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2pile Nim with a Restricted Number of Movesize Imitations, submitted to Integers, available at http://arxiv.org/abs/0710.3632
 Amer. Math. Monthly
, 1927
"... Peter Hegarty ..."
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The Rat game and the Mouse game
, 2008
"... We define three new takeaway games, the Rat game, the Mouse game and the Fat Rat game. Three winning strategies are given for the Rat game and outlined for the Mouse and Fat Rat games. The efficiencies of the strategies are determined. Whereas the winning strategies of nontrivial takeaway games ar ..."
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Cited by 8 (3 self)
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We define three new takeaway games, the Rat game, the Mouse game and the Fat Rat game. Three winning strategies are given for the Rat game and outlined for the Mouse and Fat Rat games. The efficiencies of the strategies are determined. Whereas the winning strategies of nontrivial takeaway games are based on irrational numbers, our games are based on rational numbers. Another motivation stems from a problem in combinatorial number theory. 1 Description of the Game The Rat game is played on 3 piles of tokens by 2 players who play alternately. Positions in the game are denoted throughout in the form (x, y, z), with 0 ≤ x ≤ y ≤ z, and moves in the form (x, y, z) → (u, v, w), where of course also 0 ≤ u ≤ v ≤ w (see below). The player first unable to move — because the position is (0, 0, 0) — loses; the opponent wins. There are 3 types of moves: (I) Take any positive number of tokens from up to 2 piles.
Another bridge between Nim and Wythoff
"... The P positions of both twoheap Nim and Wythoff’s game are easy to describe, more so in the former than in the latter. Calculating the actual G values is easy for Nim but seemingly hard for Wythoff’s game. We consider what happens when the rules for removing from both heaps are modfied in various w ..."
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Cited by 7 (3 self)
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The P positions of both twoheap Nim and Wythoff’s game are easy to describe, more so in the former than in the latter. Calculating the actual G values is easy for Nim but seemingly hard for Wythoff’s game. We consider what happens when the rules for removing from both heaps are modfied in various ways.
The game of EndWythoff
"... ABSTRACT. Given a vector of finitely many piles of finitely many tokens. In EndWythoff, two players alternate in taking a positive number of tokens from either endpile, or taking the same positive number of tokens from both ends. The player first unable to move loses and the opponent wins. We char ..."
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Cited by 6 (2 self)
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ABSTRACT. Given a vector of finitely many piles of finitely many tokens. In EndWythoff, two players alternate in taking a positive number of tokens from either endpile, or taking the same positive number of tokens from both ends. The player first unable to move loses and the opponent wins. We characterize the Ppositions.ai; K; bi / of the game for any vector K of middle piles, where ai; bi denote the sizes of the endpiles. A more succinct characterization can be made in the special case where K is a vector such that, for some n 2 Z 0,.K; n / and.n; K / are both Ppositions. For this case the (noisy) initial behavior of the Ppositions is described precisely. Beyond the initial behavior, we have bi ai D i, as in the normal 2pile Wythoff game. 1.
Harnessing the unwieldy MEX function
"... A pair of integer sequences that split Z>0 is often–especially in the context of combinatorial game theory–defined recursively by An = mex{Ai,Bi: 0 ≤ i < n}, Bn = An+Cn (n ≥ 0), where mex (Minimum EXcludant) of a subset S of nonnegative integers is the smallest nonnegative integer not in S, an ..."
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Cited by 4 (1 self)
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A pair of integer sequences that split Z>0 is often–especially in the context of combinatorial game theory–defined recursively by An = mex{Ai,Bi: 0 ≤ i < n}, Bn = An+Cn (n ≥ 0), where mex (Minimum EXcludant) of a subset S of nonnegative integers is the smallest nonnegative integer not in S, and C: Z≥0 → Z≥0. Given x, y ∈ Z>0, a typical problem is to decide whether x = An, y = Bn. For general functions Cn, the best algorithm for this decision problem is exponential in the input size Ω(log x + log y). We prove constructively that the problem is actually polynomial for the wide class of approximately linear functions Cn. This solves constructively and efficiently the complexity question of a number of previously analyzed takeaway games of various authors.
Complementary Iterated Floor Words and the Flora Game
"... Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ 2. We use them to formulate an algebraic polynomialtime winning strategy for a new 4pile takeaway game Flora, which is motivate ..."
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Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ 2. We use them to formulate an algebraic polynomialtime winning strategy for a new 4pile takeaway game Flora, which is motivated by partitioning the set of games into subsets CompGames and PrimGames. We present recursive, arithmetic and wordmapping winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. We further show how to generate the floor words induced by the iterations using wordmappings and and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences.
Unbounded Iterations of Floor Functions and the Flora Game
, 2009
"... Partition the set of games into subsets CompGames and PrimGames, precipitating a new 4pile takeaway game Flora. Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ 2. We use them t ..."
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Partition the set of games into subsets CompGames and PrimGames, precipitating a new 4pile takeaway game Flora. Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ 2. We use them to formulate an algebraic polynomialtime winning strategy for Flora, and also present recursive, arithmetic and morphic winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. The four strategies differ in their computational efficiencies. We further show how to generate the sequences induced by the iterations using morphisms and and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences.
Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.1 Interspersions and Fractal Sequences Associated with Fractions c j /d k
"... Suppose c ≥ 2 and d ≥ 2 are integers, and let S be the set of integers ⌊ c j /d k ⌋ , where j and k range over the nonnegative integers. Assume that c and d are multiplicatively independent; that is, if p and q are integers for which c p = d q, then p = q = 0. The numbers in S form interspersions in ..."
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Suppose c ≥ 2 and d ≥ 2 are integers, and let S be the set of integers ⌊ c j /d k ⌋ , where j and k range over the nonnegative integers. Assume that c and d are multiplicatively independent; that is, if p and q are integers for which c p = d q, then p = q = 0. The numbers in S form interspersions in various ways. Related fractal sequences and permutations of the set of nonnegative integers are also discussed. 1
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, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Another bridge between Nim and Wythoff E. Duchêne,