Abstracts

Octal games are Nim games in which players are allowed to split a heap into two in some options, in addition to taking some tokens, and these games are denoted by octal codes. When we allow the players to split the heaps into more than two heaps, the resulting games are still denoted by codes (by using digits larger than or equal to 8), and are called Code games. In these codes, the precise options when taking d tokens are given in the d-th digit of the code. In this talk, we introduce a Negative Code game, where the players are allowed to add n tokens (or take away (-n) tokens) and split the heap into more than n+1 heaps. Under this convention, the game always ends in finitely many steps. We present some interesting examples, such as (16)(8)(4). (2)(1) and (16)4., together with calculations or conjectures of the Nim-values. Some Negative Code Games can be reduced to Code Games, for example (16)(4). can be reduced to (0).(7)(31), but not all of them.

When S is a finite set of positive integers, we can consider classical Subtraction Nim with S as the set of removable numbers. Even when S consists of three elements, many questions remain unanswered. For example, we do not have a period formula of the Nim value. In this talk, we generalize S to be a finite set of positive real numbers. We found that in some regions, we can give concrete formulae for the period and the Nim value function. To be more precise, when the set S of removable numbers is S={a, b, 1} satisfying the conditions 0<a<b<1 with b<=2a, we have determined exactly that for which value (a, b, 1), the Nim value function is purely periodic with the periods a+b, a+1, b+1 or otherwise, as shown in our material. Even when S consists of integers, these results seem to be new.

We will discuss Another Cutcake Variant, which appears in Winning Ways, Volume 1, Chapter 2.
In this game, we start from a rectangle cake of size n x m, and Left chooses one piece of the cake and cuts it vertically v times along the grids in one move, and Right chooses one piece of the cake and cuts it horizontally h times along the grids in one move.
The game values is stated in the text, but is not correct, and we give the correct value in this presentation.

John Horton Conway (1937–2020) was an unconventional mathematician. He made highly sophisticated contributions across a wide range of fields, demonstrating remarkable skill in connecting them in astonishing ways. A paradigmatic example is the discovery of the surreal numbers, which, without exaggeration, can be regarded as a landmark moment in the history of mathematics. This work aims to explore that moment by providing an intuitive analysis of the crucial aspects of the discovery, as well as by presenting videos in which Conway himself explains and comments on each of them in his own words. The result is a non-technical talk, accessible to undergraduate students with some background in mathematics.

Omni-Fission is a variation of a ruleset implemented in CG Suite played with black and white stones on an mxn grid with a fixed initial configuration. On a player's turn, they choose one of their stones and pick a subset of orthogonal empty squares (minimum 2) to expand their stones on, leaving their original square empty. We present some intriguing results on 1xn boards, along with a method for generating arbitrarily hot positions. We propose some conjectures regarding game values on a 2xn board. Inspired by ‘Conway’s Game of Life’, by limiting to maximum expansion, we determine the winner of certain starting positions purely from the dimensions of the grid. Such cellular automata-like two-player games induced by this ruleset pose several interesting research directions.

In (i,j)-biased combinatorial games, the first player makes i moves per turn, while the second player makes j moves per turn. An (i,j)-biased game can also be interpreted as an n-player game in which the players are divided into two teams．
In this work, we investigate the computational complexity of (i,j)-biased Undirected Vertex Geography. The classical ((1,1)-biased) Undirected Vertex Geography is known to be solvable in polynomial time, and we are interested in determining when a game that is solvable in polynomial time in the standard (1,1) setting becomes PSPACE-hard as the parameters i and j vary. As a result, we show that (i,j)-biased Undirected Vertex Geography is PSPACE-complete for all cases except (i,j)=(1,1)．

Slime Trail is a two-player combinatorial game created by Bill Taylor in 1992 that has gained significant popularity, particularly in Portugal where it has been featured in annual mathematical game tournaments organized by Ludus since 2008. The game is played on a graph where players alternate moving a token between vertices, leaving a “slime trail” that prevents future moves to visited vertices.

It has been previously proven that it is PSPACE-complete on planar graphs, but this talk covers an extension to demonstrate it is PSPACE-complete even on grids.

We show that Misère Partizan Arc Kayles is PSPACE -complete on planar graphs via a reduction from Bounded Two-Player Constraint Logic. With the planarity, it is easy to extend the result to colored edge graphs embedded in the square and triangular grids.

This paper introduces Heavenly Domineering, a constrained variant of the classical partisan game Domineering, in which both players are restricted to placing their dominoes only at the topmost playable edge of a connected region on a square grid. This limitation simplifies the combinatorial structure of many traditionally complex Domineering positions, leading to games that are predominantly cold and easy to calculate the value of explicitly. We compute exact values for a range of positions, including dyadic rationals as small as 1/8, switches, tinies, and other infinitesimals such as up, down, and their combinations, and provide thermographic analysis to illustrate thermal behavior.

