Related papers: P-positions in Modular Extensions to Nim
In this paper, we introduce and examine a variant of the game of Nim (Sharing Nim), where players can either remove or transfer objects from 1 pile to another. The only restriction is that players may not transfer objects from a pile of…
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…
In this paper, we analyze the mis\`ere versions of two impartial combinatorial games: k-Bounded Greedy Nim and Greedy Nim. We present a complete solution to both games by showing necessary and sufficient conditions for a position to be…
Given $n$ piles of tokens and a positive integer $k \leq n$, the game Nim$^1_{n, =k}$ of exact slow $k$-Nim is played as follows. Two players move alternately. In each move, a player chooses exactly $k$ non-empty piles and removes one token…
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…
We compare to different extensions of the ancient game of nim: Moore's nim$(n, \leq k)$ and exact nim$(n, = k)$. Given integers $n$ and $k$ such that $0 < k \leq n$, we consider $n$ piles of stones. Two players alternate turns. By one move…
We study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens $(x,y)$. Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the…
Partially-ordered set games, also called poset games, are a class of two-player combinatorial games. The playing field consists of a set of elements, some of which are greater than other elements. Two players take turns removing an element…
We play a variation of Nim on stacks of tokens. Take your favorite increasing sequence of positive integers and color the tokens according to the following rule. Each token on a level that corresponds to a number in the sequence is colored…
The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes…
The game of Antonim is a variant of the game Nim, with the additional rule that heaps are not allowed to be the same size. A winning strategy for three heap Antonim has been solved. We will discuss the solution to three-heap Antonim and…
In this paper, we study an impartial game called Delete Nim. In this game, there are two heaps of stones. The player chooses one of the heaps and delete the other heap. Next, she takes away one stone from the chosen heap and optionally…
In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a…
Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), $C$, for Wythoff's Game is given by $C :=…
The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial…
Given $k\ge 3$ heaps of tokens. The moves of the 2-player game introduced here are to either take a positive number of tokens from at most $k-1$ heaps, or to remove the {\sl same} positive number of tokens from all the $k$ heaps. We analyse…
Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and…
We study a variant of 3-pile Nim in which a move consists of taking tokens from one pile and, instead of removing then, topping up on a smaller pile provided that the destination pile does not have more tokens then the source pile after the…
Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones. The game is appealing due to its surprising connections with the Fibonacci numbers and the Zeckendorf representation. In this article, we…
This paper introduces a variant of the impartial combinatorial game nim, called tree nim, as well as a particular case of tree nim called tripod nim. A certain existence-uniqueness result and a periodicity result are proven about the…