[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.
The second diagram is a most favourable magic square position. It will
be seen that prisoners 4, 8, 13, and 14 are left in their original
cells. This position may be reached in as few as thirty-seven moves.
Here are the moves: 15, 14, 10, 6, 7, 3, 2, 7, 6, 11, 3, 2, 7, 6, 11,
10, 14, 3, 2, 11, 10, 9, 5, 1, 6, 10, 9, 5, 1, 6, 10, 9, 5, 2, 12, 15,
3. This short solution will probably surprise many readers who may not
find a way under from sixty to a hundred moves. The clever prisoner was
No. 6, who in the original illustration will be seen with his arms
extended calling out the moves. He and No. 10 did most of the work, each
changing his cell five times. No. 12, the man with the crooked leg, was
lame, and therefore fortunately had only to pass from his cell into the
next one when his time came round.