A FENCE PROBLEM.
(
Money Puzzles)
The practical usefulness of puzzles is a point that we are liable to
overlook. Yet, as a matter of fact, I have from time to time received
quite a large number of letters from individuals who have found that the
mastering of some little principle upon which a puzzle was built has
proved of considerable value to them in a most unexpected way. Indeed,
it may be accepted as a good maxim that a puzzle is of little real value
unless, as well as being amusing and perplexing, it conceals some
instructive and possibly useful feature. It is, however, very curious
how these little bits of acquired knowledge dovetail into the
occasional requirements of everyday life, and equally curious to what
strange and mysterious uses some of our readers seem to apply them.
What, for example, can be the object of Mr. Wm. Oxley, who writes to me
all the way from Iowa, in wishing to ascertain the dimensions of a field
that he proposes to enclose, containing just as many acres as there
shall be rails in the fence?
The man wishes to fence in a perfectly square field which is to contain
just as many acres as there are rails in the required fence. Each
hurdle, or portion of fence, is seven rails high, and two lengths would
extend one pole (161/2 ft.): that is to say, there are fourteen rails
to the pole, lineal measure. Now, what must be the size of the field?
Answer:
Though this puzzle presents no great difficulty to any one possessing a
knowledge of algebra, it has perhaps rather interesting features.
Seeing, as one does in the illustration, just one corner of the proposed
square, one is scarcely prepared for the fact that the field, in order
to comply with the conditions, must contain exactly 501,760 acres, the
fence requiring the same number of rails. Yet this is the correct
answer, and the only answer, and if that gentleman in Iowa carries out
his intention, his field will be twenty-eight miles long on each side,
and a little larger than the county of Westmorland. I am not aware that
any limit has ever been fixed to the size of a "field," though they do
not run so large as this in Great Britain. Still, out in Iowa, where my
correspondent resides, they do these things on a very big scale. I have,
however, reason to believe that when he finds the sort of task he has
set himself, he will decide to abandon it; for if that cow decides to
roam to fresh woods and pastures new, the milkmaid may have to start out
a week in advance in order to obtain the morning's milk.
Here is a little rule that will always apply where the length of the
rail is half a pole. Multiply the number of rails in a hurdle by four,
and the result is the exact number of miles in the side of a square
field containing the same number of acres as there are rails in the
complete fence. Thus, with a one-rail fence the field is four miles
square; a two-rail fence gives eight miles square; a three-rail fence,
twelve miles square; and so on, until we find that a seven-rail fence
multiplied by four gives a field of twenty-eight miles square. In the
case of our present problem, if the field be made smaller, then the
number of rails will exceed the number of acres; while if the field be
made larger, the number of rails will be less than the acres of the
field.
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