[Illustration]

In order to make the manner of moving perfectly clear I will point out that the first knight’s move can only be made by 1 or by 2 or by 10.  Supposing 1 moves, then the next move must be by 23, 4, 8, or 21.  As there is never more than one square vacant, the order in which the counters move may be written out as follows:  1—­21—­14—­18—­22, etc.  A rough diagram should be made on a larger scale for practice, and numbered counters or pieces of cardboard used.

343.—­EXERCISE FOR PRISONERS.

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.

[Illustration]

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.

344.—­THE KENNEL PUZZLE.

[Illustration]

A man has twenty-five dog kennels all communicating with each other by doorways, as shown in the illustration.  He wishes to arrange his twenty dogs so that they shall form a knight’s string from dog No. 1 to dog No. 20, the bottom row of five kennels to be left empty, as at present.  This is to be done by moving one dog at a time into a vacant kennel.  The dogs are well trained to obedience, and may be trusted to remain in the kennels in which they are placed, except that if two are placed in the same kennel together they will fight it out to the death.  How is the puzzle to be solved in the fewest possible moves without two dogs ever being together?