arXiv subjects

Koki Suetsugu

Publications and source records attributed to Koki Suetsugu.

At least 19 recordsLinked to original sources

arXiv2026

Tasty Chocolate Games

In this paper, we investigate Chocolate Games. In Chocolate Games, each player cuts a chocolate bar into two chocolate bars and eats one of them so that they do not eat an indicated bitter block. The player who eats the bitter part loses the game. This game can be considered as a generalization of Nim, and previous studies consider the condition that for what kind of the shape of chocolate bar, the Sprague--Grundy value of the position can be calculated by Nim--sum (XOR) of the width and height of the chocolate bar. Higher dimensional cases were also studied. In this paper, we show that the ``Tasty condition,'' which was introduced in previous work for increasing staircase chocolate bars, can also be used for various other shapes of chocolate bars as a sufficient condition, or in some cases a necessary and sufficient condition, for the Sprague--Grundy value to be given by the Nim-sum of the sizes of the chocolate bar in each dimension.

math.CO↗

arXiv2026

A duality of mis\`{e}re games and play with a pass

In combinatorial game theory, the choice of play convention is a fundamental aspect of the theory. The two most widely studied conventions are normal play, in which the player who makes the last move wins, and mis\`{e}re play, in which the player who makes the last move loses. Another well-known variant is play with a single shared pass. In such games, at most one pass may be used in total during the game: once the pass has been used, neither player may pass thereafter. Moreover, once a terminal position has been reached, passing is no longer allowed. Recall that, for mis\`{e}re play, one sometimes defines SG values (or Sprague-Grundy values) in the same recursive manner as in normal play but assigns the terminal position the value 1. Also, SG values can be considered for normal-play games with a pass. In this work, we propose transformations that allow both mis\`{e}re play and play with a pass to be treated as normal-play games. As a consequence, we show that there is a certain duality between a generalization of mis\`{e}re play and a generalization of play with a pass.

math.CO↗

arXiv2025

Cyclic impartial games with carry-on moves

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 devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, 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. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush.

math.CO↗

arXiv2025

$\mathcal{L}\mathcal{R}$-Ending partisan rulesets

In this paper, we consider $\mathcal{L}\mathcal{R}$-ending partisan rulesets as a branch of combinatorial game theory. In these rulesets, the sets of options of both players are the same. However, there are two kinds of terminal positions. If the game ends in one kind of terminal position, then a player wins, and if the game ends in the other kind of terminal position, the other player wins. We introduce notations for positions in $\mathcal{L}\mathcal{R}$-ending partisan rulesets including disjunctive sum of terminal positions depending on the parity and basic definitions and show their algebraic structures. We also introduce some examples of $\mathcal{L}\mathcal{R}$-ending partisan rulesets and show how our results can be used for analyzing the rulesets.

math.CO↗

arXiv2025

Digraph Yama Nim

{\sc Yama Nim} is a variant of two piles {\sc Nim}. In this ruleset, the player chosses one of the piles and removes at least two tokens from the pile. In the same move, the player adds one token to the other pile. We show the winning strategies and SG-values of this ruleset. In addition, we introduce a generalization of {\sc Yama Nim}, named {\sc Digraph Yama Nim}. In this ruleset, a digraph is given and there are some tokens on each vertex of the digraph. Each player, in their turn, chooses one vertex and removes at least its out-degree plus one tokens from the vertex. Furthermore, one token is added to each vertex to which a directed edge from the chosen vertex is connected. We show that the winner determination problem of {\sc Digraph Yama Nim} is PSPACE-complete even when the input graph is bipartite and directed acyclic. Despite this, there are some cases that can be solved easily and we show them.

math.CO↗

arXiv2025

Combinatorial games and the golden ratio on digraphs

We introduce a new combinatorial game called Triangle Game. In this game, a directed $3$-cycle graph is given, and stones are placed on each vertex. The player chooses a directed edge and takes at least one stone from the initial vertex. At the same time, the player is allowed to return some stones to the terminal vertex of the edge, as long as the total number of stones decreases. We describe the set of \Pps~under both normal play and mis\`ere play. The golden ratio $\phi=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. We also show that Triangle Game is tame.

math.CO↗

arXiv2025

On the Sprague-Grundy values of games with a pass

In this paper, we consider two-player impartial games with a pass-move. A disjunctive compound of games is a position in which, on each turn, the current player chooses one of the components and makes a legal move in it. For disjunctive compounds, it is known that the time to determine which player has a winning strategy is bounded by the time to compute the SG-values of the components plus the time for their XOR. However, if we allow a pass-move during the play, the analysis of such games becomes much more difficult. A pass-move allows each player to skip exactly one turn in non-terminal positions during the game, after which neither player may use a pass-move again. We establish a homomorphism on the SG-values of games with a pass-move. That is, if every component satisfies a condition called one-move game, the SG-value of the disjunctive compound of the components with a pass-move is the same as the SG-value of nim with a pass-move where the size of every pile is the same as the SG-value of every component of the compound. This guarantees that the time to determine which player has a winning strategy in a disjunctive compound with a pass can be bounded by the sum of the time to determine SG-values of all components without a pass and a position in nim with a pass. We also show how the homomorphism is used for determining SG-values of some chocolate games.

math.CO↗

arXiv2025

Extended circular nim

Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of $m$ and $k$, closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim $ECN(m_S, k)$, there are $m$ piles of tokes arranged in a circle. $S$ is a set of positive integers less than or equal to half of $m$. In each turn, a player chooses an integer $s \in S$. Then the player selects at most $k$ piles among those located every $s$-th position on the circle, and removes an arbitrary number of tokens from each selected pile. We show some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight, and for a few generalized cases.

