There are eighty different arrangements of the numbers in the form of a
perfect knight's path, but only forty of these can be reached without
two men ever being in a cell at the same time. Two is the greatest
number of men that can be given a complete rest, and though the knight's
path can be arranged so as to leave either 7 and 13, 8 and 13, 5 and 7,
or 5 and 13 in their original positions, the following four
arrangements, in which 7 and 13 are unmoved, are the only ones that can
be reached under the moving conditions. It therefore resolves itself
into finding the fewest possible moves that will lead up to one of these
positions. This is certainly no easy matter, and no rigid rules can be
laid down for arriving at the correct answer. It is largely a matter for
individual judgment, patient experiment, and a sharp eye for revolutions
and position.
A
+--+--+--+--+
| 6| 1|10|15|
+--+--+--+--+
| 9|12| 7| 4|
+--+--+--+--+
| 2| 5|14|11|
+--+--+--+--+
|13| 8| 3|**|
+--+--+--+--+
B
+--+--+--+--+
| 6| 1|10|15|
+--+--+--+--+
|11|14| 7| 4|
+--+--+--+--+
| 2| 5|12| 9|
+--+--+--+--+
|13| 8| 3|**|
+--+--+--+--+
C
+--+--+--+--+
| 6| 9| 4|15|
+--+--+--+--+
| 1|12| 7|10|
+--+--+--+--+
| 8| 5|14| 3|
+--+--+--+--+
|13| 2|11|**|
+--+--+--+--+
D
+--+--+--+--+
| 6|11| 4|15|
+--+--+--+--+
| 1|14| 7|10|
+--+--+--+--+
| 8| 5|12| 3|
+--+--+--+--+
|13| 2| 9|**|
+--+--+--+--+
[Illustration: A, B, C, D]
As a matter of fact, the position C can be reached in as few as
sixty-six moves in the following manner: 12, 11, 15, 12, 11, 8, 4, 3, 2,
6, 5, 1, 6, 5, 10, 15, 8, 4, 3, 2, 5, 10, 15, 8, 4, 3, 2, 5, 10, 15, 8,
4, 12, 11, 3, 2, 5, 10, 15, 6, 1, 8, 4, 9, 8, 1, 6, 4, 9, 12, 2, 5, 10,
15, 4, 9, 12, 2, 5, 3, 11, 14, 2, 5, 14, 11 = 66 moves. Though this is
the shortest that I know of, and I do not think it can be beaten, I
cannot state positively that there is not a shorter way yet to be
discovered. The most tempting arrangement is certainly A; but things
are not what they seem, and C is really the easiest to reach.
If the bottom left-hand corner cell might be left vacant, the following
is a solution in forty-five moves by Mr. R. Elrick: 15, 11, 10, 9, 13,
14, 11, 10, 7, 8, 4, 3, 8, 6, 9, 7, 12, 4, 6, 9, 5, 13, 7, 5, 13, 1, 2,
13, 5, 7, 1, 2, 13, 8, 3, 6, 9, 12, 7, 11, 14, 1, 11, 14, 1. But every
man has moved.