arXiv subjects

Urban Larsson

Publications and source records attributed to Urban Larsson.

At least 55 records · Page 3Linked to original sources

arXiv2016

Chromatic Nim finds a game for your solution

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 red; if the level does not correspond to a number in the sequence, color it green. Now play Nim on a arbitrary number of stacks with the extra rule: if all top tokens are green, then you can make any move you like. On two stacks, we give explicit characterizations for winning the normal play version for some popular sequences, such as Beatty sequences and the evil numbers corresponding to the 0s in the famous Thue-Morse sequence. We also propose a more general solution which depends only on which of the colors `dominates' the sequence. Our construction resolves a problem posed by Fraenkel at the BIRS 2011 workshop in combinatorial games.

math.CO↗

arXiv2015

Finding Golden Nuggets by Reduction

We introduce a class of normal play partizan games, called Complementary Subtraction. Let $A$ denote your favorite set of positive integers. This is Left's subtraction set, whereas Right subtracts numbers not in $A$. The Golden Nugget Subtraction Game has the $A$ and $B$ sequences, from Wythoff's game, as the two complementary subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we find a polynomial time bit characterization for them, via the ternary Fibonacci representation.

math.CO↗

arXiv2015

Global Fibonacci Nim

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 investigate some properties of a variant played with multiple piles of stones, and solve the 2-pile case. A player chooses one of the piles and plays as in Fibonacci nim, but here the move-size restriction is a global parameter, valid for any pile.

math.CO↗

arXiv2015

Building Nim

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture.

cs.DM↗

arXiv2015

A Cellular Automaton for Blocking Queen Games

We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as {\em Blocking Wythoff Nim}, consists of moving a queen as in chess, but always towards (0,0), and it may not be moved to any of $k-1$ temporarily "blocked" positions specified on the previous turn by the other player. The game ends when a player wins by blocking all possible moves of the other player. The value of $k$ is a parameter that defines the game, and the pattern of winning positions can be very sensitive to $k$. As $k$ becomes large, parts of the pattern of winning positions converge to recurring chaotic patterns that are independent of $k$. The patterns for large $k$ display an unprecedented amount of self-organization at many scales, and here we attempt to describe the self-organized structure that appears.

math.CO↗

arXiv2015

Guaranteed Scoring Games

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by `switching the players'. The order relation between two games is defined by comparing their stops in \textit{any} disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

math.CO↗

arXiv2015

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 in 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.

math.CO↗

arXiv2015

When waiting moves you in scoring combinatorial games

Combinatorial Scoring games, with the property `extra pass moves for a player does no harm', are characterized. The characterization involves an order embedding of Conway's Normal-play games. Also, we give a theorem for comparing games with scores (numbers) which extends Ettinger's work on dicot Scoring games.

math.CO↗

arXiv2015

An Aperiodic Subtraction Game of Nim-dimension Two

In a recent arXiv-manuscript Fox studies infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the subtraction set $\{F_{2n+1}-1\}$, where $F_i$ is the $i$th Fibonacci number, and where $n$ ranges over the positive integers. Our definition of nim-dimension reflects the precise number of power-of-two-components generated by the games; the group of nim-values is of order four so the dimension is two (in the classical definition this dimension would have been one). Thanks to Carlos Santos for an enlightening discussion on this matter.

math.CO↗

arXiv2013

Impartial games whose rulesets produce given continued fractions

We study 2-player impartial games of the form take-away which produce P-positions (second player winning positions) corresponding to complementary Beatty sequences, given by the continued fractions (1;k,1,k,1,...) and (k+1;k,1,k,1,...). Our problem is the opposite of the main field of research in this area, which is to, given a game, understand its set of P-positions. We are rather given a set of (candidate) P-positions and look for "simple" rules. Our rules satisfy two criteria, they are given by a closed formula and they are invariant, that is, the available moves do not depend on the position played from (for all options with non-negative coordinates).

math.CO↗

arXiv2012

Comply subtraction games avoiding arithmetic progressions

Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games, similar to those introduced by Holshouser, Reiter, Smith, Stănică, can be defined as having its sets of winning positions identical to well-known sets avoiding arithmetic progressions such as $x+z=2y$, studied by Szerkeres, Erd\H os and Turán, and many others, thus exploring a new territory combining ideas from combinatorial games and combinatorial number theory. The sets we have in mind are greedy, that is, for our example: recursively a new nonnegative integer is included to the set if and only if it does not form a three term arithmetic progression with the smaller entries. It is known that the set thus obtained is equivalent to the following log-linear time closed expression: each winning position contains exclusively the digits 0 and 1 in base 3 expansion. In fact this set is impossible as a set of winning positions for a classical subtraction game, in a sense introduced recently by Duchêne and Rigo. Therefore our comply-rule generalization of the subtraction games can be seen to resolve new classes of sets as winning positions for heap games. In this context the $\star$-operator for invariant subtraction games was introduced by Larsson, Hegarty and Fraenkel. We define a similar operator for our game. Our comply games generalize into several dimensions. In two dimensions the winning positions can be represented by certain greedy permutations avoiding arithmetic progressions, one of which was recently introduced by Hegarty; while others generalize classical combinatorial games such as Nim and Wythoff Nim.

