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A Generalized Diagonal Wythoff Nim

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Published/Copyright: October 2, 2012
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Integers
From the journal Integers Volume 12 Issue 5

Abstract.

The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope and respectively. We study extensions to this game where a player may also remove simultaneously tokens from either of the piles and from the other, where are given positive integers and where t ranges over the positive integers. We prove that for certain pairs (p,q) the P-positions are identical to those of Wythoff Nim, but for they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs (p,q) for which Wythoff Nim's beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim.

Keywords: Combinatorial Game; Integer Sequence; Wythoff Nim; Splitting of P-Beam
Received: 2010-05-10
Revised: 2011-07-22
Accepted: 2012-05-10
Published Online: 2012-10-02
Published in Print: 2012-10-01

© 2012 by Walter de Gruyter Berlin Boston

Articles in the same Issue

  1. Masthead
  2. Two-Color Babylon
  3. On Abelian and Additive Complexity in Infinite Words
  4. Approximations of Additive Squares in Infinite Words
  5. The Googol-th Bit of the Erdős–Borwein Constant
  6. Odd Repdigits to Small Bases Are Not Perfect
  7. Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
  8. The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
  9. Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
  10. Sum-Products Estimates with Several Sets and Applications
  11. The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
  12. A Proof of Catalan's Convolution Formula
  13. Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
  14. The Impossibility of Certain Types of Carmichael Numbers
  15. Integer Subsets with High Volume and Low Perimeter
  16. Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
  17. A Generalized Diagonal Wythoff Nim
  18. A Recursive Process Related to a Partizan Variation of Wythoff
  19. The (r1,...,rp)-Stirling Numbers of the Second Kind
  20. Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
  21. On k-Lehmer Numbers
https://doi.org/10.1515/integers-2012-0020
Keywords for this article
Combinatorial Game; Integer Sequence; Wythoff Nim; Splitting of P-Beam

Articles in the same Issue

  1. Masthead
  2. Two-Color Babylon
  3. On Abelian and Additive Complexity in Infinite Words
  4. Approximations of Additive Squares in Infinite Words
  5. The Googol-th Bit of the Erdős–Borwein Constant
  6. Odd Repdigits to Small Bases Are Not Perfect
  7. Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
  8. The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
  9. Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
  10. Sum-Products Estimates with Several Sets and Applications
  11. The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
  12. A Proof of Catalan's Convolution Formula
  13. Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
  14. The Impossibility of Certain Types of Carmichael Numbers
  15. Integer Subsets with High Volume and Low Perimeter
  16. Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
  17. A Generalized Diagonal Wythoff Nim
  18. A Recursive Process Related to a Partizan Variation of Wythoff
  19. The (r1,...,rp)-Stirling Numbers of the Second Kind
  20. Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
  21. On k-Lehmer Numbers
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