One of the key themes of the paper is the interaction of geometry with the value of the game: whereas a 90-degree rotation of a classical Domineering position corresponds to negation, the global-top constraint breaks the symmetry, and we investigate to what extent rotation, possibly together with reflection, can still perform negation or other simple value changes in Heavenly.

Criss-Cross is a two-player partizan game played on an m x n grid and is a variant of Domineering in which players take turns placing non-overlapping diagonal dominoes. We construct an explicit affine correspondence between Criss-Cross and standard Domineering positions and show that every Criss-Cross position is equivalent to the disjunctive sum of two Domineering positions. We conjecture that every empty m x n board with both dimensions odd is an N-position, with game value always equal to ∗, a switch, or ∗ plus switch, and no other values appear in the cases we have examined. We also show that the construction underlying the classical snake positions in Domineering carries over to Criss-Cross, producing a closely related family of dyadic values.

We investigate conditions under which positions in combinatorial games admit simple values.
We introduce a unified diamond framework, called the diamond property for integers and dyadic rational numbers.
Under certain conditions, this framework guarantees that all values of positions closed under options are pairs of the form {m | n}, where m and n are integers or dyadic rational numbers.
As an application, we establish that every position in the Yashima game on bipartite graphs has an integer pair value.

There is a well-known game called Chopsticks, also known as Survival Finger. In this report, we study two finite variations of it, called Top Match and Tap Match. Players move by tapping an opponent’s hand, thereby increasing the number of pieces until it exceeds the vanish level. While Tap Match remains faithful to the classical all-small variant, Top Match introduces a restriction on tapping based on strength.
In Top Match, we prove that the temperature of every game position is bounded above by 1 and identify a subclass of positions that are always either P or R. In Tap Match, we show that for any vanish level n, the atomic weights are strictly less than n. Furthermore, we construct a sequence of positions capable of achieving any arbitrarily large atomic weight.

Fission is a simple combinatorial game played on a rectangular grid with stones placed in an alternating pattern. Left plays by "vertically" splitting a stone into two that land on empty spaces above and below that stone, and Right does the same but "horizontally". One can see CGSuite for reference, where the ruleset is already implemented. In this presentation, we'll explore some basic results about the game values in normal play, in particular, values of 1-by-n and 2-by-n grids, and results about larger grids. We will also look into an all-small variation, called CoolFission and the equivalence of some of its positions with positions in the famous Dawson's Kayles.

We investigate Full Support Subtraction (FSS), a subtraction game in which the players can remove any number of pebbles from the heap up to certain bounds that are typically different for Left and Right. The player with the richer move set always wins for all but finitely many heap sizes. We confirm this advantage by finding the general canonical form and the atomic weights of this game.

To restore fairness (and peace), we introduce Truncated Support Subtraction (TSS), which trims the larger subtraction set from below. As the cuts deepen, the game begins producing infinitely many P- and N-positions. However, if the truncation is shallow, the unfairness persists above a certain heap size, and one player continues ruling. We also explore the atomic weights for these lightly trimmed scenarios.

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.

Chain Reaction is a well-known combinatorial game played on a finite grid. In this report, we study two terminating variations of the game, called Stacking Attackers and Blocking Attackers. While Blocking Attackers is an all-small game, Stacking Attackers exhibits partizan behavior. In the partizan version, attackers are formed according to the threshold value of a tile, whereas in the all-small version, opponents block each other’s attackers by means of a common shared piece.

In the partizan version, we analyze the game values arising from small board configurations and show that dyadic rational values occur up to 1/8, but no dyadic values beyond 1/8 are attainable. Furthermore, we derive a systematic method for constructing infinitesimal game values, generating the sequence from tiny_2 through tiny_9.

In the all-small version, we investigate the atomic weights of positions and present a construction that produces boards with arbitrary integral atomic weight.

We study a game of Nim with two different rules. If the number of stones satisfies a specific condition, players play according to the first rule; otherwise, they play according to the second rule. Let n,m be natural numbers. We consider a three-pile NIM, and we denote by x,y,z the number of stones in the first, second, and third pile. Without any loss of generality, we assume that x <= y <= z . In the first game, we play according to the rule of the traditional three-pile NIM when x,y,z = 2^n. Then, the set of P-positions is the union of three sets {(0,x,x):x=0,1,… }, {(x,y,z): the nim sum of x, y, z is 0 and x,y,z <2^n }, and {(x,y,z):x= 2^n }. Therefore, the set of P-positions is a simple mixture of the traditional NIM and the Greedy NIM. In the second game, we play according to the traditional three-pile NIM when x,y,z = 2^n+1. Then, the set of P-positions is the union of four sets {(0,x,x):x=0,1,… }, {(x,y,z): the nim-sum of x,y,z, = 0 and x,y,z <2^n+1}, {(x,y,z):x= 2^n+1 }- {(m,2^n+1,2^n+1):m=1,2,..,2^n}, and {(m,2^n,2^n+1):m=2,3,…,2^n-1}. We also study a game of Nim with two different rules by introducing a pass move. Then, we get a game that is very similar to the traditional NIM, and this game has a formula for the set of P-positions when a pass is allowed.

