quotient for Octal code 4 · 0 · · · 01

Abstract: Ending Partizan Nim is a ruleset, which was introduced by Hiyu Inoue and Shin-nosuke Kadowaki in 2024 (and will appear in this conference), where the both players are allowed exactly the same options for each phase of the game, but is partizan at the game ending, for example the Left wins if the number of remaining tokens is even and the Right wins if the number is odd. In this talk, let k > 0 be an odd integer, and we consider the Octal code 4 · 0 · · · 01 with 1 at the k-th place, namely the players can choose one heap and separate into two, or the other option is to remove exactly k tokens from one heap.

We will describe the winning strategy in terms of a Partizan quotient, which is a generalization of misère quotient in impartial misère games. Let G_k be the set of all positions for this Ending Partizan Nim. For H ∈ G_k, the outcome o(H) equals to L, R, N and P if there is the winning strategy for Left, Right, Next player or Previous player, respectively. For H₁, H₂ ∈ G_k, we denote H₁ ∼ H₂ if the outcomes o(H₁ + X) and o(H₂ + X) are equal for all X ∈ G_k, where «+» means the disjunctive sum. «∼» is an equivalence relation and we denote equivalence class of H ∈ G_k under «∼» by [H]. For  H₁, H₂ ∈ G_k, we define [H₁] · [H₂] = [H₁ + H₂], where «+» means the disjunctive sum, then (G_k/ ∼, ·) forms a semi-group. Let (n) be one heap position with n tokens, and let a, b, c, d, and e be equivalence class, where a = [(1)], b = [(2)], c = [(3k)], d = [(k)], and e = [(2k)]. Then Q is generated by a, b, c, d, e, and Ending Partizan quotient (Q,P,L, R) is represented as

Q = ⟨a, b, c, d, e | a2=1, b2=1, c2=1, d4=d2, e5=e3, ae3=e3, be4=e3, ce3=e3, d2e=e⟩

G_k is devided into |Q| = 68 classes, with

P = {d2, ad2, e2, ae2, e4}

L = {1, b, ac, abc, acd2, acd3, be, ace, abce, ade, ace2, acde2}

R = {a, ab, c, bc, cd2, cd3, abe, ce, bce, de, ce2, cde2}

and the remaining 39 classes are in N.

Hironori Kiya, Osaka Metropolitan University, Japan

Title: Traffic Jam with various car sizes

Abstract: Traffic Jam, proposed by Urban Larsson, is an affine normal-play game (Santos et al. 2024) that involves transporting dominoes, representing cars, through a multi-lane crossing. In this presentation, we introduce a Traffic Jam game with various sizes of cars (i.e., non-domino-size cars) and its best or better strategy and winning positions. We prove that most of these variants are N-positions.

(Joint work with Hiroki Inazu and Urban Larsson)

Hiyu Inoue, Hiroshima University, Japan

Title: Ending Partizan Subtraction Nim

Abstract: We consider Subtraction Nim, where two players have exactly same options, but are partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. The example we consider is the following: Let the set of removable numbers S be a non-empty subset of positive integers greater than or equal to 2, which is applied for both players. At the end of the game, Left wins if the number of remaining tokens is even, and Right wins if the number of remaining tokens is odd.  We computed the outcomes for many S, and found a surprinsing phenomenon that in many examples of S (to be precise, 5474 cases out of 8192 examples of S with 2 ∈ S and the maximum of S less than or equal to 15), the outcomes are L-position for all large enough n. In comparison, R-positions apper occasionally, if any.  Our theorems explain why that phenomenon occurs. We prove that n±1 are L-positions when n is R-position. We also prove that P-positions last only min(S)−1 at most, and both side of them are L-positions. In addition, if S is finite, N-position last only max(S)-1 at most, and both side of them are L-positions. Only L-positions can last forever.

 (Joint work with Shin-nosuke Kadowaki)

Martin Mueller, University of Alberta, Canada

Title: A search-based approach for solving sum games

Abstract: In combinatorial games, the most fundamental question is «who wins?». We study general purpose algorithms for efficiently answering this question for specific short game positions, especially for the case when the given position is a sum of independent subgames. The most popular existing software tools, such as Siegel’s CGSuite, are built around the fundamental concept of canonical form. However, for solving sum games, computing canonical forms can very quickly become a major bottleneck – it is often a much more expensive procedure than «just» finding the winner. We report ongoing work on a new Minimax-based Combinatorial Game Solver (MCGS), a tool for finding the winner of a sum game by search. MCGS answers questions of the form «does player P win game G?». We develop specific algorithms for this purpose, which avoid computing canonical forms. Subgame decomposition and simplification techniques based on minimax search and on basic principles of combinatorial games greatly increase the efficiency of MCGS. After some motivational examples, we define our search framework and show first computational results for one-dimensional strips of combinatorial games such as Clobber and NoGo.

