arXiv subjects

Ryohei Miyadera

Publications and source records attributed to Ryohei Miyadera.

At least 19 recordsLinked to original sources

arXiv2026

A Subtraction Nim with a Pass

We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.

math.CO↗

arXiv2026

Variants of Wythoff's Games with Different Terminal Sets

We study 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. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally in any number of steps. The winner is the player who moves the queen to the terminal position in the upper-left corner, the position (0,0) in our coordinate system. Let k be a positive integer, and we consider the variant of Wythoff's game with the terminal set {(x,y):x,y are non-negative integers and x+y <=k}. The set of P-positions of this variant is described by the Fibonacci sequence without using recursion.

math.CO↗

arXiv2025

A Variant of Wythoff's Game Defined by Hofstadter's G-Sequence

In this paper, we study 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. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally, in any number of steps. The winner is the player who moves the queen into the upper-left corner, the position (0,0) in our coordinate system. We call (0,0) the terminal position of Wythoff's game. In our variant of Wythoff's game, we have a set of positions {(0,0),(1,0),(0,1),(1,1),(2,0),(0,2)} as the terminal set. If a player moves the queen into this terminal set, that player is the winner of the game. The P-positions of this variant are described by the P-positions of Wythoff's game and Hofstadter's G-Sequence. This variant has two remarkable properties. For a position (x,y) with x >= 8 or y >= 8, the Grundy number of the position (x,y) is 1 in this variant if and only if (x,y) is a P-position of Wythoff's game. There is another remarkable property.For a position (x,y) with x >= 8 or y >= 8, (x,y) is a P-position of of the misere version of this variant if and only if (x,y) is a P-position of of Wythoff's game.

math.CO↗

arXiv2025

Two Dimensional Silver Dollar Game

We define a variant of the two-dimensional Silver Dollar game. 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. Jumping a coin over another coin or on another coin is illegal. We add another operation: moving a coin and pushing another coin. For non-negative integers w,x,y,z, we denote the positions of the two coins by (w,x,y,z), where (w,x) is the position of one coin and (y,z) is the position of the other coin. Then, the set of P-positions (the previous player's winning positions) of this game is {(w,x,y,z):the nim-sum of (w-1),(x-1),(y-1), and (z-1) is 0} . Next, we make another game by omitting the rule of pushing another coin and permitting a jump over another coin. Then, there exist two relatively small sets A and B such that 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} cup A) -B.

math.GM↗

arXiv2025

A Variant of Game of Sliding Coins

Here, we present a variant of the sliding coins game. Two coins are placed on distinct squares of a semi-infinite linear board with squares numbered $0, 1, 2, dots, $. Two players take turns and move a coin to a lower unoccupied square. When a coin is pushed to the outside of the linear board, the players cannot use this coin anymore. In this game, we have another operation of moving coins: moving the coin on the right leftward and pushing the coin on the left. This last operation complicates this game's mathematical structure, but we have managed to make formulas for Grundy numbers. Since all the positions of this game of two coins are the next player's winning position, this game of two coins is trivial as a game. However, by making the sum of two games, we get a meaningful game in which the player who plays for the last time is the winner. With these Grundy numbers formulae, we get the winning strategy.

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↗

arXiv2024

A New 0(klog n) Algorithm for Josephus Problem

We present a new O(k log n) algorithm of the Josephus problem. The time complexity of our algorithm is O(k log n), and this time complexity is on a par with the existing O(k log n) algorithm. We do not have any recursion overhead or stack overflow because we do not use any recursion. Therefore, the space complexity of our algorithm is O(1), and ours is better than the existing O(k log n) algorithm in this respect. When k is small and n is large, our algorithm is better than the existing O(k log n) algorithm. This new algorithm is based on a relation between the Josephus problem and a maximum Nim of combinatorial game theory.