math.NT↗

arXiv2012

Maharaja Nim, Wythoff's Queen meets the Knight

New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim in which two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner. Here, in addition to the classical rules a player may also move the Queen as the Knight of Chess moves. We prove that the second player's winning positions of Maharaja Nim are close to the ones of Wythoff Nim, namely they are within a bounded distance to the lines with slope $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$ respectively. For a close relative to Maharaja Nim, where the Knight's jumps are of the form $(2,3)$ and $(3,2)$ (rather than $(1,2)$ and $(2,1)$), we also demonstrate polynomial time complexity to the decision problem of the outcome of a given position.

math.CO↗

arXiv2012

$(1,2)$-GDWN splits

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 game terminates when a player in turn is unable to move. We follow the normal play convention, which means that a player who cannot move loses. In the game of Wythoff Nim, a player is allowed to remove either any number of tokens from precisely one of the piles or the same number of tokens from both. Let $ϕ= \frac{1+\sqrt{5}}{2}$ and for all nonnegative integers $n$, $A_n=\lfloorϕn \rfloor$ and $B_n=A_n+n$. The P-positions of Wythoff Nim are all pairs of piles with $A_n$ and $B_n$ tokens respectively. We study a generalization of this game called $(1,2)\G$ where, in addition to the rules of Wythoff Nim, a player has the choice to remove a positive number of tokens from one of the piles and twice that number from the other pile. We show that there is an infinite sector $α\le y/x \le α+ε$, for given real numbers $α>1$ and $ε> 0$, for which each $(x,y)$ is an N-position, but that there are infinitely many P-positions for both $1\le y/x <α$ and $α+ε< y/x $. This proves a conjecture from a recent paper. Namely, the adjoined set of moves in $(1,2)\G$ \emph{splits} the beam of slope $ϕ$ P-positions of Wythoff Nim. We also provide a lower bound on the lower asymtotic density of lower pile heights of P-positions for extensions of Wythoff Nim. Suppose that $(a_i)$ and $(b_i)$, $i>0$, is a pair of so-called complementary sequences on the natural numbers which satisfy $(a_i)$ is increasing and for all $i$, $a_i<b_i$, for all $i\ne j$, $b_i-a_i\ne b_j-a_j$. Then $\liminf_{n\rightarrow \infty}\frac{#\{i\mid a_i < n\}}{n} \ge ϕ^{-1}$.

math.CO↗

arXiv2012

Rational Heap Games

We study variations of classical combinatorial games on two finite heaps of tokens, a.k.a. \emph{subtraction games}. Given non-negative integers $p_1,q_1, p_2,q_2$, where $p_1q_2 > q_1p_2$, $p_1>0$ and $q_2>0$, two players alternate in removing $(m_1,m_2)\ne (0,0)$ tokens from the respective heaps, where the allowed ordered pairs of non-negative integers are given by a certain move set $(m_1,m_2)\in\M$. There is a restriction imposed on the allowed heap sizes $(X, Y)$, they must satisfy $Xq_1\le Yp_1$ and $Yp_2\le Xq_2$. A player who cannot move loses and the other player wins. For a certain restriction of these games, namely where each allowed move option $(m_1,m_2)$ is of the form $(sp_1+tp_2,sq_1+tq_2)$, for some ordered pair of non-negative integers $(s,t)\ne (0,0)$, we show that all games have equivalent outcomes via a certain surjective map to a canonical subtraction game. Other interests in our games are various interactions with classical combinatorial games such as \emph{Nim} and \emph{Wythoff Nim}.

math.CO↗

arXiv2012

From heaps of matches to the limits of computability

We study so-called invariant games played with a fixed number $d$ of heaps of matches. A game is described by a finite list $\mathcal{M}$ of integer vectors of length $d$ specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in $\mathcal{M}$, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap game, the vector $(1,-2)$ would mean adding one match to the first heap and removing two matches from the second heap. If $(1,-2) \in \mathcal{M}$, such a move would be permitted provided there are at least two matches in the second heap. Two players take turns, and a player unable to make a move loses. We show that these games embrace computational universality, and that therefore a number of basic questions about them are algorithmically undecidable. In particular, we prove that there is no algorithm that takes two games $\mathcal{M}$ and $\mathcal{M}'$ (with the same number of heaps) as input, and determines whether or not they are equivalent in the sense that every starting-position which is a first player win in one of the games is a first player win in the other.

cs.CC↗

arXiv2012

Impartial games emulating one-dimensional cellular automata and undecidability

We study two-player \emph{take-away} games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its behavior are algorithmically undecidable. We show that similar questions are undecidable for our \emph{rule 110} game.

math.CO↗

arXiv2011

Black & White Nim games

We present a new family of Nim games where the rules depend on a given `coloring' of the tokens, each token being either black or white. The rules are as in Nim with the restriction that a white token on top of each heap is not allowed. We resolve the winning strategies of two disjoint game families played on two heaps. The heap-sizes with black tokens correspond to the numbers $\lfloor βn \rfloor$, where $β>2$ is an integer for one of the families and irrational for the other, and where $n$ ranges over the positive integers. In the process, new notions of \emph{invariant games} are introduced.

math.CO↗

arXiv2011

A Variant $β$-Wythoff Nim on Beatty's Theorem

We give short rules for two-pile take-away games satisfying that a pair of complementary homogeneous Beatty sequences together with $(0,0)$ constitute a complete set of $P$-positions.

math.CO↗