THE MAGIC KNIGHT'S TOUR.
(
Magic Squares Problem.)
Here is a problem that has never yet been solved, nor has its
impossibility been demonstrated. Play the knight once to every square of
the chessboard in a complete tour, numbering the squares in the order
visited, so that when completed the square shall be "magic," adding up
to 260 in every column, every row, and each of the two long diagonals. I
shall give the best answer that I have been able to obtain, in which
there is a slight error in the diagonals alone. Can a perfect solution
be found? I am convinced that it cannot, but it is only a "pious
opinion."
Answer:
+----+----+----+----+----+----+----+----+
| 46 | 55 | 44 | 19 | 58 | 9 | 22 | 7 |
+----+----+----+----+----+----+----+----+
| 43 | 18 | 47 | 56 | 21 | 6 | 59 | 10 |
+----+----+----+----+----+----+----+----+
| 54 | 45 | 20 | 41 | 12 | 57 | 8 | 23 |
+----+----+----+----+----+----+----+----+
| 17 | 42 | 53 | 48 | 5 | 24 | 11 | 60 |
+----+----+----+----+----+----+----+----+
| 52 | 3 | 32 | 13 | 40 | 61 | 34 | 25 |
+----+----+----+----+----+----+----+----+
| 31 | 16 | 49 | 4 | 33 | 28 | 37 | 62 |
+----+----+----+----+----+----+----+----+
| 2 | 51 | 14 | 29 | 64 | 39 | 26 | 35 |
+----+----+----+----+----+----+----+----+
| 15 | 30 | 1 | 50 | 27 | 36 | 63 | 38 |
+----+----+----+----+----+----+----+----+
Here each successive number (in numerical order) is a knight's move from
the preceding number, and as 64 is a knight's move from 1, the tour is
"re-entrant." All the columns and rows add up 260. Unfortunately, it is
not a perfect magic square, because the diagonals are incorrect, one
adding up 264 and the other 256--requiring only the transfer of 4 from
one diagonal to the other. I think this is the best result that has ever
been obtained (either re-entrant or not), and nobody can yet say whether
a perfect solution is possible or impossible.
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