math.CO↗

arXiv2024

Relationship between mis\`ere NIM and two-player GOISHI HIROI

In combinatorial game theory, there are two famous winning conventions, normal play and mis\`ere play. Under normal play convention, the winner is the player who moves last and under mis\`ere play convention, the loser is the player who moves last. The difference makes these conventions completely different, and usually, games under mis\`ere play convention is much difficult to analyze than games under normal play convention. In this study, we show an interesting relationship between rulesets under different winning conventions; we can determine the winner of two-player GOISHI HIROI under normal play convention by using NIM under mis\`ere play convention. We also analyze two-player GOISHI HIROI under mis\`ere play convention.

cs.GT↗

arXiv2024

Enforce and selective operators of combinatorial games

We consider an {\em enforce operator} on impartial rulesets similar to the Muller Twist and the comply/constrain operator of Smith and St\u anic\u a, 2002. Applied to the rulesets A and B, on each turn the opponent enforces one of the rulesets and the current player complies, by playing a move in that ruleset. If the outcome table of the enforce variation of A and B is the same as the outcome table of A, then we say that A dominates B. We find necessary and sufficient conditions for this relation. Additionally, we define a {\em selective operator} and explore a distributive-lattice-like structure within applicable rulesets. Lastly, we define nim-values under enforce-rulesets, and establish that the Sprague-Grundy theory continues to hold, along with illustrative examples.

math.CO↗

arXiv2024

What happens when we add impartial loopy games and impartial entailing games?

The disjunctive sum of impartial games is analyzed by Sprague-Grundy theory. The theory has been extended to loopy games and entailing games by early results. In this study, we consider further extension of this theory and show partial algebraic structure of the sum of loopy positions and entailing positions.

math.CO↗

arXiv2024

Improving upper and lower bounds of the number of games born by day 4

In combinatorial game theory, the lower and upper bounds of the number of games born by day $4$ have been recognized as $3.0 \cdot 10^{12}$ and $10^{434}$, respectively. In this study, we improve the lower bound to $10^{28.2}$ and the upper bound to $4.0 \cdot 10^{184}$, respectively.

math.CO↗

arXiv2023

New universal partizan rulesets and a new universal dicotic partisan ruleset

Universal partizan ruleset is a ruleset in which every game value of partizan games can be appear as a position. So far, generalized konane and turning tiles have been proved to be universal partizan rulesets. In this paper, we introduce two rulesets go on lattice and beyond the door and prove that they are universal partizan rulesets by using game tree preserving reduction. Further, we consider dicotic version of beyond the door, and we prove the ruleset is a universal partisan dicotic ruleset.

math.CO↗

arXiv2023

A complete solution for the partisan chocolate game

The class of Poset Take-Away games includes many interesting and difficult games. Playing on an $n$-dimensional positive quadrant (the origin being the bottom of the poset) gives rise to nim, wythoff's nim and chomp. These are impartial games. We introduce a partisan game motivated by chomp and the recent chocolate-bar version. Our game is played on a chocolate bar with alternately flavored pieces (or a checkerboard). We solve this game by showing it is equivalent to blue-red hackenbush strings. This equivalence proves that the values of game are numbers and it gives an algorithm for optimal play when there is more than one chocolate bar. The checkerboard interpretation leads to many natural questions.

math.CO↗

arXiv2023

Turning Tiles is PSPACE-complete

In combinatorial game theory, the winning player for a position in normal play is analyzed and characterized via algebraic operations. Such analyses define a value for each position, called a game value. A game (ruleset) is called universal if any game value is achievable in some position in a play of the game. Although the universality of a game implies that the ruleset is rich enough (i.e., sufficiently complex), it does not immediately imply that the game is intractable in the sense of computational complexity. This paper proves that the universal game Turning Tiles is PSPACE-complete.

cs.DM↗

arXiv2023

Hardness of braided quantum circuit optimization in the surface code

Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the number of physical qubits, and the number of physical gates. While optimizing fault-tolerant circuits to minimize this overhead is critical, the computational complexity of such optimization problems remains unknown. This ambiguity leaves room for doubt surrounding the most effective methods for compiling fault-tolerant circuits for a large-scale quantum computer. In this paper, we show that the optimization of a special subset of braided quantum circuits is NP-hard by a polynomial-time reduction of the optimization problem into a specific problem called Planar Rectilinear 3SAT.

quant-ph↗

arXiv2023

Some extensions of Delete Nim

Nim is a well-known combinatorial game with several variants, e.g., Delete Nim and Variant Delete Nim. In Variant Delete Nim, the player deletes one of the two heaps of stones and splits the other heap on his/her turn. In this paper, we discuss generalized Variant Delete Nim, which generalizes the number of stone heaps to three or more, All-but-one-delete Nim, Half-delete Nim, No-more-than-half-delete Nim, and Single-delete Nim. We study the win-loss conditions for each of these games.

math.CO↗

arXiv2022

Playing impartial games on a simplicial complex as an extension of the emperor sum theory

In this paper, we considered impartial games on a simplicial complex. Each vertex of a given simplicial complex acts as a position of an impartial game. Each player in turn chooses a face of the simplicial complex and, for each position on each vertex of that face, the player can make an arbitrary number of moves. Moreover, the player can make only a single move for each position on each vertex, not on that face. We show how the P-positions of this game can be characterized using the P-position length. This result can be considered an extension of the emperor sum theory. While the emperor sum only allowed multiple moves for a single component, this study examines the case where multiple moves can be made for multiple components, and clarifies areas that the emperor sum theory did not cover.

math.CO↗