Makino defined 2-pile divisor Nim as a variation of 2-pile Nim.
Its rule is that the player removes d stones from a pile where d divides the number of stones in the other pile.
In this talk, we generalize it to m-pile divisor Nim and show that the Sprague-Grundy value is periodic.

Conway proved that the class of all ordinal numbers forms a field of characteristic 2 by introducing the Nim sum and the Nim product operations. This lecture will present algorithms for performing these operations and discuss methods for solving algebraic equations in this field.

We define a real-valued distance metric wd on the space C of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure of C, which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of (C,wd).

We will discuss Crushcar Nim. There are some piles of tokens with indices, and the player chooses a pile and takes at least 1 token from the pile. Then, she/he can add tokens to the piles with smaller index. For example, one can play (2,3,1,5) to (2,0,3,2026). We show that the Grundy number takes transfinite value, and to express it in base (ω2) is more natural than to express it in base ω.
We also show that this game is tame. In other words, if Normal Grundy number g(x)>1, then misère Grundy number g’(x)=g(x).

I consider a generalization of Wythoff's Game, where the two piles are treated unequally. Let m and n be integers greater than or equal to 1. In this game, there are two piles of tokens, and a player is allowed to take one or more tokens from a single pile, or to take i tokens from the first pile and j tokens from the second pile such that i-n < j < i+m. The case n=m=1 corresponds to the classical Wythoff's Game. I investigated the P-positions under both normal play and misère play and proved that the P-positions can be described by two Beatty Sequences. Furthermore, as in previous studies, the P-positions can have a mex discription and also a Zeckendorf style discription.

Triangular Nim is a variant of two-heaps Nim in which the players take at least two tokens from one heap and return at least one tokens to the other heap, so that the total number of tokens decreases. When we also allow the Wythoff option, namely the players can take the same number of tokens from both heaps, then the P-positions of the game with x 0, and the players are allowed to take i tokens and j tokens respectively from each heap, when |i-j|<s_Min(i, j), then not only triangular numbers but also various interesting sequences appear in the description of P-positions. Conversely when we give a sequence of positive integers satisfying certain conditions, then we can apply an algorithm to produce S from which the given sequence appears. When S={1, 1, 1, …}, we have the triangular numbers, when S={1, 2, 3, 4, …}, we have the Mersenne numbers {1, 3, 7, 15, …}. We can apply our algorithm also to many polynomial functions like power sums, and the factorial function n!, and a lot more. If we allow 0 for the elements of S, then we can also make a game where the Fibonacci Sequence appears in the description.

Triangular c-Wythoff Nim is a variant of two-heaps Nim. This game allows two types of options: (1) 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 decreases; (2) the players take tokens from both heaps in such a way that the difference between the removed tokens is less than a constant c. The P-positions of this game are described by consecutive (c+2)-gonal numbers, say (0, 0), (0, 1), (1, 3), (3, 6), …. when c=1 and x<=y,

In this talk, we introduce, still impartial but an asymmetric variant of the Triangular Wythoff Nim with two positive integers, say (a, b), where the legal options depend on the heap. In this game, the P-positions are described by (a+2)-gonal numbers and (b+2)-gonal numbers. For example when a=1 and b=2, the P-positions are (0, 0),(0, 1), (1, 4), (3, 9), (6, 16), (10, 25), …, for 2x=y. Moreover, for some choices of the parameters (a, b), for example (1, 3), the total number of tokens may increase, and this game becomes loopy, even though all positions are either P or N-positions, and the winning side can finish the game in finite moves. If the players do not care about winning, they can play this game forever for such parameters.

We consider a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner.
In this variant, the terminal set is {(x, y): x + y =8 or y >= 8. The second  one is that there is an essential relation between this variant and Hofstadter's G-Sequence.
Let a_1(n) and a_2(n) are integer sequences such that 
{(a_1(n),a_2(n)):n=0,1,2,…} cup {(a_2(n), a_1(n)):n=0,1,2…} is the set of P-positions of the classical Wythoff's game. 
We define a variant of Wythoff's game by using {(x,y):x+y <=2} as the terminal set.
Then there exists an integer-valued function g(n) and 
if we let b_1(n) =a_1(n)+g(n)-1 and   b_2(n)) = a_2(n)+g(n), then
{(b_1(n),b_2(n)):n=0,1,2,…} cup {(b_2(n), b_1(n)):n=0,1,2…} are the set of P-positions of the variant. By using Hofstadter's G-Sequence, we can define this function g(n).
The third one is that the sequence {b_(n+1)-b_(n):n=1,2,…} has a self-similarity.

