no. 2
A morphic approach to combinatorial games : the Tribonacci case
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 375-393.

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.

DOI : 10.1051/ita:2007039
Classification : 91A46, 68R15, 68Q45
Mots clés : two-player combinatorial game, combinatorics on words, numeration system, Tribonacci sequence 
@article{ITA_2008__42_2_375_0,
     author = {Duch\^ene, Eric and Rigo, Michel},
     title = {A morphic approach to combinatorial games : the {Tribonacci} case},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {375--393},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     doi = {10.1051/ita:2007039},
     mrnumber = {2401268},
     zbl = {1143.91314},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2007039/}
}
TY  - JOUR
AU  - Duchêne, Eric
AU  - Rigo, Michel
TI  - A morphic approach to combinatorial games : the Tribonacci case
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2008
SP  - 375
EP  - 393
VL  - 42
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita:2007039/
DO  - 10.1051/ita:2007039
LA  - en
ID  - ITA_2008__42_2_375_0
ER  - 
%0 Journal Article
%A Duchêne, Eric
%A Rigo, Michel
%T A morphic approach to combinatorial games : the Tribonacci case
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2008
%P 375-393
%V 42
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita:2007039/
%R 10.1051/ita:2007039
%G en
%F ITA_2008__42_2_375_0
Duchêne, Eric; Rigo, Michel. A morphic approach to combinatorial games : the Tribonacci case. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 375-393. doi : 10.1051/ita:2007039. http://www.numdam.org/articles/10.1051/ita:2007039/

[1] E. Barcucci, L. Bélanger, S. Brlek, On Tribonacci sequences. Fibonacci Quart. 42 (2004) 314-319. | MR | Zbl

[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. | MR | Zbl

[4] L. Carlitz, R. Scoville, V.E. Hoggatt Jr., Fibonacci representations of higher order. Fibonacci Quart. 10 (1972) 43-69. | MR | Zbl

[5] A. Cobham, Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164-192. | MR | Zbl

[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). | MR | Zbl

[7] A. Fraenkel, I. Borosh, A generalization of Wythoff's game. J. Combin. Theory Ser. A 15 (1973) 175-191. | MR | Zbl

[8] A. Fraenkel, How to beat your Wythoff games' opponent on three fronts. Amer. Math. Monthly 89 (1982) 353-361. | MR | Zbl

[9] A. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR | Zbl

[10] A. Fraenkel, Heap games, numeration systems and sequences. Ann. Comb. 2 (1998) 197-210. | MR | Zbl

[11] A. Fraenkel, The Raleigh game, to appear in INTEGERS, Electron. J. Combin. Number Theor 7 (2007) A13. | MR

[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). | MR | Zbl

[14] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982) 147-178. | Numdam | MR | Zbl

[15] M. Rigo and A. Maes, More on generalized automatic sequences. J. Autom. Lang. Comb. 7 (2002) 351-376. | MR | Zbl

[16] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, see http://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. | MR | Zbl

[18] W.A. Webb, The length of the four-number game. Fibonacci Quart. 20 (1982) 33-35. | MR | Zbl

[19] W.A. Wythoff, A modification of the game of Nim. Nieuw Arch. Wisk. 7 (1907) 199-202. | JFM

[20] E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liège 41 (1972) 179-182. | MR | Zbl

Cité par Sources :