arXiv subjects
Publications and source records attributed to Masato Shinoda.
We consider a variant of Nim in which, for a fixed set $S$ of nonnegative integers, a move is forbidden if the total number of remaining stones belongs to $S$. This game coincides with normal-play Nim when $S=\emptyset$, and with mis\`ere Nim when $S=\{0\}$. In this paper, we focus in particular on the case where $S$ is the set of multiples of $d$, and show that simple criteria for determining the outcome can be obtained for $d=2,3,4$.
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 paper, 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 study the theoretical aspects of this extended game and analyze its fundamental properties, such as optimal strategies and payoff functions.
The classic game of Nim has been well-known for many years, inspiring numerous variations. One such variant is Delete Nim, where players take turns eliminating one pile of stones and splitting the remaining pile into two smaller piles. In this paper we generalize the game to include the case of n piles. On each turn, a player eliminates one pile and splits one of the remaining piles into two smaller piles. We specifically analyze the case where n=4, deriving the conditions for a winning strategy.
The game of Nim, which has been well known for many years, has numerous variations. One such variation is Circular Nim, where piles of stones are arranged on a circumference, and players take stones from consecutive adjacent piles in one move. In this paper, we propose a new variant called Shrinking Circular Nim, in which the size of the circle decreases as the game progresses. We also examine the winning conditions in this variant. This paper discusses that Shrinking Circular Nim is solved for initial configurations with five or fewer piles. In addition, we derive the winning condition for the specific case with eight piles.
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.
We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution. The connection between uniform spanning trees and loop-erased random walk is then exploited to prove convergence of the latter to a continuous stochastic process. Some geometric properties of this limit process, such as the Hausdorff dimension, are investigated as well. The method is also applicable to other self-similar graphs with a sufficient degree of symmetry.