The illustration is supposed to represent an arrangement of sixty-four kennels.  It will be seen that five kennels each contain a dog, and on further examination it will be seen that every one of the sixty-four kennels is in a straight line with at least one dog—­either horizontally, vertically, or diagonally.  Take any kennel you like, and you will find that you can draw a straight line to a dog in one or other of the three ways mentioned.  The puzzle is to replace the five dogs and discover in just how many different ways they may be placed in five kennels in a straight row, so that every kennel shall always be in line with at least one dog.  Reversals and reflections are here counted as different.

312.—­THE FIVE CRESCENTS OF BYZANTIUM.

When Philip of Macedon, the father of Alexander the Great, found himself confronted with great difficulties in the siege of Byzantium, he set his men to undermine the walls.  His desires, however, miscarried, for no sooner had the operations been begun than a crescent moon suddenly appeared in the heavens and discovered his plans to his adversaries.  The Byzantines were naturally elated, and in order to show their gratitude they erected a statue to Diana, and the crescent became thenceforward a symbol of the state.  In the temple that contained the statue was a square pavement composed of sixty-four large and costly tiles.  These were all plain, with the exception of five, which bore the symbol of the crescent.  These five were for occult reasons so placed that every tile should be watched over by (that is, in a straight line, vertically, horizontally, or diagonally with) at least one of the crescents.  The arrangement adopted by the Byzantine architect was as follows:—­

[Illustration]

Now, to cover up one of these five crescents was a capital offence, the death being something very painful and lingering.  But on a certain occasion of festivity it was necessary to lay down on this pavement a square carpet of the largest dimensions possible, and I have shown in the illustration by dark shading the largest dimensions that would be available.

The puzzle is to show how the architect, if he had foreseen this question of the carpet, might have so arranged his five crescent tiles in accordance with the required conditions, and yet have allowed for the largest possible square carpet to be laid down without any one of the five crescent tiles being covered, or any portion of them.

313.—­QUEENS AND BISHOP PUZZLE.

It will be seen that every square of the board is either occupied or attacked.  The puzzle is to substitute a bishop for the rook on the same square, and then place the four queens on other squares so that every square shall again be either occupied or attacked.

[Illustration]

314.—­THE SOUTHERN CROSS.

[Illustration]

In the above illustration we have five Planets and eighty-one Fixed Stars, five of the latter being hidden by the Planets.  It will be found that every Star, with the exception of the ten that have a black spot in their centres, is in a straight line, vertically, horizontally, or diagonally, with at least one of the Planets.  The puzzle is so to rearrange the Planets that all the Stars shall be in line with one or more of them.