A morphic approach to combinatorial games : the Tribonacci case
Duchêne, Eric ; Rigo, Michel
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008), p. 375-393 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2008-01-01
DOI : https://doi.org/10.1051/ita:2007039
Classification:  91A46,  68R15,  68Q45 
@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},
     volume = {42},
     year = {2008},
     pages = {375-393},
     doi = {10.1051/ita:2007039},
     mrnumber = {2401268},
     zbl = {1143.91314},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 375-393. doi : 10.1051/ita:2007039. http://gdmltest.u-ga.fr/item/ITA_2008__42_2_375_0/

[1] E. Barcucci, L. Bélanger, S. Brlek, On Tribonacci sequences. Fibonacci Quart. 42 (2004) 314-319. | MR 2110084 | Zbl 1138.11309

[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 613413 | Zbl 0456.10005

[4] L. Carlitz, R. Scoville, V.E. Hoggatt Jr., Fibonacci representations of higher order. Fibonacci Quart. 10 (1972) 43-69. | MR 304293 | Zbl 0236.05002

[5] A. Cobham, Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164-192. | MR 457011 | Zbl 0253.02029

[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 1970385 | Zbl 1014.11015

[7] A. Fraenkel, I. Borosh, A generalization of Wythoff's game. J. Combin. Theory Ser. A 15 (1973) 175-191. | MR 339824 | Zbl 0265.90065

[8] A. Fraenkel, How to beat your Wythoff games' opponent on three fronts. Amer. Math. Monthly 89 (1982) 353-361. | MR 660914 | Zbl 0504.90087

[9] A. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR 777556 | Zbl 0568.10005

[10] A. Fraenkel, Heap games, numeration systems and sequences. Ann. Comb. 2 (1998) 197-210. | MR 1681514 | Zbl 0942.91015

[11] A. Fraenkel, The Raleigh game, to appear in INTEGERS, Electron. J. Combin. Number Theor 7 (2007) A13. | MR 2337047 | Zbl pre05181750

[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 1475463 | Zbl 0874.20040

[14] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982) 147-178. | Numdam | MR 667748 | Zbl 0522.10032

[15] M. Rigo and A. Maes, More on generalized automatic sequences. J. Autom. Lang. Comb. 7 (2002) 351-376. | MR 1957696 | Zbl 1033.68069

[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 2339496 | Zbl 1120.11009

[18] W.A. Webb, The length of the four-number game. Fibonacci Quart. 20 (1982) 33-35. | MR 660757 | Zbl 0477.10021

[19] W.A. Wythoff, A modification of the game of Nim. Nieuw Arch. Wisk. 7 (1907) 199-202. | JFM 37.0261.03

[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 308032 | Zbl 0252.10011