Knight’s Circuit

Problem To find a Knight’s circuit of a chess board. Can we visit every square of a chessboard, once and only once, and return to the starting square.

Systematic Search We could write a brute force search program (e.g. in Java) 64 squares. From each position 2, 4, 6, 8 or 16 moves!!! Explosion and implosion of choice of moves.

Invent a new problem Allow any moves This problem is trivial Lets consider restrictive moves.

Horizontal and Vertical Lets consider once square. Horizontal or vertical. Call these straight moves. Can we find a solution. Do this as an exercise.

8x8 Or arbitrary board EX 8.1 What size boards can we cover with straight moves. What are the base case and inductive case? See diagram 8.3

8.2 Supersquares Imagine dividing into 2 by 2 squares – 8.6 Knight moves are either straight or diagonal. Straight moves are either vertical or horizontal to a Supersquare. Diagonal moves are not straight.

Crucial observation Observe vertical moves and horizontal moves. Vertical moves flip around vertical axis, Horizontal moves flip around horizontal axis.

Notation Denote v, the operation of flipping around vertical axis. Denote h, the operation of flipping around horizontal axis. ; is used to separate actions. e.g. v;h means??? Does v;h = h;v Both are equivalent to c, rotating thru 180 degrees. There is a 4 th operations n do nothing. (n = no change) SO v;v = h;h = c;c = n;n This allows us to simplify a sequence of actions to one action. Do e.g. 124 on board In words…

Exercise 8.6 Construct a 4 by 4 table showing the single operation of each of the rows and cols

Partitioning the board Suppose a square is labelled n. Other squares can also be labelled. All squares are labelled into disjoint sets Two squares having the same colour, means they can be reached from each other by straight Knight moves. Two squares of different colour cannot be reached from each other by straight moves. We can construct straight moves for each of the colours We have 4 disjoint circuits. These need to be combined into one circuit.

Combining circuits To combine 4 circuits, combine a pair, then combine another pair. Combine red and blue Combine green and yellow Combine red-blue and green-yellow 8.12