We study impartial subtraction games played on circular game boards by defining various widths of {\em sinks}, which correspond to terminating P-positions. This approach allows positions to undergo multiple cycles before settling on definitive game values. We examine these games in the context of Smith-Fraenkel-Yesha generalized Sprague-Grundy values and demonstrate that patterns ultimately stabilize as we expand the cycles at regular intervals, a phenomenon we term expansion periodicity. Furthermore, we conjecture that these expansion periods are independent of sink size and cycle lengths, and that they settle at a divisor of the period length induced by the sink width of non-cyclic play.

Subtraction games belong to the folklore of Combinatorial Game Theory (CGT). In particular, a classical result of Golomb (1966) shows that every subtraction game with a finite move set has an eventually periodic nimber sequence. From this perspective, subtraction games are often regarded as “solved”. However, this result provides only an exponential upper bound on the period length, while most studied rulesets have polynomial (small degree) period lengths. Very few general results are known concerning explicit formulas for given ruleset families. Through extensive computational experiments, remarkably, Flammenkamp (PhD thesis, 1996) identifies subtraction sets S with as few as five moves whose outcome-period lengths appear to grow as large as ∼2^{0.3 max S}. He also formulates a conjecture for three-move subtraction games, for which a striking experimental classification emerges: non-additive sets exhibit linear period lengths of the form “the sum of two moves”, although the choice of which two moves displays fractal-like behavior, whereas additive sets S = {a, b, a + b} have purely periodic outcomes with conjectured linear or quadratic period lengths. Additive subtraction games receive special attention already in *Winning Ways* (1982), where a specific family with linear period lengths is solved. However, contrary to a statement in Flammenkamp’s thesis, to the best of our knowledge the general additive case in the standard setting remains open (see also Ward’s more recent arXiv post).

In this talk, we introduce and analyze a dual setting, which we call sink subtraction. In contrast to the standard “wall” convention, where moves to negative positions are forbidden, the sink convention declares a player the winner as soon as they move to any non-positive position (normal play). This winning convention has not yet received much attention, except for a related generalization appearing in Miklós et al. (2024), where three-move games under specific terminal “seeds” are shown to attain super-polynomial, though still sub-exponential, period lengths. A fundamental structural difference is that the wall convention enforces finite play, whereas the sink convention allows circular play for finite sinks; see Bhagat’s talk at this conference. In celebration of this added generality, we demonstrate that additive sink subtraction, with an infinite sink, admits a complete solution: the nimber sequence is purely periodic with an explicit (linear or) quadratic period formula. Strikingly, this formula exhibits “dual” properties to the conjectured period formula for additive wall subtraction; see Manabe’s talk at this conference.

Standard subtraction games, played under the classical 'wall' convention (where moves to negative positions are forbidden), have been studied for decades. While the eventual periodicity of their nimber sequences have proved, the explicit structure of these periods remains an open problem. As highlighted in the talk by Prof. Larsson, the periodicity of additive subtraction sets S={a,b,a+b} under the wall convention has been the subject of conjectures by Flammenkamp (1996), yet a complete proof for the general case has remained open.
In this talk, we present a fundamental correspondence between the wall convention and the 'sink' convention introduced in the previous talks. We show that, remarkably, the nimber sequence of a subtraction game under the wall convention exhibits a precise structural duality with that of the corresponding sink game. By leveraging Prof. Larsson's complete solution for the additive sink subtraction games, this duality allows us to determine the period structure of some additive wall subtraction games, thereby affirming and extending the long-standing conjectures regarding their behavior.
We will discuss the mechanism of this duality and outline the proof strategy that bridges the gap between the solvable sink setting and the classical wall setting.

We define a new game "Drop Two Rooks". Two coins are placed on a chessboard of unbounded size, and two players take turns choosing one of the coins and moving it. Coins are to be moved to the left or upward vertically as far as desired. If a coin is dropped off the board, players cannot use this coin. For non-negative integers w,x,y,z, by (w,x,y,z) we denote the positions of the two coins, where (w,x) is the position of one coin and (y,z) is the position of the other coin. If xy=0 or zw=0, then the coin denoted by (x,y) or the coin denoted by (z,w) is outside of the chessboard. We permit a jump over another, but not on another coin. We consider two games. In the first game, if two coins are on the same vertical or horizontal line, a player can push a coin by another coin and drop both of coins outside of chessboard to win the game. Then, the set of P-positions of this game is
{(w,x,y,z):the nim-sum of (w-1),(x-1), (y-1), and (z-1) is 0}. In the second game, if two coins are side by side on the same vertical or horizontal line, a player can push a coin by another and drop both of coins outside of chessboard to win the game. Let N_0={(x +- 1,y +- 1, x -+ 1, y -+ 1): x,y = 2,3 mod 4 } cup {(x +- 1,y -+ 1, x -+ 1, y +- 1): x,y = 2,3 mod 4 },
P_1={(x-1,y,x+1,y): x = 2,3 mod 4 } cup {(x, y-1, x, y+1): y = 2,3 mod 4 }
cup {(x, y+1, x, y-1): y = 2,3 mod 4 } cup {(x, y+1, x, y-1): y = 2,3 mod 4 }.
Then, the set of P-positions of this game is
P={(w,x,y,z): the nim-sum of (w-1),(x-1), (y-1), and (z-1) is 0 and wxyz>0}cup P_1 – N_0.

