THE SPANISH DUNGEON.
(
Magic Squares Problem.)
Not fifty miles from Cadiz stood in the middle ages a castle, all traces
of which have for centuries disappeared. Among other interesting
features, this castle contained a particularly unpleasant dungeon
divided into sixteen cells, all communicating with one another, as shown
in the illustration.
Now, the governor was a merry wight, and very fond of puzzles withal.
One day he went to the dungeon and said to the prisoners, "By my
halidame!" (or its equivalent in Spanish) "you shall all be set free if
you can solve this puzzle. You must so arrange yourselves in the sixteen
cells that the numbers on your backs shall form a magic square in which
every column, every row, and each of the two diagonals shall add up the
same. Only remember this: that in no case may two of you ever be
together in the same cell."
One of the prisoners, after working at the problem for two or three
days, with a piece of chalk, undertook to obtain the liberty of himself
and his fellow-prisoners if they would follow his directions and move
through the doorway from cell to cell in the order in which he should
call out their numbers.
He succeeded in his attempt, and, what is more remarkable, it would seem
from the account of his method recorded in the ancient manuscript lying
before me, that he did so in the fewest possible moves. The reader is
asked to show what these moves were.
Answer:
+-----+-----+-----+-----+ +-----+-----+-----+-----+
| | | | | | | | | |
| 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.
Informational
