Chess King

A chess king is placed on a 8x8 chessboard. It has to make 64 moves, visiting each (of 64) squares only once, and to return there where it started. The path has to be with no intersections (a path looking like '8' is no good). For a loop, one can count the total number of horizontal + vertical (i.e. excluding diagonal) moves; let's call this number M. 1. Give an example of at least one such loop. 2. Give an example of a loop with the largest possible M. 3. Give an example of a loop with M=28. 4. Prove that 28 is the smallest possible M.

Originally posted by GinoJ
A chess king is placed on a 8x8 chessboard. It has to make 64 moves, visiting each (of 64) squares only once, and to return there where it started. The path has to be with no intersections (a path looking like '8' is no good). For a loop, one can count the total number of horizontal + vertical (i.e. excluding diagonal) moves; let's call this number M. 1. Give ...[text shortened]... sible M. 3. Give an example of a loop with M=28. 4. Prove that 28 is the smallest possible M.

Originally posted by UmbrageOfSnow
Explain the loops and M thing more clearly, sounds like an interesting problem, but I'm not sure what you're asking yet.

1-2
a1-a2-...-a8-b8-b7-...-b1-c1-...-g1-...-g8-h8-...-h1

That doesn't return to the starting square. If we look at not returning to the starting square the minimum number would be 15.

Originally posted by XanthosNZ
That doesn't return to the starting square. If we look at not returning to the starting square the minimum number would be 15.

Ka1-a8-b8-b2-c2-c8-d8-d2-e2-e8-f8-f2-g2-g8-h8-h1-a1. 17 loops, if you count the one between finish and start. Not proof exactly but I couldn't find any way with fewer loops at the moment.

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