We introduce a method for analyzing the misère version of variants of Nim from a novel perspective and present results obtained from an analysis of several variants of Greedy Nim in misère play, for which this method proved particularly effective.

We consider an Ending Partizan Subtraction Nim, whose options are the same as impartial Subtraction Nim, but the winner is determined in a partizan way by the terminal position at which the game ends.
We concentrate on the LR-type convention, namely, Left wins if the total number of remaining tokens is even, and Right wins otherwise.

To analyze game positions, for impartial games under normal convention, Conway used the notation like 0 = {|}, 1 = {0|} etc., and defined game values as the equivalence class of game positions where G and H are identified if and only if the outcome of G + X is same as H + X for each game position X. We have a criterion that says G and H have the same game value if and only if the outcome of G – H is P, hence we don't really need to test with X. For Misere convention, Plambeck and Siegel proposed Misere Quotient, where the test positions X are restricted. In this talk, we generalize both Conway type notations/game values and Quotients, for LR-type convention. Some games can be completely analyzed by combining these two techniques. We also compare these “game values” for LR-type with Nowakowski normal interpretation, where, for example the game ends with even number of tokens, instead of declaring Left as the winner, we allow only Left the final extra move. We can use G – H criterion for this interpretation, and its behavior is very similar to our game values but is decisively different when it comes to addition of games.

Inspired by the works of Nanako Omiya at Tohoku University, we investigated several variations of Greedy Nim, incluiding Nezumi-Kozo Nim, named after Japanese Robin Hood, who steals money from rich people and distributs it to poor people (is this really greedy?). We also consider Second Greedy Nim, where the player can take tokens not only from the piles of the most tokens, but also from the piles with the second most tokens.

In this joint work with Hiroki Inazu inspired by his joint work of Koki Suetsugu, we tried to formalize Ending Partizan Nim with inner disjunctive sum, and discovered amazing phenomenon that disjunctive sum can be not commutative, and sometimes not even associative.

We study the game of Nim with a forced pass. In combinatorial games, if the standard rules of the game are modified to allow a one-time pass, that is, a passing move that may be used at most once in the game and not from a terminal position, and once either player has used a pass, it is no longer available, the game's underlying structure changes significantly, increasing its complexity. The authors studied games with a pass and published their findings. In this paper, we present a variant of pass-move, namely a forced pass. A forced pass is a pass that can be forced on the opponent by one of the players. In this paper, we study traditional Nim, traditional Nim with a restriction on the number of stones to be removed, and Greedy Nim under the forced-pass rule. We also study traditional Nim with a restricted forced pass. If Player A forced a pass on Player B, Player A can play twice. To play twice is a mighty move, and understandably, the set of P-positions (the previous player's winning positions) is relatively small compared to the set of N-positions (the next player's winning positions). First, we study the traditional three-pile Nim with a forced pass, and prove that the set of P-positions is very simple. Next, we study the three-pile Nim, in which the number of stones to be removed is at least one and at most m, where m is a fixed natural number. When we introduce a forced pass for this Nim, the set of P-positions is simple when there is no natural number k such that m=2^k. When there is a natural number k such that m=2^k, the set of P-positions is complicated. Next, we study the case where we remove only an even number of stones twice when we force a pass to the opponent. Then, we can describe the set of P-positions with nim-sum.

Capture go is a variant of Go in which the first player to capture a stone wins. On a finite Go board, we can make "win in k" puzzles — positions where Black is winning and White can prolong the game for at most k moves. If we let the board be arbitrarily large, we can construct win in k puzzles for arbitrarily large k. On an infinite Go board, there are even more possibilities. We can imagine a position on an infinite board in which Black will win in a finite number of moves, but on White's first move White decides what that finite number is. We could describe such a situation as a "win in omega" for Black. Similarly, each position in infinite capture go which will end in a finite number of moves under optimal play can be assigned an ordinal number. The omega one of capture go is the supremum of this set of ordinals. We will present the problem of finding the omega one of infinite capture go as well as some progress towards solving it. Joint work with Isobel Shaw.

Professor Berlekamp and his research group started to apply Combinatorial Game Theory to mathematical Go endgames in the 1990s. The author joined the group around 1991. We studied small-area (room) analysis without kos and with kos. Small-area analysis without kos is relatively easy.

1. Example without kos
Fig.1 shows an easy example of endgame of Go. Both Black and White may play a or b.
The outcome of chilled game of G is G-1. G*1 is defined by Y{ G^L-1 I G^R +1 Y}.

