Entrepreneurial Chess

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Abstract

Although the combinatorial game EntrepreneurialChess (or Echess) was invented around 2005, this is our first publication devoted to it. A single Echess position begins with a Black king vs. a White king and a White rook on a quarter-infinite board, spanning the first quadrant of the xy-plane. In addition to the normal chess moves, Black is given the additional option of “cashing out”, which removes the board and converts the position into the integer \(x + y\), where [xy] are the coordinates of his king’s position when he decides to cash out. Sums of Echess positions, played on different boards, span an unusually wide range of topics in combinatorial game theory. We find many interesting examples.

Keywords

Combinatorial games Chess 

Mathematics Subject Classification

91A46 

Notes

Acknowledgements

In 2005, Mark Pearson, then a student in Berlekamp’s graduate course on Combinatorial Game Theory, determined the means and temperatures shown in Figs. 15 and 26. All of our subsequent calculations have been done by hand. The authors wish to thank Eugenia Lee for creating the beautiful figures found throughout this paper. We are also indebted to David Berlekamp, both for his technical prowess in dealing with several mutually incompatible document preparation technologies, and for his help in debugging the mathematics in some of the sections of this paper. We are grateful for the valuable comments made by the referee and Editor. Finally, the authors wish to thank Bernhard von Stengel for his help in resolving some unforeseen typesetting problems as this paper went to press.

References

  1. Albert MH, Nowakowski RJ (2009) Games of no chance 3 (Berkeley, CA). Math. Sci. Res. Inst. Publ., vol 56, Cambridge Univ. Press, CambridgeGoogle Scholar
  2. Albert MH, Nowakowski RJ, Wolfe D (2007) Lessons in play: an introduction to combinatorial game theory. A. K. Peters, USAGoogle Scholar
  3. Berlekamp ER (2002) The 4G4G4G4G4 problems and solutions. In: More games of no chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ., vol 42, Cambridge Univ. Press, Cambridge, pp 231–241Google Scholar
  4. Berlekamp ER (1988) Blockbusting and domineering. J Comb Theory Ser A 49:67–116CrossRefGoogle Scholar
  5. Berlekamp ER, Conway JH, Guy RK (2001) Winning ways for your mathematical plays, Volume 1 (of 4), AK. Peters, USAGoogle Scholar
  6. Berlekamp ER, Wolfe D (1994) Mathematical go: chilling gets the last point. AK Peters, USAGoogle Scholar
  7. Conway JH (1976) On numbers and games. Academic, LondonGoogle Scholar
  8. Fraenkel AS (1996) Combinatorial games: selected biography with a succinct gourmet introduction. In: Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol 29. Cambridge Univ. Press, CambridgeGoogle Scholar
  9. Guy RK, Nowakowski RJ (2002) Unsolved problems in combinatorial games. In: More games of no chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ., vol 42. Cambridge Univ. Press, Cambridge, pp 457–473Google Scholar
  10. Guy RK, Nowakowski RJ (2002) More games of no chance (Berkeley, CA 2000), Math. Sci. Res. Inst. Publ., vol 42. Cambridge Univ. Press, CambridgeGoogle Scholar
  11. Low RM, Stamp M (2006) King and rook vs. king on a quarter-infinite board. Integers. 6, #G3:8 (electronic) Google Scholar
  12. Nowakowski RJ (2015) Games of no chance 4 (Berkeley, CA), Math. Sci. Res. Inst. Publ., vol 63. Cambridge Univ. Press, CambridgeGoogle Scholar
  13. Nowakowski RJ (1996) Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol 29. Cambridge Univ. Press, CambridgeGoogle Scholar
  14. Pearson M, Berlekamp ER (2005) Entrepreneurial chess: some king and rook versus king combinatorial game theory problems (unpublished)Google Scholar
  15. Snatzke RG (2002) Exhaustive search in the game Amazons. In: More games of no chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ., vol 42. Cambridge Univ. Press, Cambridge, pp 261–278Google Scholar
  16. Siegel AN (2013) Combinatorial game theory, graduate studies in mathematics, vol 146. American Mathematical Society, Providence, RICrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Elwyn and Jennifer Berlekamp FoundationOaklandUSA
  2. 2.Department of MathematicsSan Jose State UniversitySan JoseUSA

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