(Joint work with Henry Du, Taylor Folkersen, Zahra Bashir, and Fatemeh Tavakoli)

Michael Fisher, West Chester University , USA

Title: Atomic variations of Roll the Lawn and Cricket Pitch

Abstract: Nowakowski and Ottaway introduced two games in a 2011 paper as examples of option-closed games.  The first game is Roll the Lawn, and it uses a row of bumps (nonnegative integers) and a roller that is either between two bumps or at one end of the row. Left moves the roller to the left flattening each bump by 1 unless the bump has been flattened to 0 in which case nothing happens to that bump. For a move to be legal, at least one bump must be flattened by 1. Right moves the roller to the right, with the same effect and constraint. In Cricket Pitch, there’s an extra constraint: the roller cannot go over a bump that has already been flattened to 0. For two of the variations considered in this talk, we imagine that the bumps are green Hackenbush stalks.  In addition to the rules above, we allow each player to make a Nim move on any one Hackenbush stalk of nonzero height.  As the canonical forms become complicated very quickly, we instead provide a formula for the atomic weight of a given position. The next generalization considered is a variant of Roll the Lawn.  In this variant, we replace the roller with a fence which Left may move to the right over any number of stalks, reducing each nonzero stalk by one edge as it moves over it. Left may also make a Nim move on any one nonzero stalk on her side of the fence. Right’s moves are similar. Our final variation is like the one above, but it includes the Cricket Pitch restriction.  As with the other two variations, canonical forms quickly become messy. Thus, we turn to atomic weight to make sense of the game.

Miloš Stojaković, University of Novi Sad, Serbia

Title: Generalized saturation game

Abstract: We study a game version of the generalized graph Turán problem. For two fixed graphs F and H, two players, Max and Mini, alternately claim unclaimed edges of the complete graph such that the graph with the claimed edges must remain F-free throughout the game. The game ends when no further edges can be claimed, i.e. when the graph becomes F-saturated. The H-score of the game is the number of copies of H. Max aims to maximize the H-score, while Mini wants to minimize it. We look for the H-score for several natural choices of F and H, when both players play optimally.

(Joint work with Balázs Patkós, Jelena Stratijev, and Máté Vizer)

Kanae Yoshiwatari, Nagoya University, Japan

Title: Universal and polynomially decidable rulesets on grid graphs

Abstract: In combinatorial game theory, a ruleset is called universal if it is a habitat of the short Conway group. Several games, such as Portuguese Konane, Turning Tiles, and others, have been proven to be universal, and they all are PSPACE-complete in terms of the winner determination. This study demonstrates that there exist universal and polynomially decidable rulesets on grid graphs. The rulesets are variants of partisan Generalized Geography, on which arbitrary game value can be expressed as a position of a grid graph with the restriction that the winner determination can be done in polynomial time. These are the first rulesets that realize the universality and the polynomial decidability. As a byproduct, we show that the rulesets under the restriction help prove the universality of other games. To illustrate this, we give an alternate proof of the universality of Turning Tiles and demonstrate that a variant of Chess, called Inertially Capturing Chess, is also universal.

(Joint work with Koki Suetsugu and Hirotaka Ono)

Koki Suetsugu, Hiroshima University and Waseda University, Japan

Title: A survey on universalities

Abstract: In these days, the universalities of rulesets are discussed in many studies. A universal ruleset R on a set of combinatorial game values S is a ruleset such that every position in R is equal to a value in S, and every value in S is equal to a position in R. For example, NIM is a universal ruleset of all impartial values, Blue-Red-Hackenbush is a universal ruleset of all dyadic numbers, Portuguese Konane (or Generalized Konane) is a universal ruleset of all partisan values. Universality is a similar concept to completeness in computational complexity. In this talk, I will show recent notable studies on the universality of rulesets. In particular, I will highlight the result on the first universal ruleset of all partisan dicotic values, which is shown by myself. I will also introduce how the methods of induction and reduction work on the proof of universalities.

Koki Suetsugu, Hiroshima University and Waseda University, Japan

Title: A book by Abuku, Sakai and Suetsugu

Abstract: We, Abuku, Sakai, and Suetsugu, wrote a Japanese CGT book and it was published in 2024. This talk will provide an overview of the book and how it was being written. We also introduce how we organize Japanese CGT community and conferences.