-1 or + 1 is TAX for white (black) because it is usually get one point each for one move in end
stage of Endgame of Go. Under this hypothesis, Fig 1 is analyzed that

G.L=¥{2+1/8,2+up 13/4,1/2¥}=¥{2+1/811/2¥}=1+1/4.

2. Simple 1-point ko
Fig.2 shows a simple 1-point ko. If Black’s turn, he may play the ko and take a stone.
White may not take back the ko immediately. If White plays another place X and Black
responds there, then White may take back the ko. X is called a ko-threat.

3. Ko-master
When there is a ko, Black or White eventually filles the ko. The filling side is the winner
of the ko and cailed ko-master. Note that ko-master must fill the ko.

4. Hidden Ko position
Fig. 3 shows a hidden-ko. There are no kos. But, after couple of plays, there might be a
ko. Thie type of ko is calied hidden ko.

5. Rogue Positions
There are interesting positions called Rogue Positions including hidden Ko

5.1 Wolfe’s Rogue posirion
Fig.4 shows Wolfe’S Rogue position.
5.2 Takizawa’s Rogue position

Fig. 5 shows Takizawa’s Rogue position.

Snort is a partizan game played on a finite graph in which the players alternate colouring vertices (Left blue and Right red) such that two vertices of the opposite colours are not adjacent. And the temperature of a position intuitively represents the urgency of moving first. It is known that the temperature of Snort is unbounded (e.g. the temperature of the star $K_{1,n}$ is n). We will show that the temperature can even be unboundedly larger than the maximum degree of the graph and will discuss a few other questions discussing temperature compared to maximum degree.

How complex can a game of stacking boxes be? We introduce Climb Over, a partisan game where pieces traverse
a 2D grid by climbing over or falling onto adjacent stacks. We show that this interplay between vertical stacking
and horizontal movement is capable of generating all dyadic rational values. We present a constructive proof, detailing a recursive algorithm to build game positions with canonical values 1/2^k. Additionally, we solve the game for fundamental positions with few pieces, revealing that victory in these sparse settings relies on controlling specific parity invariants and managing the relative gaps between pieces.

In this talk, we consider games in which, some components are given like disjunctive sum, and only one of them is in the special situation, called "superposition" and the others are standard positions. The current player makes a move in the superposition and change it to a standard position. Furthermore, the player selects one of the all standard positions and makes a move. Then, the position changes to a superposition and the turn ends.

These games cannot be analyzed by the classical Sprague-Grundy theory, however, we show that we can define Sprague-Grundy values for positions in such rulesets by extending Sprague-Grundy theory.
In addition, we introduce some natural rulesets belong to this framework, for example, "Box split game", and "Corner the ninja".

In an impartial combinatorial game, both players have exactly the same options at every position of the game. Classical Sprague–Grundy theory provides a fundamental framework for the analysis of short impartial games, where play is finite, the number of options is limited, and no special moves occur. Over the years, several extensions of this theory have been developed to address more general settings. Notably, the Smith–Frankel–Perl theory applies to games in which infinite play is possible, while the Larsson–Nowakowski–Santos theory accommodates entailing moves that disrupt the usual behavior of the disjunctive sum.

This talk introduces a generalization that combines these two approaches, making it possible to analyze cyclic impartial games with carry-on moves. Carry-on moves constitute a special class of entailing moves in which the responding player has no freedom of choice. The theory is illustrated through GREEN-LIME HACKENBUSB, a game inspired by the classical game of GREEN HACKENBUSB.

This presentation considers the following framework in the normal-play convention. An impartial game is assigned independently to each element x of a finite poset (P, <=). On a turn, a player selects one element of P and plays the game assigned to it. At that time, for every element y of P such that y < x, the game assigned to y is replaced with a game H_y, which is fixed in advance for each element of P.
We show some results on the outcome classes and Grundy numbers for several simple cases. In particular, we analyze a generalization of ordinal sums (G:H) in which, each time G is played, H is replaced with a game J that is fixed in advance.

In the world of all-small games, many games have integer-valued atomic weight. The first known family of atomic games are named ups defined as ↑_n={↑_(n-1) |*} (↑_0=0), where n>0 is a natural number. Thereafter, the family of ups was extended to including ↑_d={↑_(d^L ) |↑_(d^R ),*}, where d>0 is a number. These ups are totally ordered and each has atomic weight one. Sums of ups and star are called sumbers, whose outcomes can be determined by a simple rule. This paper extends the family of ups to including ↑_d^n for all natural number n and number d. The properties of total ordering of ups, unit atomic weight for each up and simple outcome rule for a sum of ups are all preserved in this extended family of ups. This paper further claims that if A and B are two sumbers and A-B<| ↑_ω+* (ω={0,1,2,…|}) then {A|B} is a sumber.