cs.DS↗

arXiv2024

A Problem of Knot

In this article, the authors give the correct answer to the following problem, which is presented in the well-known problem book "CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLUTIONS"? by A. M. Yaglom and L. M. Yaglom. There are six long blades of grass with the ends protruding above and below, and you will tie together the six upper ends in pairs and then tie together the six lower ends in pairs. What is the probability that a ring will be formed when the blades of grass are tied at random in this fashion? The solution in the above book needs to be corrected, and we will present a correct answer in this article. Therefore, we are the first persons to present a correct?answer to a problem in a book published in the USSR? in 1954. By following the original idea of this problem book, we present the correct answer without using knowledge of higher knowledge, although we used a very basic knowledge of the Knot theory.

math.GM↗

arXiv2024

Maximum Nim and Josephus Problem algorithm

In this study, we study a Josephus problem algorithm. Let $n,k$ be positive integers and $g_k(n) = \left\lfloor \frac{n}{k-1} \right\rfloor +1$, where $ \left\lfloor \ \ \right\rfloor$ is a floor function. Suppose that there exists $p$ such that $g_{k}^{p-1}(0) < n(k-1) \leq g_{k}^{p}(0)$, where $g_{k}^p$ is the $p$-th functional power of $g_k$. Then, the last number that remains is $nk-h2_{k}^{p}(0)$ in the Josephus problem of $n$ numbers, where every $k$-th numbers are removed. This algorithm is based on Maximum Nim with the rule function $f_k(n)=\left\lfloor \frac{n}{k} \right\rfloor$. Using the present article's result, we can build a new algorithm for Josephus problem.

math.CO↗

arXiv2024

Maximum Nim and Josephus Problem

In this study, we study the relation between Grundy numbers of a Maximum Nim and Josephus problem. Let f(x) = floor(x/k), where floor( ) is the floor function and k is a positive integer. We prove that there is a simple relation with a Maximum Nim with the rule function f and the Josephus problem in which every k-th numbers are to be removed.

math.CO↗

arXiv2023

Josephus Nim

Here, we present a variant of Nim with two piles. In the first pile, we have stones with a weight of 1, and in the second pile, we have stones with a weight of -2. Two Players take turns to take stones from one of the piles, and the total weight of stones to be removed should be equal to or less than half of the total weight of the stones in the pile. The player who removed the last stone or stones is the winner of the game. The authors discovered that when (n,m) is a previous player's winning position, 2m+1 is the last remaining number of the Josephus problem, where there are n -numbers, and every second number is to be removed. There are similar relations between the position of which the Grundy number is s and the (n-s)-th removed number.

math.CO↗

arXiv2023

Games of Nim with Dynamic Restrictions

The authors present formulas for the previous player's winning positions of two variants of restricted Nim. In both of these two games, there is one pile of stones, and in the first variant, we investigate the case that in k-th turn, you can remove f(k) stones at most, where f is a function whose values are natural numbers. In the second variant, there are two kinds of stones. The Type 1 group consists of stones with the weight of one, and the Type 2 group consists of stones with the weight of two. When the total weight of stones is a, you can remove stones whose total weight is equal to or less than half of a.

math.CO↗

arXiv2023

Curious Properties of Iterative Sequences

In this study, several interesting iterative sequences were investigated. First, we define the iterative sequences. We fix function f(n). An iterative sequence starts with a natural number n, and calculates the sequence f(n),f(f(n)), ...f(f(f(f(n)))),... We then search for interesting features in this sequence. We study Kaprekar's routine, the digit factorial process, and the digit power process. The authors presented new variants of Kaprekar's routine.

math.GM↗

arXiv2023

Impartial and Partizan Restricted Chocolate Bar Games

In this paper, we consider impartial and partizan restricted chocolate bar games. In impartial restricted chocolate bar games, players cut a chocolate bar into two pieces along any horizontal or vertical line and eat whichever piece is smaller. If the two pieces are the same size, a player can eat either one. In constrast, partizan restricted chocolate bar games include players designated as Left and Right and chocolate bars with black and white stripes. Left cuts the chocolate bar in two as above and eats the part with fewer black blocks. Similarly, Right cuts the bar and eats the part with fewer white blocks. A player loses when they cannot eat the remaining chocolate bar. We provide formulas that describe the winning positions of the previous player, Right, and Left players. We also present an interesting similarity in the graphs of previous players' winning positions for impartial and partizan chocolate bar games.

