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 Knight's Tour (3 & 4) (Posted on 2004-04-12)
Please see Knight's Tour (2) for the rules of a Knight's Tour.

A Magic Tour is a tour where, if you number each square with the corresponding knight's step, the result *is* a magic square.

A Semi-Magic Tour is a tour where, if you number each square with the corresponding knight's step, the result *is* a semimagic square.

(A magic square, as you may already know, is one in which the respective sums of the numbers in all the rows, columns, and long diagonals, add up to the same number -- whereas -- a semimagic square is one in which the respective sums of the numbers in all the rows, columns, but not necessarily the diagonals, add up to the same number.)
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The problem:
Find a Magic Tour, on a standard 8x8 chessboard, or prove that it is impossible.

If you find that is is impossible, find a Semimagic Tour, on a standard 8x8 chessboard or prove that it is impossible. Show your work!

So, the first square the knight is on, is marked (1). The next square the knight jumps to is marked (2), and so on... until (64).

* For extra credit, make sure that, at the end, the knight is exactly one knight's move away from the starting square.
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Since "Knight's Tour" is a term used outside the scope of this problem, I'm sure you can find an answer on the internet. Please find an independent solution.

This may require a computer program (hence the category).

 No Solution Yet Submitted by SilverKnight Rating: 3.0000 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution (With Great Assistance from Others) Comment 9 of 9 |

We can (thanks to Dudeny) come up with a solution that has a chess knight visit each square once and only once and also have the total of each row and column be the same number (260).  The diagonals do not have the same total. Chess masters (with assistance from IBM) have been able to prove that no true magic square (ie with diagonals also having the same total) is possible.

ROW

Col   A     B     C     D     E     F     G     H     Total
I    1    30    47    52    5    28    43    54     260

J    48   51     2     29   44   53     6    27     260

K   31    46   49      4    25  8      55    42    260

L   50      3   32     45   56   41    26      7    260

M  33    62   15      20     9   24   39     58    260

N  16    19    34     61    40   57   10    23     260

O  63   14    17      36    21   12   59    38     260

P  18   35    64       13    60   37   22    11     260

-----------------------------------------------------------

260  260  260    260    260  260  260  260

=======================================

Gordon S.

 Posted by Gordon Steel on 2004-04-13 15:32:47

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