Sprague-Grundy Research Articles

Graph convexity impartial games: Complexity and winning strategies

Accordingly to Duchet (1987), the first paper of convexity on general graphs, in english, is the 1981 paper “Convexity in graphs”. One of its authors, Frank Harary, introduced in 1984 the first graph convexity games, focused on the geodesic convexity, which were investigated in a sequence of five papers that ended in 2003. In this paper, we continue this research line, extend these games to other graph convexities, and obtain winning strategies and complexity results. Among them, we obtain winning strategies for general convex geometries and winning strategies for trees from the Sprague-Grundy theory on impartial games. We also obtain the first PSPACE-hardness results on convexity games, by proving that the normal play and the misère play of the impartial hull game on the geodesic convexity is PSPACE-complete even in graphs with diameter two.

Impartial games with decreasing Sprague–Grundy function and their hypergraph compound

AbstractThe Sprague–Grundy (SG) theory reduces the disjunctive compound of impartial games to the classical game of NIM. We generalize this concept by introducing hypergraph compounds of impartial games. An impartial game is called SG-decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce hypergraph compounds of SG-decreasing games to hypergraph compounds of single-pile NIM games. We show that this reduction works only if all games involved in the compound are SG-decreasing. A hypergraph is called SG-decreasing if the corresponding hypergraph compound of single-pile NIM games is an SG-decreasing game. We provide some necessary and some sufficient conditions for a hypergraph to be SG-decreasing. In particular, for hypergraphs with hyperedges of size at most 3 we obtain a necessary and sufficient condition verifiable in polynomial time.

Sprague–Grundy theory in bounded arithmetic

We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node Kayles position is provably equivalent to some NIM heap.

Some i-Mark games

Let S be a set of positive integers, and let D be a set of integers larger than 1. The game ▪ is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract s∈S from the pile, or divide the size of the pile by d∈D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1,t−1] and D={d} for d≢1(modt), but left the case d≡1(modt) open.We solve this problem by calculating the Sprague–Grundy function of ▪ for d≡1(modt), for all t,d≥2. We also calculate the Sprague–Grundy function of ▪ for all k, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with |D|>1, we derive some partial results for the game ▪, whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value 0,1,2 occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

On pruning search trees of impartial games

In this paper we study computing Sprague-Grundy values for short impartial games under the normal play convention. We put forward new game-agnostic methods for effective pruning search trees of impartial games. These algorithms are inspired by the α-β, a well-known pruning method for minimax trees. However, our algorithms prune trees whose node values are assigned by the mex function instead of min/max.We have verified the effectiveness of our algorithms experimentally on instances of some standard impartial games (that is Nim, Chomp, and Cram). Based on the results of our experiments we have concluded that: (1) our methods generally perform well, especially when transposition table can store only a small fraction of all game positions (which is typical when larger games are concerned); (2) one of our algorithms constitutes a more universal alternative to the state-of-the-art algorithm proposed by Lemoine and Viennot.

On the Sprague–Grundy function of extensions of proper Nim

We consider the game of proper Nim, in which two players alternately move by taking stones from n piles. In one move a player chooses a proper subset (at least one and at most $$n-1$$) of the piles and takes some positive number of stones from each pile of the subset. The player who cannot move is the loser. Jenkyns and Mayberry (Int J Game Theory 9(1):51–63, 1980) described the Sprague–Grundy function of these games. In this paper we consider the so-called selective compound of proper Nim games with certain other games, and obtain a closed formula for the Sprague–Grundy functions of the compound games, when $$n\ge 3$$. Surprisingly, the case of $$n=2$$ is much more complicated. For this case we obtain several partial results and propose some conjectures.

Sprague–Grundy function of symmetric hypergraphs

We consider a generalization of the classical game of Nim called hypergraph Nim. Given a hypergraph H on the ground set V={1,…,n} of n piles of stones, two players alternate in choosing a hyperedge H∈H and strictly decreasing all piles i∈H. The player who makes the last move is the winner. In 1980 Jenkyns and Mayberry obtained an explicit formula for the Sprague–Grundy function of the hypergraph Nim whose hypergraph contains as the hyperedges all proper subsets of V (that is, all except ∅ and V itself). Somewhat surprisingly, the same formula works for a very wide family of hypergraphs. In this paper we characterize symmetric hypergraphs in this family.

Proper 2-coloring game on some trees

Let Pn denote the path with exactly n vertices and let Tn,i denote the tree obtained from Pn−1 by adding a pendant vertex to the (i+1)-th vertex of Pn−1, where n≥4 and 1≤i≤n−3. In this paper we study proper 2-coloring game on Tn,i by using Sprague–Grundy function and give the optimal strategy of proper 2-coloring game on Tn,i.

A generalization of Arc-Kayles

The game Arc-Kayles is played on an undirected graph with two players taking turns deleting an edge and its endpoints from the graph. We study a generalization of this game, Weighted Arc Kayles (WAK for short), played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left. The last player to play a move wins. We give a winning strategy for WAK on trees of depth 2. Moreover, we show that the Grundy values of WAK and Arc-Kayles are unbounded. We also prove a periodicity result on the outcome of WAK when the number of counters is fixed for all the vertices but one. Finally, we show links between this game and a variation of the non-attacking queens game on a chessboard.

The Sprague–Grundy function for some nearly disjunctive sums of Nim and Silver Dollar games

We introduce and analyse an extension of the disjunctive sum operation on some classical impartial games. Whereas the disjunctive sum describes positions formed from independent subpositions, our operation combines positions that are not completely independent but interact only in a very restricted way. We extend the games Nim and Silver Dollar, played by moving counters along one-dimensional strips of cells, by joining several strips at their initial cell. We prove that, in certain cases, computing the Sprague–Grundy function can be simplified to that of a simpler game with at most two tokens in each strip. We give an algorithm that, for each Sprague–Grundy value g, computes the positions of two-token Star Nim whose Sprague–Grundy values are g. We establish that the sequence of differences of coordinates of g-positions is ultimately periodic.

On the Sprague–Grundyfunction of Exact[formula omitted]-Nim

Moore’s generalization of the game of Nim is played as follows. Let n and k be two integers such that 1≤k<n. Given n piles of tokens, two players move alternately, removing tokens from at least one and at most k of the piles. The player who makes the last move wins. The game was solved by Moore in 1910; an explicit formula for its Sprague–Grundy function was given by Jenkyns and Mayberry in 1980, but only for the case n=k+1. We introduce another generalization of Nim, called Exactk- Nim, in which exactly k piles are reduced by each move. We give an explicit formula for the Sprague–Grundy function of Exactk- Nim for the case 2k≥n. If n=2k, our formula is surprisingly similar to Jenkyns and Mayberry’s one.

Algebraic games—playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some $$0 \ne a \in A$$ . The game then continues with the quotient group $$A/\langle a \rangle $$ . We prove that under the normal play rule, the second player has a winning strategy if and only if A is a square, i.e. $$A \cong B \times B$$ for some abelian group B. Under the misere play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as R[X], where R is a principal ideal domain.

Relation between number of kernels (and generalizations) of a digraph and its partial line digraphs

Let D=(V,A) be a digraph, an arc subset A′⊆A and a surjective mapping ϕ:A→A′ such that the set of heads of A′ is V and ϕ|A′=Id and for every vertex j∈V, ϕ(ω−(j))⊂ω−(j)∩A′. The partial line digraph of D, LD, is the digraph with vertex set V(LD)=A′ and set of arcs A(LD)={(ij,ϕ(j,k)):(j,k)∈A}. In this paper we prove the following results: Let k,l be two natural numbers such that 1≤l≤k, and D a digraph with δ−(D)≥1. Then the number of (k,l)-kernels of D is less than or equal to the number of (k,l)-kernels of LD. Moreover, if l<k and the girth of D is at least l+1, then these two numbers are equal. The number of semikernels of D is equal to the number of semikernels of LD. Also we introduce the concept of (k,l)-Grundy function as a generalization of the concept of Grundy function and we prove that the number of (k,l)-Grundy functions of D is equal to the number of (k,l)-Grundy functions of every partial line digraph LD.

Three-pile Sharing Nim and the quadratic time winning strategy

We study a variant of 3-pile Nim in which a move consists of taking tokens from one pile and, instead of removing them, topping up on a smaller pile provided that the destination pile does not have more tokens than the source pile after the move. We discover a situation in which each column of two-dimensional array of Sprague–Grundy values is a palindrome. We establish a formula for P-positions by which winning moves can be computed in quadratic time. We prove a formula for positions whose Sprague–Grundy values are 1 and estimate the distribution of those positions whose nim-values are g. We discuss the periodicity of nim-sequences that seem to be bounded.

Sprague-Grundy Values of the $\mathcal{R}$-Wythoff Game

We examine the Sprague-Grundy values of the game of $\mathcal{R}$-Wythoff, a restriction of Wythoff's game introduced by Ho, where each move is either to remove a positive number of tokens from the larger pile or to remove the same number of tokens from both piles. Ho showed that the $P$-positions of $\mathcal{R}$-Wythoff agree with those of Wythoff's game, and found all positions of Sprague-Grundy value $1$. We describe all the positions of Sprague-Grundy value $2$ and $3$, and also conjecture a general form of the positions of Sprague-Grundy value $g$.

Grundy values of Fibonacci nim

In this article, we investigate the Grundy values of the popular game of Fibonacci nim. The winning strategy, which amounts to understanding positions of Grundy value 0, was known since Whinihan (Fibonacci Quart 1:9–13, 1963). In this paper, we extend Whinihan’s analysis by computing all the positions of Grundy value at most 3. Furthermore, we show that, when we delete the Fibonacci numbers (which have Grundy value 0), the Grundy values of the starting positions are increasing, and we give upper and lower bounds on the growth rate.

The Max–Welter game

On a semi-infinite strip of squares rightward numbered 0,1,2,… with at most one coin in each square, in Welter’s game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The player who first cannot move loses. We examine a variant of Welter’s game, that we call Max–Welter, in which players are allowed to move only the coin furthest to the right. We solve the winning strategy and describe the positions of Sprague–Grundy value 1. We propose two theorems classifying some special cases where calculating the Sprague–Grundy value of a position of size k becomes easier by considering another position of size k−1. We establish two results on the periodicity of the Sprague–Grundy values. We then show that the Max–Welter game is classified in a proper subclass of tame games that Gurvich calls strongly miserable.

Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k>0 colors. Each different ruleset specifies that game’s coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function.

On the Sprague-Grundy Values of the $\mathcal{F}$-Wythoff Game

We examine the Sprague-Grundy values of $\mathcal{F}$-Wythoff, a restriction of Wythoff's game introduced by Ho, where the integer ratio of the pile sizes must be preserved if the same number of tokens is removed from both piles. We answer two conjectures raised by Ho. First, we show that each column of Sprague-Grundy values is ultimately additively periodic. Second, we prove that every diagonal of Sprague-Grundy values contains all the nonnegative integers. We also investigate the asymptotic behavior of the sequence of positions attaining a given Sprague-Grundy value.

A RESTRICTION OF EUCLID

AbstractEuclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.