It is a well-known result of Mesdal and Ottaway that there are no non-zero invertible games in misère. When we consider restricted misère, however, we can sometimes birth invertible elements. Most recent work has focussed on studying restrictions to special sets of games called universes; it is known that the dicot and dead-ending universes both have many invertible games. In this talk, we consider what are called weak universes, which have no invertible elements (and in fact are even more restrictive than just this). We then use what we have seen to study the invertible elements of the monoids of Amazons positions and Kōnane positions. In particular, we show that there are no non-zero invertible games in both the Amazons and Kōnane monoids.

Under the misère play condition, the first player who cannot make a move wins. Most of the structure that exists in normal play, like additive inverses, does not exist in misère play. Mathlib is a collaborative effort to formalize mathematics using the Lean 4 theorem prover. We have built upon Mathlib and formalized various results in misère play, yielding more general theorems that capture transfinite games.

In the domain of Shogi, ANI (Artificial Narrow Intelligence) has already reached a level that far exceeds human cognitive capabilities. This presentation reports on how the Shogi community has adapted to this "supra-human ANI" and analyzes it as a preliminary case study for the coming era of ASI (Artificial Super Intelligence).

Specifically, we discuss two main points. First is the transformation of learning environments and spectator experiences driven by ANI. The widespread adoption of advanced analysis technologies has led to a "democratization of skills," eliminating geographical and environmental disparities. Additionally, the visualization of evaluation scores has opened the "black box" of the game, creating new entertainment value.

Second are the ethical and existential challenges posed by the ubiquity of ANI. The risk of "cheating via AI assistance" has forced a shift from traditional environments based on mutual trust to those requiring strict surveillance systems (Zero Trust). Furthermore, the existence of AI possessing "absolute answers" compels professional players to redefine their value beyond mere victory and defeat, questioning the very meaning of human play.

Through the process of friction and adaptation observed in the Shogi community, this research offers insights into the challenges and possibilities humanity will face in the approaching ASI era.

At CGTC V in Lisbon 2025, we introduced Version 1 of MCGS, our Minimax-based Combinatorial Game Solver. In this update, we describe the work done over the last year, leading up to the current version 1.5. This includes support for many more games, more algorithms, and many efficiency improvements.

We consider the P and N-positions of the Nim-type game similar to Wythoff. The game starts with $k$ piles of tokens, the players take turns removing 1 token from one pile or removing one token from each pile (Version B of our 3 versions). The last player to move wins. In this talk, we'd like to present the automated proof methodology. First we conjecture of the P and N-positions using the computer program. Then we also write the program to prove the conjectures rigorously.

We introduce an oracle model for imperfect-information games, with a focus on Nim. In our model, heap sizes are hidden, while the oracle reveals position signals, e.g., the Sprague–Grundy number. We ask which oracle information suffices to choose an optimal move from observation alone, and we report basic properties on the feasibility of optimal play in Nim and its variants under this model.

Modular Nim is a classical impartial combinatorial game with many unresolved instances. For a finite move set \(M=\{m_1,\dots,m_k\}\subset\mathbb{Z}\) and \(n\ge 1\), \(\Gamma(m_1,\dots,m_k;n)\) denotes a game of Modular Nim played on a directed cycle with \(n\) vertices. In 2014, Tan and Ward proposed several conjectures about winning player of Modular Nim games with a two-element move set, including conjectures for the families \(\Gamma(a,b;2b)\), \(\Gamma(a,b;2(a+b))\), and \(\Gamma(2,b;2+2b)\).

In this work, we prove these conjectures and establish a general result that yields the complete solution of \(\Gamma(a,b;2b)\) as a special case. We also introduce and completely solve the family \(\Gamma(1,b;2b-1)\). In addition, we give an explicit winning strategy for the first player for games of the form \(\Gamma(M;n)\), where \(M\) consists only of odd integers and \(n\) is even. Finally, we complete the classification of the family \(\Gamma(2,b-1;b)\), resolving the remaining cases of a conjecture posed by Tan and Ward that was partially solved by Srinivas Arun in 2025.

