Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-03T05:23:16.939Z Has data issue: false hasContentIssue false

A morphic approach to combinatorial games: the Tribonacci case

Published online by Cambridge University Press:  13 December 2007

Eric Duchêne
Affiliation:
Institute of Mathematics, University of Liège, Grande Traverse 12 (B37), 4000 Liège, Belgium; [email protected]; [email protected]
Michel Rigo
Affiliation:
Institute of Mathematics, University of Liège, Grande Traverse 12 (B37), 4000 Liège, Belgium; [email protected]; [email protected]
Get access

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.

Type
Research Article
Copyright
© EDP Sciences, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barcucci, E., Bélanger, L., Brlek, S., Tribonacci, On sequences. Fibonacci Quart. 42 (2004) 314319.
E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning ways (two volumes). Academic Press, London (1982).
Boshernitzan, M., Fraenkel, A., Nonhomogeneous spectra of numbers. Discrete Math. 34 (1981) 325327. CrossRef
Carlitz, L., Scoville, R., Hoggatt Jr, V.E.., Fibonacci representations of higher order. Fibonacci Quart. 10 (1972) 4369.
Cobham, A., Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164192. CrossRef
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).
Fraenkel, A., Borosh, I., A generalization of Wythoff's game. J. Combin. Theory Ser. A 15 (1973) 175191. CrossRef
Fraenkel, A., How to beat your Wythoff games' opponent on three fronts. Amer. Math. Monthly 89 (1982) 353361. CrossRef
Fraenkel, A., Systems of numeration. Amer. Math. Monthly 92 (1985) 105114. CrossRef
Fraenkel, A., Heap games, numeration systems and sequences. Ann. Comb. 2 (1998) 197210. CrossRef
A. Fraenkel, The Raleigh game, to appear in INTEGERS, Electron. J. Combin. Number Theor 7 (2007) A13.
A. Fraenkel, The rat game and the mouse game, preprint.
M. Lothaire, Combinatorics on words. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997).
Rauzy, G., Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982) 147178. CrossRef
Rigo, M. and Maes, A., More on generalized automatic sequences. J. Autom. Lang. Comb. 7 (2002) 351376.
N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences, see http://www.research.att.com/~njas/sequences/
Tan, B., Wen, Z.-Y., Some properties of the Tribonacci sequence. Eur. J. Combin. 28 (2007) 17031719. CrossRef
Webb, W.A., The length of the four-number game. Fibonacci Quart. 20 (1982) 3335.
Wythoff, W.A., A modification of the game of Nim. Nieuw Arch. Wisk. 7 (1907) 199202.
Zeckendorf, E., Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liège 41 (1972) 179182.