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 .
