LION-HUNTING.
(
Chessboard Problems)
My friend Captain Potham Hall, the renowned hunter of big game, says
there is nothing more exhilarating than a brush with a herd--a pack--a
team--a flock--a swarm (it has taken me a full quarter of an hour to
recall the right word, but I have it at last)--a _pride_ of lions. Why a
number of lions are called a "pride," a number of whales a "school," and
a number of foxes a "skulk" are mysteries of philology into which I will
not enter.
Well, the captain says that if a spirited lion crosses your path in the
desert it becomes lively, for the lion has generally been looking for
the man just as much as the man has sought the king of the forest. And
yet when they meet they always quarrel and fight it out. A little
contemplation of this unfortunate and long-standing feud between two
estimable families has led me to figure out a few calculations as to the
probability of the man and the lion crossing one another's path in the
jungle. In all these cases one has to 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.
Answer:
There are 6,480 ways of placing the man and the lion, if there are no
restrictions whatever except that they must be on different spots. This
is obvious, because the man may be placed on any one of the 81 spots,
and in every case there are 80 spots remaining for the lion; therefore
81 x 80 = 6,480. Now, if we deduct the number of ways in which the lion
and the man may be placed on the same path, the result must be the
number of ways in which they will not be on the same path. The number of
ways in which they may be in line is found without much difficulty to be
816. Consequently, 6,480 - 816 = 5,664, the required answer.
The general solution is this: 1/3n(n - 1)(3n squared - n + 2). This is, of
course, equivalent to saying that if we call the number of squares on
the side of a "chessboard" n, then the formula shows the number of
ways in which two bishops may be placed without attacking one another.
Only in this case we must divide by two, because the two bishops have no
distinct individuality, and cannot produce a different solution by mere
exchange of places.
