The idea behind this extraordinary image below is the construction of an 8x8 magic square capable of describing BOTH the Knight's tour (" a sequence of moves of a knight on a chessboard such that the knight visits every square only once...if the knight ends on a square that is one knight's move from the beginning square) WITH the resulting moves forming a series of magic squares. It is the product of a Kiwi engineer named Sturmer, and appeared in the Scientific American Supplement for 1888.
Eric Weinstein says in his article "There Are No Magic Knight's Tours on the Chessboard" on Wolfram's Mathworld site says, well, such a thing is not possible. "After 61.40 days of computation, a 150-year-old unsolved problem has finally been answered. The problem in question concerns the existence of a path that could be traversed by a knight on an empty numbered 8 x 8 chessboard."
Weinstein is concise: "Not surprisingly, a knight's tour is called a magic tour if the resulting arrangement of numbers forms a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. It has long been known that magic knight's tours are not possible on n x n boards for n odd. It was also known that such tours are possible for all boards of size 4k x 4k for k > 2. However, while a number of semimagic knight's tours were known on the usual 8 x 8 chessboard, including those illustrated above, it was not known if any fully magic tours existed on the 8 x 8 board."
And this, from the British Chess Maagzine, 1888, page 278: