Although the combinatorial game Entrepreneurial Chess (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 [x, y] 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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Albert MH, Nowakowski RJ (2009) Games of no chance 3 (Berkeley, CA). Math. Sci. Res. Inst. Publ., vol 56, Cambridge Univ. Press, Cambridge
Albert MH, Nowakowski RJ, Wolfe D (2007) Lessons in play: an introduction to combinatorial game theory. A. K. Peters, USA
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 231–241
Berlekamp ER (1988) Blockbusting and domineering. J Comb Theory Ser A 49:67–116
Berlekamp ER, Conway JH, Guy RK (2001) Winning ways for your mathematical plays, Volume 1 (of 4), AK. Peters, USA
Berlekamp ER, Wolfe D (1994) Mathematical go: chilling gets the last point. AK Peters, USA
Conway JH (1976) On numbers and games. Academic, London
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, Cambridge
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 457–473
Guy RK, Nowakowski RJ (2002) More games of no chance (Berkeley, CA 2000), Math. Sci. Res. Inst. Publ., vol 42. Cambridge Univ. Press, Cambridge
Low RM, Stamp M (2006) King and rook vs. king on a quarter-infinite board. Integers. 6, #G3:8 (electronic)
Nowakowski RJ (2015) Games of no chance 4 (Berkeley, CA), Math. Sci. Res. Inst. Publ., vol 63. Cambridge Univ. Press, Cambridge
Nowakowski RJ (1996) Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol 29. Cambridge Univ. Press, Cambridge
Pearson M, Berlekamp ER (2005) Entrepreneurial chess: some king and rook versus king combinatorial game theory problems (unpublished)
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 261–278
Siegel AN (2013) Combinatorial game theory, graduate studies in mathematics, vol 146. American Mathematical Society, Providence, RI
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.
About this article
Cite this article
Berlekamp, E., Low, R.M. Entrepreneurial Chess. Int J Game Theory 47, 379–415 (2018). https://doi.org/10.1007/s00182-017-0580-z
- Combinatorial games
Mathematics Subject Classification