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Báo cáo toán học: "Which Chessboards have a Closed Knight’s Tour within the Rectangular Prism"

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: Which Chessboards have a Closed Knight’s Tour within the Rectangular Prism? | Which Chessboards have a Closed Knight s Tour within the Rectangular Prism Joe DeMaio Department of Mathematics and Statistics Kennesaw State University Kennesaw Georgia 30144 USA j demaio@kennesaw.edu Bindia Mathew Department of Mathematics and Statistics Kennesaw State University Kennesaw Georgia 30144 USA bmathew@students.kennesaw.edu Submitted Aug 17 2010 Accepted Dec 6 2010 Published Jan 5 2011 Mathematics Subject Classification 05C45 00A08 Abstract A closed knight s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m X n rectangular chessboards that admit a closed knight s tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk s paper this article extends his result by classifying the i X j X k rectangular prisms that admit a closed knight s tour. 1 Introduction The closed knight s tour of a chessboard is a classic problem in mathematics. Can the knight use legal moves to visit every square on the board and return to its starting position The two dimensional movement of the knight makes its tour an intriguing problem which is trivial for other chess pieces. Euler presents solutions for the 8 X 8 board in a 1759 paper 4 . Martin Gardner discusses the knight s tour on rectangular boards and other mathematical problems involving the knight in his October 1967 column in Scientific American 5 . Papers exist analyzing the closed knight s tour on variant chessboards such as the cylinder 12 the torus 13 the sphere 1 the exterior of the cube 9 and the interior of the cube 3 . Donald Knuth generalizes the study of the 1 2 -knight on a rectangular board to the r s -leaper on a rectangular board 8 . Across the Board The Mathematics of Chessboard Problems by John Watkins is an indispensable collection of knight s tour results as well as many other mathematically themed chessboard problems 11 . Generalizing away from the chessboard