(Joint work with Tomoaki Abuku and Ko Sakai)

Kosaku Watanabe, Hiroshima University, Japan

Title: The fractal structure in the P-positions of Wythoff’s game variationsu

Abstract: In this talk, we consider generalizations of the classical two heaps Wythoff Nim: Letting a be a positive integer and c be a non-negative integer, in (a,c)-Wythoff Nim, the player can take either a or more tokens from one heap, or take tokens from both heaps, say i tokens from the first heap and j tokens from the second heap, so that both i and j are positive, and |i–j|≤c holds. (1,0)-Wythoff Nim is the classical Wythoff Nim.

We describe the set of P-positions of the normal play of those games. Interestingly, fractal structure by symbolic substitution appears, which is the key observation in the following description. For example for (2,0)-Wythoff Nim, the P-positions (x,y) with x≤y are (0,0),(0,1),(2,4),(2,5),(3,7),(3,8),(6,12),(6,13),⋯, in other words, (e₀,e₀),(e₀,e₀+1),(e₁,e₁+2),(e₁,e₁+3),⋯, (e_k,e_k+2k),(e_k,e_k+2k+1),(e_k+1,e_k+1+2(k+1)), ⋯ so that all positive integers appear exactly once as the x-coordinate or y-coordinate. Then the difference sequence (d₁,d₂,d₃,⋯)=(e₁–e₀, e₂–e₁, e₃–e₂,⋯) is (2,1,3,3,1,1,3,1,1,3,3,⋯), then by sending 2→2, 1, 1→3, and 3→3, 1, 1,

and one can see that the difference sequence is stable by this substitution, which describes the fractal picture below.

For this picture, The horizontal axis represents the index , and the line chart is plotted starting from 0, so that if the value of d_i is 1, it goes down by 1, and if it is 3, it goes up by 1. And this chart coincident to the plot of the values of e_i-2i. If you invert the vertical axis and double the horizontal axis, it will appear to match the original chart. For example, it follows that lim(i → ∞) e_i/i = 2.

(Joint work with Shun-ichi Kimura and Takahiro Yamashita)

Kyle Burke, Florida Southern College, USA

Title: Triangular-Grid Col is PSPACE-complete

Abstract: The computational complexity of Col has enjoyed a lot of attention in the past decade.  Col is a 2-player  placement game on graphs where a turn consists of painting an uncolored vertex with your own color, with the restriction that you are not allowed to choose a vertex adjacent to one already in your color.  It was shown to be PSPACE-complete on uncolored non-planar graphs in 2015 and on planar graphs in 2018.  We show that Col is PSPACE-complete on triangular grids via a reduction from Bounded (2-player) Constraint Logic.

(Joint work with Craig Tennenhouse)

Paul Ellis,  Manhattanville College, USA

Title: The Penults of Tak: Adventures in impartial, normal-play, positional games

Abstract: For normal play, impartial games, we define penults as those positions in which every option results in an immediate win for the other player. We explore the number of tokens in penults of two positional games, Impartial Tic and Impartial Tak. We obtain a complete classification in the former case. We then explore winning strategies and further directions.

(Joint work with Boris Alexeev, Michael Richter, and Thotsaporn Aek Thanatipanonda)

Paul Ellis,  Manhattanville College, USA

Title: Categories of impartial rulegraphs and gamegraphs

Abstract: The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization rulegraph, based on the natural description of a game as a digraph where the vertices are positions and the arrows represent possible moves. Such digraphs form a category where the morphisms are option preserving maps. We study several versions of this category. Our development includes congruence relations, quotients, and isomorphism theorems and is analogous to the corresponding notions in universal algebra. The quotient by the maximum congruence relation produces an object that is essentially equivalent to the traditional model. After the development of the general theory, we count the number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the number of positions.

(Joint work with Dana Ernst, Nandor Sieben, Bojan Basic, and Danijela Popovich)

Prem Kant, Indian Institute of Technology Bombay, India

Title: Numbers and infinitesimals in bidding combinatorial games

Abstract: Discrete Richman Bidding Combinatorial Games that generalize alternating normal play were introduced by Kant, Larsson, Rai, and Upasany (2022). In this framework, the properties of integers, dyadic rationals, and defined zugzwang positions were explored in subsequent work by Kant, Larsson, Rai, and Upasany (2024). In the current study, we extend this work to investigate the structure of numbers and infinitesimals within the same bidding setup.

