Here are the moves for getting the men into one or other of the above two positions.  The numbers are those of the men in the order in which they move into the cell that is for the time being vacant.  The pair is shown in brackets:—­

Place 5 on 1.  Then, 6, 9, 8, 6, 4, (6), 2, 4, 9, 3, 4, 9, (6), 7, 6, 1.

Place 5 on 9.  Then, 4, 1, 2, 4, 6, (14), 8, 6, 1, 7, 6, 1, (14), 3, 4, 9.

Place 5 on 3.  Then, 6, (8), 2, 6, 4, 7, 8, 4, 7, 1, 6, 7, (8), 9, 4, 3.

Place 5 on 7.  Then, 4, (12), 8, 4, 6, 3, 2, 6, 3, 9, 4, 3, (12), 1, 6, 7.

The first and second solutions produce Diagram A; the second and third produce Diagram B. There are only sixteen moves in every case.  Having found the fewest moves, we had to consider how we were to make the burdened man do as little work as possible.  It will at once be seen that as the pair have to go into the centre before separating they must take at fewest two moves.  The labour of the burdened man can only be reduced by adopting the other method of solution, which, however, forces us to take another move.

403.—­THE SPANISH DUNGEON.

[Illustration]

+-----+-----+-----+-----+    +-----+-----+-----+-----+
|     |     |     |     |    |     |     |     |     |
|  1  |  2  |  3  |  4  |    | 10  |  9  |  7  |  4  |
|_____|_____|_____|_____|    |_____|_____|_____|_____|
|     |     |     |     |    |     |     |     |     |
|  5  |  6  |  7  |  8  |    |  6  |  5  | 11  |  8  |
|_____|_____|_____|_____|    |_____|_____|_____|_____|
|     |     |     |     |    |     |     |     |     |
|  9  | 10  | 11  | 12  |    |  1  |  2  | 12  | 15  |
|_____|_____|_____|_____|    |_____|_____|_____|_____|
|     |     |     |     |    |     |     |     |     |
| 13  | 14  | 15  |     |    | 13  | 14  |     |  3  |
|     |     |     |     |    |     |     |     |     |
+-----+-----+-----+-----+    +-----+-----+-----+-----+

This can best be solved by working backwards—­that is to say, you must first catch your square, and then work back to the original position.  We must first construct those squares which are found to require the least amount of readjustment of the numbers.  Many of these we know cannot possibly be reached.  When we have before us the most favourable possible arrangements, it then becomes a question of careful analysis to discover which position can be reached in the fewest moves.  I am afraid, however, it is only after considerable study and experience that the solver is able to get such a grasp of the various “areas of disturbance” and methods of circulation that his judgment is of much value to him.