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Amusements in Mathematics

start on certain more or less arbitrary assumptions. That is why in the above illustration I
have thought it necessary to represent the paths in the desert with such rigid regularity.
Though the captain assures me that the tracks of the lions usually run much in this way, I
have doubts.
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.
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.
 
 
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