(Joint work with Urban Larsson)

Richard Nowakowski, Dalhousie University, Canada

Title: All-small Toppling Dominoes

Abstract: Toppling Dominoes is played with a strip of red and blue upright dominoes – Left topples a blue and Right a red domino, in either direction, and all the dominoes in that direction also topple. The game was discovered to have very interesting properties. In the all-small version, a move can only be played in a strip that has dominoes of both colours. A strip is a G=*:g where g is another version of Toppling Dominoes played with the layout of G.

(Joint work with Alex Meadows, Svenja Huntemann, and Urban Larsson)

Ryan B. Hayward, University of Alberta, Canada

Title: Alternating Linear Clobber

Abstract: Clobber is an alternate-turn two-player game introduced in 2001 by Albert, Grossman, Nowakowski and Wolfe. The board is a  graph with each node colored black (here denoted with the token x), white (here the token o), or empty; player Left has black stones, player Right has white stones; on a turn, a player takes one of their stones that is adjacent to an opponent stone and clobbers the opponent’s stone (replaces it with theirs); whoever cannot move loses. Linear Clobber is Clobber played on a path, for example, one row of a go board. In 2004 Albert et al. conjectured that, for every non-empty even-length alternating-color linear Clobber position except oxoxox, the first player has a winning strategy. We prove their conjecture.

(Joint work with Xinyue Chen, Taylor Folkersen, Kamillah Hasham, Owen Randall, Luke Schultz, and Emily Vandermeer)

Shun-ichi Kimura, Hiroshima University, Japan

Title: Ending Partizan Nim with S={2, 3}

Abstract: Ending Partizan Nim was introduced by Hiyu Inoue and Shin-nosuke Kadowaki in 2024.  Let S be a set of integers larger than or equal to 2, and two players that can take s tokens from one heap as far as s is in S.  At the end of the game, Left wins if the remaining number of tokens is even, and Right wins if the number is odd.  We give a detailed analysis of this game with multiple heaps in the case S={2, 3}.

(Joint work with Koki Suetsugu, Hiyu Inoue, Shin-nosuke Kadowaki, Hiroki Inazu, Takahiro Yamashita, and Kosaku Watanabe)

Svenja Huntemann, Mount Saint Vincent University, Canada

Title: Degrees are useless in Snort when measuring temperature

Abstract: Snort is a colouring game played on any simple graph where Left colours vertices blue and Right red such that opposite colours are not adjacent. The temperature of Snort, intuitively the urgency of going first, is known to be unbounded in general. We show that further the difference between the temperature and the degree of the graph is also unbounded.

(Joint work with Tomasz Maciosowski)

Takahiro Yamashita, Hiroshima University, Japan

Title: Triangular Nim and its Wythoff variations

Abstract: In classical two heaps Nim, (x,y) is a P-position if and only if x=y. The Wythoff variation prevents this winning strategy by allowing to take i tokens from both heaps, and it is well-known that interesting mathematics appears to describe its P-positions. Triangular Nim is a variation of two heaps Nim introduced by the authors together with Prof. Urban Larsson, Tomoaki Abuku, Hironori Kiya, Indrajit Saha, and Koki Suetsugu in 2023. The players take at least two tokens from one heap, and return at least one token to the other heap, so that the total number of tokens in the heaps decreases. (x,y) is a P-position of Triangular Nim if and only if |x−y|≤1 and it is natural to apply Wythoff variations to see if interesting mathematics appears for the description of their P-positions. Our main theorem says that triangular numbers (and other polygonal numbers) appear in the description. To be more precise, in addition to Triangular Nim, if the players are allowed to take (i,i) tokens with i>0 , then the P-positions (x,y) with x≤y are listed as {(0,0), (0,1), (1,3), (3,6), (6,10), (10,15), (15,21), (21,28),⋯} , namely (t_n,t_n+1) with t_n=n(n-1)/2 where n=0,1,2,⋯. When the players are allowed to take (i,j) tokens with, |i−j|≤1 the P-positions are {(0,0), (0,1), (1,4), (4,9), (9,16), (16,25), (25,36),⋯}, square numbers appear, and in general when the players are allowed to take (i,j) tokens with |i−j|≤c, the (c+3)-gonal numbers appear, listed as {(0,0), (p₀,p₁), (p₁,p₂), (p₂,p₃), (p₃,p₄), (p₄,p₅),⋯} with p_n=((1+c)/2)n²+((1-c)/2)n. We can also describe the misere case and Grundy numbers, and other generalizations.

