EXERCISE FOR PRISONERS.
(
The Guarded Chessboard)
The following is the plan of the north wing of a certain gaol, showing
the sixteen cells all communicating by open doorways. Fifteen prisoners
were numbered and arranged in the cells as shown. They were allowed to
change their cells as much as they liked, but if two prisoners were ever
in the same cell together there was a severe punishment promised them.
Now, in order to reduce their growing obesity, and to combine physical
exercise with mental recreation, the prisoners decided, on the
suggestion of one of their number who was interested in knight's tours,
to try to form themselves into a perfect knight's path without breaking
the prison regulations, and leaving the bottom right-hand corner cell
vacant, as originally. The joke of the matter is that the arrangement at
which they arrived was as follows:--
8 3 12 1
11 14 9 6
4 7 2 13
15 10 5
The warders failed to detect the important fact that the men could not
possibly get into this position without two of them having been at some
time in the same cell together. Make the attempt with counters on a
ruled diagram, and you will find that this is so. Otherwise the solution
is correct enough, each member being, as required, a knight's move from
the preceding number, and the original corner cell vacant.
The puzzle is to start with the men placed as in the illustration and
show how it might have been done in the fewest moves, while giving a
complete rest to as many prisoners as possible.
As there is never more than one vacant cell for a man to enter, it is
only necessary to write down the numbers of the men in the order in
which they move. It is clear that very few men can be left throughout in
their cells undisturbed, but I will leave the solver to discover just
how many, as this is a very essential part of the puzzle.
Answer:
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.
