Results 1 
5 of
5
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
unknown title
"... Abstract Given k * 3 heaps of tokens. The moves of the 2player game introduced here are to either take a positive number of tokens from at most k \Gamma 1 heaps, or to remove the same positive number of tokens from all the k heaps. We analyse this extension of Wythoff's game and provide a poly ..."
Abstract
 Add to MetaCart
Abstract Given k * 3 heaps of tokens. The moves of the 2player game introduced here are to either take a positive number of tokens from at most k \Gamma 1 heaps, or to remove the same positive number of tokens from all the k heaps. We analyse this extension of Wythoff's game and provide a polynomialtime strategy for it. Keywords: multiheap games, efficient strategy, Wythoff game
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 ..."
Abstract
 Add to MetaCart
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
A New Heap Game
, 1998
"... Abstract. Given k ≥ 3 heaps of tokens. The moves of the 2player game introduced here are to either take a positive number of tokens from at most k − 1 heaps, or to remove the same positive number of tokens from all the k heaps. We analyse this extension of Wythoff’s game and provide a polynomialti ..."
Abstract
 Add to MetaCart
Abstract. Given k ≥ 3 heaps of tokens. The moves of the 2player game introduced here are to either take a positive number of tokens from at most k − 1 heaps, or to remove the same positive number of tokens from all the k heaps. We analyse this extension of Wythoff’s game and provide a polynomialtime strategy for it. 1.