Introduction to knight toursA knight tour is a series of steps of a chess knight that visits every square on the chess board just once. A magic knight tour only exists if all the rows, columns and diagonals sum to same amount. There are two ways to identify a knight tour. One way is called geometric which where a line is drawn through all the squares that is visited by the knight which can display an interesting graph. Another way to identify a knights tour is called numeric which is where at the start of the knights tour the square is labelled 1, then the second step is labelled 2 and this carries on to the last step of the knights tour labeleld as 64.Closed tour and open tourA closed tour is when the first step of knight tours can be linked to the last step of the knight tour by using a standard knight move. Whereas an open tour is when the first step of a knight tour cannot be linked to the last step of the knight tour thought a standard knight movement. It is claimed there are over twenty six trillion different closed tours on a standard chess board.
However it has not been found how many open tours there are on a standard chess board up to this date. Using both indicators numeric and geometric can be used to create quite an interesting graph of a closed tour as the first and last step of the knights tour will be just one knight move away from each otherRook Knight tour patternsAfter a lot of research and computing it is found there cannot be a magic night tour on a standard chess board. However it is possible for all the rows and columns to the sum to the same amount.
This is called a semi magic tour which is used to justify that magical tours don’t exist on a standard board. This was also proved by J.C Meyrignacs program which also showed 140 distinct semi magical tours. It is known the original chess can only have semi magical tour as all the rows and columns sums to 260 but the diagonals equal to different amounts.Rook polynomials pattern 1The numbers in the picture below shows the step number the knight takes on the chess board during the knight tour. The picture above shows an example of a semi magical tour where the numbers in each columns and rows sum to 260 which is very interesting. However the diagnals add up to 210 aand 282 which proves its not a magical tour.