math.CO↗

arXiv2022

Multi-Dimensional Chocolate and Nim with a Pass

Chocolate-bar games are variants of the CHOMP game. A three-dimensional chocolate bar comprises a set of cubic boxes sized 1 X 1 X 1, with a bitter cubic box at the bottom of the column at position (0,0). For non-negative integers u,w such that u < x and w \< z, the height of the column at position (u,w) is min (F(u,w),y) +1, where F is a monotonically increasing function. We denote this chocolate bar as CB(F,x,y,z). Each player, in turn, cuts the bar on a plane that is horizontal or vertical along the grooves, and eats the broken piece. The player who manages to leave the opponent with the single bitter cubic box is the winner. In this study, functions F such that the Sprague--Grundy value of CB(F,x,y,z) is x xor y xor z are characterized. We have already published the research on three-dimensional chocolate games. In this paper, the authors study a multi-dimensional chocolate game, where the dimension is more than three, and apply the theory to the problem of pass move in Nim. We modify the standard rules of the game to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once a pass has been used by either player, it is no longer available. It is well-known that in classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity. A multi-dimensional chocolate game can show a perspective on the complexity of the game of Nim with a pass. Therefore, the authors address a longstanding open question in combinatorial game theory. The authors present this paper, since their discovery seems to be significant. It appears to the authors that the relation between chocolate games and Nim with a pass will be an essential research topic soon.

math.CO↗

arXiv2022

Restricted Nim with a Pass

This paper presents a study of restricted Nim with a pass. In the restricted Nim considered in this study, two players take turns and remove stones from the piles. In each turn, when the number of stones is m, each player is allowed to remove at least one stone and at most the ceiling of m/2 stones from a pile of m stones. The standard rules of the game are modified to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once a pass has been used by either player, it is no longer available. It is well-known that in classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity. In the restricted Nim considered in this study, the pass move was found to have a minimal impact. There is a simple relationship between the Grundy numbers of restricted Nim and the Grundy numbers of restricted Nim with a pass, where the number of piles can be any natural number. Therefore, the authors address a longstanding open question in combinatorial game theory: the extent to which the introduction of a pass into a game affects its behavior. The game that we developed appears to be the first variant of Nim that is fully solvable when a pass is not allowed and remains fully solvable following the introduction of a pass move.

math.CO↗

arXiv2022

Previous Player's Positions of Impartial Three-Dimensional Chocolate-Bar Games

In this study, we investigate three-dimensional chocolate bar games, which are variants of the game of Chomp. A three-dimensional chocolate bar is a three-dimensional array of cubes in which a bitter cubic box is present in some part of the bar. Two players take turns and cut the bar horizontally or vertically along the grooves. The player who manages to leave the opponent with a single bitter block is the winner. We consider the P-positions of this game, where the P-positions are positions of the game from which the previous player (the player who will play after the next player) can force a win, as long as they play correctly at every stage. We present sufficient conditions for the case when the position {p,q,r} is a P-position if and only if the bitxor of p-1, q-1, r-1, where p, q and r are the length, height, and width of the chocolate bar, respectively.

math.CO↗

arXiv2017

Impartial Triangular Chocolate Bar Games

Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rectangular chocolate bar game is a thinly disguised form of classical multi-heap Nim. In this work, we investigate the mathematical structure of triangular chocolate bar games in which the triangular chocolate bar can be cut in three directions. In the triangular chocolate bar game, a position is a $\mathcal{P}$-position if and only if $x \oplus y \oplus z = 0$, where the numbers $x,y,z$ stand for the maximum number of times that the chocolate bar can be cut in each direction. Moreover, the Grundy number of a position $(x,y,z)$ is not always equal to $x \oplus y \oplus z $, and a generic formula for Grundy numbers in not known. Therefore, the mathematical structure of triangular chocolate bar game is different from that of classical Nim.

math.CO↗