I will talk about three topics: chocolate games, games with a pass, and Maximum Nim. I have been researching these topics with my high school students for more than 20 years. These three topics are related to each other. Arguably, I was the first in the world to take up combinatorial games as a topic for a high school math research group and publish papers with students. My high school students and I read about rectangular chocolate games twenty years ago, and rectangular chocolate games are disguised Nim. These games are mathematically the same as three-pile Nim but have representations that differ from classical Nim. It is often the case that a different representation offers a different path for research, and this rectangular chocolate gave my students and me a perfect opportunity to conduct research. By changing the shape of chocolates, we developed a variety of Nim variants and published nine articles on chocolate games. We discovered the necessary and sufficient conditions for chocolate games to have the Grundy-number formula described by the nim-sum, and some sufficient conditions for chocolate games to have a formula for P-positions described by the nim-sum. About 10 years ago, we began studying games with a pass. This pass move may be used at most once in the game and not from the terminal position. In the case of classical Nim, the introduction of the pass alters the game’s mathematical structure, considerably increasing its complexity, and finding the formula that describes the set of previous players’ winning positions remains an important open question. We wondered why a pass move makes traditional Nim too complicated and began studying many games with a pass, and found that Chocolate Game, Maximum Nim, Wythoff’s Game, and Silver Dollar Game have formulas for P-positions when a pass is permitted. We discovered a sufficient condition for chocolate games with a pass to have Grundy-number formulas. We also found a sufficient condition for games in general to have formulas for P-positions, and well-known games such as Wythoff’s Game, Silver Dollar Game, Maximum Nim, and Greedy Nim satisfy this condition. We also studied Maximum Nim without a pass and discovered a fascinating relation between Maximum Nim and the Josephus problem. We found a formula to calculate the P-positions of Maximum Nim without using recursion, which enables us to efficiently determine the last remaining number in the Josephus problem. In Chess, Go, and Shogi (Japanese Chess), there are many very young professional players, indicating the significant potential of young talent in the study of games. I hope that many young people will discover many new things in combinatorial games in the coming decades.

We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN(n,k) with n=7, k=3 and n=8,k=3.

Nim is a well-known combinatorial game, in which two players alternately remove stones from distinct piles.
A player who removes the last stone wins under the normal play rule, while a player loses under the mis\`ere play rule.
In this talk, we propose a new variant of Nim with scoring that generalizes both the normal and mis\`ere play versions of Nim as special cases.
We discuss theoretical aspects of this extended game and present some results on its fundamental properties, such as optimal strategies and payoff functions.

This study presents a simple method for linking social choice theory to combinatorial game theory. The author utilizes the framework of the Gibbard-Satterthwaite theorem in social choice theory as a primary example. While several fundamental concepts of the Gibbard-Satterthwaite theorem have previously been adapted into the framework of 2-by-2 normal-form games, this study extends that application to simple combinatorial games and demonstrates the typical consequences. From the perspective of advancing social choice theory, applying concepts from combinatorial game theory appears to be a fruitful endeavor, though the benefits may not be as significant in the reverse direction.

We define the partizan placement ruleset Expansion, and it’s all small variant Void Expansion. Expansion is played on an m×n grid, where players place stones on all orthogonal squares to any connected component of their own color. In Void Expansion, you may instead place a single stone on any disconnected empty square. Consider Expansion: we prove that all dyadic rational values can be achieved on row boards (1×n). In the case of Void Expansion we conjecture that arbitrarily large atomic weights appear on row boards; with a game value of ↑n .

Play the game here: https://adityak1729.github.io/Expansion/

We introduce and analyze two loopy combinatorial games, Blippers and Flippers,which belong to the class of placement games with dynamic elimination mechanics.
Both games are played on finite grids where players alternately place stones of their color, but differ in their elimination rules. In Blippers, pairs of orthogonally adjacent stones of the same color with no other same-color orthogonal neighbors simultaneously
disappear (”blip”). In Flippers, when two stones of the same color become orthogonally adjacent, the previously placed stone disappears (”flips”), unless there exists a third
stone of the same color orthogonally adjacent to either stone in the pair.
We establish fundamental results about the outcome classes of these games on rectangular grids, proving that for any t × x grid where t, x ∈ [2, n], the game has a P-position (second player win) when tx is even and an N-position (first player win)
when tx is odd. We further investigate the loopy nature of certain game positions and provide a comprehensive analysis using tools from combinatorial game theory,we can assign precise algebraic values to these positions. This analysis emphasizes the roles of Stoppers, the Onside/Offside rule, changed outcome classes, the Sidling Theorem, and atomic weights to reveal the strategic nature of these loops.

We introduce a general framework for positional games in which players score points by claiming a prescribed portion of each winning set, extending the notion of scoring Maker–Breaker games. We investigate the impact of strategy restrictions on the achievable score, analysing these games both under optimal play and under the additional constraint that Maker is restricted to a pairing strategy.

We comprehensively study games where the boards are regular grids, which provide a natural and uniform setting for illustrating the framework. After developing several general tools for the analysis of both scores, we complement them by a number of ad-hoc strategies tailored for particular cases of these games, to obtain both upper and lower bounds for the two scores on triangular, square, rhombus and hexagonal grids.

Grid Nim, a two-player combinatorial game played on an m*n grid, where some(all) blocks of the grid have tokens. Right can remove any number of tokens from a single row, and Left can remove any number of tokens from a single column. Although the game positions are significantly complicated due to a significantly large number of options, it is possible to predict the player's advantage using a simple formula. The main result proves that the advantage of a player depends only on the difference in the number of non-empty rows and columns in the game.

Further, attaching a link of a playable version of the game (the game was earlier called A different Nim, but the rules remain the same) https://butterknifees.github.io/different-nim/