TAILIEUCHUNG - Báo cáo toán học: "Further Hopping with Toads and Frogs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Further Hopping with Toads and Frogs. | Further Hopping with Toads and Frogs Thotsaporn Aek Thanatipanonda Research Institute for Symbolic Computation RISC Johannes Kepler University A-4040 Linz Austria thotsaporn@ Submitted Jun 15 2009 Accepted Mar 17 2011 Published Mar 24 2011 Mathematics Subject Classification 91A46 Abstract We prove some new results about the combinatorial game Toads and Frogs . We give a finite number of recurrence relations for computing the values of all positions with exactly one . We show that T DDF is an infinitesimal for a 4. At the end we make five new conjectures and describe possible future work. 1 Introduction The game Toads and Frogs invented by Richard Guy is extensively discussed in Winning Ways 1 the famous classic by Elwyn Berlekemp John Conway and Richard Guy that is the bible of combinatorial game theory. The game is played on a 1 X n strip with either Toad T Frog F or on the squares. Left plays T and Right plays F. T may move to the immediate square on its right if it happens to be empty and F moves to the next empty square on the left if it is empty. If T and F are next to each other they have an option to jump over one another in their designated directions provided they land on an empty square 1 . Throughout the paper we will use the notation Xn to denote n contiguous copies of the Toads and Frogs position X. For example D3 TF 2F is shorthand for TFTFF. Already in 1 there is some analysis of Toads and Frogs positions such as TTDFFD and TFDb. In 1996 Erickson 2 analyzed more general positions and made six conjectures about the values of some families of positions. All of them are starting positions positions where all T are rightmost and all F are leftmost . Erickson s conjectures were E1 T DDFb a 3 a b 3 b for all a b 2. E2 T DDDFF a 2 a 2 for all a 2. E3 T DDDFFF a 2 for all a 5. E4 T D F -1 1 or 1 1 for all a 1. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P67 1 E5 T DbF is an infinitesimal for all a b except a b 3 2 . E6 Toads and Frogs is .

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• báo cáo của tạp chí Department of Mathematic
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