Are there semimagic knight tours from any starting square?

Question

There are $140$ distinct semimagic knight tours on a normal chessboard ($8\ \times\ 8$). A semimagic knight tour is a knight tour (not necessarily closed) such that a semimagic square appears if the numbers 1 to 64 in increasing order are written down where the knight starts for the next move. It is known that there is no solution with correct diagonal sums. For more details search with google with the words "magic knight tour" and click on the first hit.

I tried to generate them with turbo pascal, but it is a hard task because there are so few solutions.

I also do not know a complete list of the tours. So my questions are :

• Is there a semi-magic knight tour for any starting square ?
• If yes, can a given tour be transformed in another one with different starting square ? (Unfortunately, replacing 1 by 2, 2 by 3 and so on and finally 64 by 1 does not work because the board does not stay semiagic)

Answer 1

Carl Friedrich Andreyevich Jaenisch, in 1862, wrote a book titled Mathematical Applications of the Analysis of the Game of Chess. It included a Knight's Tour which would always generate a semi-magic square when started from 10 particular squares. The next best known tour only has 8 such starting points.

With rotations and reflections of the board, this can be brought up to 48 starting points. Interestingly, the only squares that won't give you a semi-magic square are those in the 2 by 2 areas in the corners (In algebraic notation, group 1 would be a8, b8, a7, and b7, Group 2: g8, h8, g7, and h7, Group 3: a2, b2, a1, and b1, Group 4: g2, h2, g1, and h1).

Answer 2

Let me add to the previous comment. That one semi-magic tour (SMT) allows you to utilise 48 starting squares and generate a SMT. However, there are many more SMTs, and these allow you to generate a SMT from ANY starting square. In other words, there exists a SMT for any starting square. However, and this goes without saying, you can't pick any starting point and any finish point and get a SMT; the number of SMTs is to restrictive. So for example, if you start at a corner [A8 for example], you can only reach C5 or H6 in a SMT. However, if you relax the restriction that the tour has to be SM, then you can start a knight's tour (KT) on any square and finish on any opposite colored square. Even better, with very few exceptions, you can start on any square, finish on any opposite colored square, and specify any intermediate square and number! So, for example, you can have a tour from C5 to G4 hitting D2 on the 27th move [because C5 and D2 are the same color, it has to be an odd number of moves].