The puzzle is simply to find out in how many different ways the man and the lion may be placed on two different spots that are not on the same path.  By “paths” it must be understood that I only refer to the ruled lines.  Thus, with the exception of the four corner spots, each combatant is always on two paths and no more.  It will be seen that there is a lot of scope for evading one another in the desert, which is just what one has always understood.

319.—­THE KNIGHT-GUARDS.

[Illustration]

The knight is the irresponsible low comedian of the chessboard.  “He is a very uncertain, sneaking, and demoralizing rascal,” says an American writer.  “He can only move two squares, but makes up in the quality of his locomotion for its quantity, for he can spring one square sideways and one forward simultaneously, like a cat; can stand on one leg in the middle of the board and jump to any one of eight squares he chooses; can get on one side of a fence and blackguard three or four men on the other; has an objectionable way of inserting himself in safe places where he can scare the king and compel him to move, and then gobble a queen.  For pure cussedness the knight has no equal, and when you chase him out of one hole he skips into another.”  Attempts have been made over and over again to obtain a short, simple, and exact definition of the move of the knight—­without success.  It really consists in moving one square like a rook, and then another square like a bishop—­the two operations being done in one leap, so that it does not matter whether the first square passed over is occupied by another piece or not.  It is, in fact, the only leaping move in chess.  But difficult as it is to define, a child can learn it by inspection in a few minutes.

I have shown in the diagram how twelve knights (the fewest possible that will perform the feat) may be placed on the chessboard so that every square is either occupied or attacked by a knight.  Examine every square in turn, and you will find that this is so.  Now, the puzzle in this case is to discover what is the smallest possible number of knights that is required in order that every square shall be either occupied or attacked, and every knight protected by another knight.  And how would you arrange them?  It will be found that of the twelve shown in the diagram only four are thus protected by being a knight’s move from another knight.

THE GUARDED CHESSBOARD.

On an ordinary chessboard, 8 by 8, every square can be guarded—­that is, either occupied or attacked—­by 5 queens, the fewest possible.  There are exactly 91 fundamentally different arrangements in which no queen attacks another queen.  If every queen must attack (or be protected by) another queen, there are at fewest 41 arrangements, and I have recorded some 150 ways in which some of the queens are attacked and some not, but this last case is very difficult to enumerate exactly.