Knight's Tour - Oregon State University

1472215201326 2918823 2 5 10 25 16 19 12 273 6 9 24 21 141128 17 30The 3 × 10 boardGiven any 3×m board where m 6= 3, 5, 6, the above boards can be strung together wherethe numbers on the boards are incremented appropriately.Two 3 × 7 boards hooked together3.1.2 CircuitsLemma 3.4 There are no circuits on the 3 × 4, 3 × 6 or the 3 × 8 board.Proof. We will prove in theorem 3.15 that any board with dimension 4 has no circuit,thus the 3 × 4 board cannot have a circuit. By lemma 3.2, the 3 × 6 board cannot havetours, thus it cannot have a circuit. The 3 × 8 board was shown to have no circuits byexhaustive search.Definition 3.5 An n × m tab board is a board with at least one tour with specific startingand ending squares. The tour begins in the upper left square and ends in a square such thatthe knight may jump off the board to the position directly to the left of the starting square.These boards are called tab boards because the starting and ending squares can be thoughtof as "tabs" which will be used in order to connect them to other boards, creating a circuit.The figures below illustrate the tab boards.These are the tab boards that we will need for constructing the circuits.11417 20118516211236191013215189 4 73 × 7139