The first point is to make a choice of the most promising knight's
string and then consider the question of reaching the arrangement in the
fewest moves. I am strongly of opinion that the best string is the one
represented in the following diagram, in which it will be seen that each
successive number is a knight's move from the preceding one, and that
five of the dogs (1, 5, 10, 15, and 20) never leave their original
kennels.
+-----+------+------+------+------+
|1 |2 |3 |4 |5 |
| | | | | |
| 1 | 18 | 9 | 14 | 5 |
| | | | | |
+-----+------+------+------+------+
|6 |7 |8 |9 |10 |
| | | | | |
| 8 | 13 | 4 | 19 | 10 |
| | | | | |
+-----+------+------+------+------+
|11 |12 |13 |14 |15 |
| | | | | |
| 17 | 2 | 11 | 6 | 15 |
| | | | | |
+-----+------+------+------+------+
|16 |17 |18 |19 |20 |
| | | | | |
| 12 | 7 | 16 | 3 | 20 |
| | | | | |
+-----+------+------+------+------+
|21 |22 |23 |24 |25 |
| | | | | |
| | | | | |
| | | | | |
+-----+------+------+------+------+
[Illustration]
This position may be arrived at in as few as forty-six moves, as
follows: 16--21, 16--22, 16--23, 17--16, 12--17, 12--22, 12--21,7--12,
7--17, 7--22, 11--12, 11--17, 2--7, 2--12, 6--11, 8--7, 8--6, 13--8,
18--13, 11--18, 2--17, 18--12, 18--7, 18--2, 13--7, 3--8, 3--13, 4--3,
4--8, 9--4, 9--3, 14--9, 14--4, 19--14, 19--9, 3--14, 3--19, 6--12,
6--13, 6--14, 17--11, 12--16, 2--12, 7--17, 11--13, 16--18 = 46 moves. I
am, of course, not able to say positively that a solution cannot be
discovered in fewer moves, but I believe it will be found a very hard
task to reduce the number.