(Joint work with Shun-ichi Kimura)

Thotsaporn Thanatipanonda, Mahidol University International College, Thailand

Title: Two Games on Arithmetic Functions: Saliquant and Nontotient

Abstract: We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson (2022). In each game, the set of positions is N. In Saliquant, the options are to subtract a non-divisor. Here we obtain several nice number theoretic lemmas, a fundamental theorem, and two conjectures about the eventual density of Sprague-Grundy values. In Nontotient, the only option is to subtract the number of relatively prime residues. Here are able to calculate certain Sprague-Grundy values, and start to understand an appropriate class function.

(Joint work with Paul Ellis, Jason Shi and Andrew Tu)

Tomoaki Abuku, Gifu University, Japan

Title: A variant of Multiple Hook Removing Game with carry-on moves

Abstract: We consider two variations of an impartial game called the Multiple Hook Removing Game, where the starting position is a rectangular Young diagram with unimodal numbering. In one game, that allows a player to take an additional turn after making two consecutive moves of the Multiple Hook Removing Game, and in another, instead of the consecutive moves, that enables a player to force the opponent to make a certain move after their own turn. In fact, these two games can essentially be treated as the same game and games allowing such moves, known as entailing moves (particularly carry-on moves), have recently been the focus of active research. We introduce our analytical results based on this theoretical framework.

(Joint work with Koki Suetsugu and Masato Tada)

Urban Larsson, Indian Institute of Technology Bombay, India

Title: A future of GoNC

Abstract: Games of No Chance (publishers MSRI, CUP) is, alongside IJGT’s special issues, a unique forum that covers all kinds of CGT topics. Five volumes have already appeared, and R. J. Nowakowski has been the main editor over the years. I have recently been honored with taking on the job by editing the last two volumes, and GoNC6 is about to appear with 22 strong research papers and surveys.  Sometimes we must rely on this series of books to get high quality CGT paper published, while standard journals that invite CGT submissions, may reject for reasons that in summary hints at a limited understanding of our topic. Games of no Chance has shown invaluable value over the years. But it must not continue in its current form. This talk will explain why this is so, and encourage a discussion on a possible future with an online journal with the same name.

Urban Larsson, Indian Institute of Technology Bombay, India

Title: Subtraction games in more than one dimension

Abstract: Subtraction games is a popular topic in combinatorial game theory, but very little research stretches beyond one dimensional rulesets. In this presentation we solve all two move games in any dimension, by using a certain P-to-P principle. The general class, two-player impartial vector subtraction games on tuples of nonnegative integers was introduced by Golomb in 1966. Through multiple computer visualizations of outcomes of two-dimensional rulesets, we observe that they tend to partition the game board into periodic mosaics on very few regions/segments, which can depend on the number of moves in a ruleset. For example, we have found a five-move ruleset with an outcome segmentation into six semi-infinite slices. In this spirit, we develop a coloring automaton that paints P-positions in segments of the game board, independent of game play. Moreover, we prove that games in two dimensions have row/column eventually periodic outcomes. Several regularity conjectures are provided. Through visualizations of some rulesets, we pose open problems on the generic hardness of games in two dimensions. We conjecture that not all games have regular outcomes, and wonder if such games are Turing complete. (This latter question was solved by Larsson and Wästlund in 2013, with an affirmative answer, if we exchange «subtraction» for «addition» in finite dimensions).

(Joint work with I. Saha and M. Yokoo)

Vlado Uljarević, University of Novi Sad, Serbia

Title: A variation of Hats-in-a-line game

Abstract: Hat-guessing games gained the scientific community’s attention at the end of the 20th century when, in 1998, Todd Ebert formulated the so-called n-prisoners puzzle. Despite being formulated in a way that brings it into recreational mathematics, Ebert’s puzzle is closely related to certain problems and notions of coding and information theory. The different variations of this and similar games were developed and investigated in the years after. One of the most known hat-guessing games is the Hats-in-a-line game, in which a warden arranges n prisoners in a line, wearing hats colored in one of two colors (white and black for example), so that each of them can see all the hats of the prisoners before him, but can’t see the hats of anybody behind. Then, each of them guesses (one by one, starting from the back of the line) whether the hat on his head is white or black. The goal is to maximize the total number of correct guesses. We introduce a new variant of this game in which the amount of information prisoners can transmit is significantly reduced, and we discuss the maximum number of correct guesses depending on the number of colors the hats are colored with. In the end, we briefly show how this problem can be translated into an instance of the SAT problem, allowing us to use modern SAT-solving algorithms as a last resort.  

(Joint work with Bojan Bašić)