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Euclid, Calkin & Wilf — playing with rationals. (English) Zbl 1193.91033

Summary (translated from the German): The combinatorial game Euclid was introduced by A. J. Cole and A. J. T. Davie [Math. Gaz. 53, 354–357 (1969; Zbl 0186.25303)]. Two players take turns drawing from a position \((a, b)\in\mathbb N^2\), where the drawing player has to subtract a multiple of the smaller number from the larger number. The player making the move has to subtract a multiple of the smaller number from the larger number, so that the result is still in \(\mathbb N^2\). The first player who can no longer move has lost. Up to now, we know elementary (Cole & Davie), geometrical (Lengyel) and number (Lengyel) and number-theoretic (Schwartz) descriptions of the winning positions for Euclid (optimal play of both players is assumed).
The authors add a graph-theoretic winning strategy by relating Euclid to the Calkin-Wilf. This remarkable tree, introduced by N. J. Calkin and {it H. S. Wilf} [Am. Math. Mon. 107, No. 4, 360–363 (2000; Zbl 0983.11009)], generates all positive rational numbers in abbreviated form by a simple rule exactly once.

MSC:

91A46 Combinatorial games
91A43 Games involving graphs
11B75 Other combinatorial number theory
05C05 Trees

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References:

[1] Alkauskas, G.; Steuding, J.: Statistical properties of the Calkin-Wilf tree and applications to finite contin- ued fractions. Preprint.
[2] Bird, R.; Gibbons, J.; Lester, D.: Functional pearl: enumerating the rationals. J. Funct. Programming 16 (2006), 281-291. 117 · Zbl 1103.68444 · doi:10.1017/S0956796806005880
[3] Bradley, D.M.: Counting the Positive Rationals: A Brief Survey. Preprint available at http://germain.umemat.maine.edu/faculty/bradley/papers/pub.html
[4] Calkin, N.; Wilf, H.: Recounting the rationals. Amer. Math. Monthly 107 (2000), 360-363. · Zbl 0983.11009 · doi:10.2307/2589182
[5] Cole, A.J.; Davie, A.J.T.: A game based on the Euclidean algorithm and a winning strategy for it. Math. Gaz. 53 (1969), 354-357. · Zbl 0186.25303 · doi:10.2307/3612461
[6] Lehmer, D.H.: On Stern’s diatomic series. Amer. Math. Monthly 36 (1929), 59-67.
[7] Lengyel, T.: A nim-type game and continued fractions. Fibonacci Quart. 41 (2003), 310-320. · Zbl 1046.05500
[8] Lengyel, T.: On calculating the Sprague-Grundy function for the game Euclid . Proceedings of the 11th International Conference on Fibonacci Numbers and Their Applications 2004, 169-175 (to appear).
[9] Newman, M.: Recounting the rationals, Continued. Credited in Amer. Math. Monthly 110 (2003), 642-643.
[10] Reznick, B.: Regularity properties of the Stern enumeration of the rationals. Preprint available at http://de.arxiv.org/pdf/math.NT/0610601.pdf?front · Zbl 1204.11027
[11] Sloane, N.; Plouffe, S.: The On-line Encyclopedia of integer sequences . http://www.research.att.com/\sim njas/sequences/ · Zbl 0845.11001
[12] Stern, M.A.: Über eine zahlentheoretische Funktion. J. Reine Angew. Math. 55 (1858), 193-220.
[13] Steuding, J.: Diophantine Analysis . CRC Press/Chapman-Hall 2005. · Zbl 1101.11022
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