A morphic approach to combinatorial games: the Tribonacci case

Eric Duchêne; Michel Rigo

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 42, Issue: 2, page 375-393
  • ISSN: 0988-3754

Abstract

top
We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time.

How to cite

top

Duchêne, Eric, and Rigo, Michel. "A morphic approach to combinatorial games: the Tribonacci case." RAIRO - Theoretical Informatics and Applications 42.2 (2007): 375-393. <http://eudml.org/doc/92877>.

@article{Duchêne2007,
abstract = { We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time. },
author = {Duchêne, Eric, Rigo, Michel},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Two-player combinatorial game; combinatorics on words; numeration system; Tribonacci sequence},
language = {eng},
month = {12},
number = {2},
pages = {375-393},
publisher = {EDP Sciences},
title = {A morphic approach to combinatorial games: the Tribonacci case},
url = {http://eudml.org/doc/92877},
volume = {42},
year = {2007},
}

TY - JOUR
AU - Duchêne, Eric
AU - Rigo, Michel
TI - A morphic approach to combinatorial games: the Tribonacci case
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/12//
PB - EDP Sciences
VL - 42
IS - 2
SP - 375
EP - 393
AB - We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time.
LA - eng
KW - Two-player combinatorial game; combinatorics on words; numeration system; Tribonacci sequence
UR - http://eudml.org/doc/92877
ER -

References

top
  1. E. Barcucci, L. Bélanger, S. Brlek, On Tribonacci sequences. Fibonacci Quart.42 (2004) 314–319.  
  2. E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning ways (two volumes). Academic Press, London (1982).  
  3. M. Boshernitzan, A. Fraenkel, Nonhomogeneous spectra of numbers. Discrete Math.34 (1981) 325–327.  
  4. L. Carlitz, R. Scoville, V.E. Hoggatt Jr., Fibonacci representations of higher order. Fibonacci Quart.10 (1972) 43–69.  
  5. A. Cobham, Uniform tag sequences. Math. Syst. Theor.6 (1972) 164–192.  
  6. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Lect. Notes Math.1794, Springer-Verlag, Berlin (2002).  
  7. A. Fraenkel, I. Borosh, A generalization of Wythoff's game. J. Combin. Theory Ser. A15 (1973) 175–191.  
  8. A. Fraenkel, How to beat your Wythoff games' opponent on three fronts. Amer. Math. Monthly89 (1982) 353–361.  
  9. A. Fraenkel, Systems of numeration. Amer. Math. Monthly92 (1985) 105–114.  
  10. A. Fraenkel, Heap games, numeration systems and sequences. Ann. Comb.2 (1998) 197–210.  
  11. A. Fraenkel, The Raleigh game, to appear in INTEGERS, Electron. J. Combin. Number Theor7 (2007) A13.  
  12. A. Fraenkel, The rat game and the mouse game, preprint.  
  13. M. Lothaire, Combinatorics on words. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997).  
  14. G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France110 (1982) 147–178.  
  15. M. Rigo and A. Maes, More on generalized automatic sequences. J. Autom. Lang. Comb.7 (2002) 351–376.  
  16. N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, see  URIhttp://www.research.att.com/~njas/sequences/
  17. B. Tan, Z.-Y. Wen, Some properties of the Tribonacci sequence. Eur. J. Combin.28 (2007) 1703–1719.  
  18. W.A. Webb, The length of the four-number game. Fibonacci Quart.20 (1982) 33–35.  
  19. W.A. Wythoff, A modification of the game of Nim. Nieuw Arch. Wisk.7 (1907) 199–202.  
  20. E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liège41 (1972) 179–182.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.